MEMS Resonators

Abstract

The MEMS (microelectromechanical systems) resonator assembly (100) comprises a support structure (102), a resonator element (101) suspended on the support structure (102), and an actuator for exciting the resonator element (101) into a resonant mode. The resonator element (101) has a resonant frequency f 0 The MEMS resonator assembly vibrates at a frequency of 1000 Hz and includes at least one bulk acoustic resonator (110a, 110b). The equivalent series resistance of the MEMS resonator assembly is ESR, and the area of ​​the resonator element is A. The equivalent series resistance of the MEMS resonator assembly is ESR*A*f 0 is 12Ωmm 2 MHz to 83Ωmm 2 It is in the MHz range.

Description

[Technical field]

[0001] The present invention relates to microelectromechanical systems (MEMS) resonators.

[0002] Please note that this section provides useful background information, but is not an admission that the technology described herein represents the state of the art.

[0003] Achieving low equivalent series resistance (ESR) in MEMS resonators is challenging, especially for resonators with small size. For bulk acoustic resonators, the resonant frequency is inversely proportional to the lateral dimension, which can be defined as the frequency-determining dimension. For Length-Extended (LE) resonators, the beam length is the frequency-determining dimension. To reach high resonant frequencies, the frequency-determining dimension is scaled down. However, frequency requirements force the length of LE resonators to be shortened while at the same time increasing their width to achieve a sufficiently low equivalent series resistance.

[0004] The need to shrink one dimension while expanding another creates design challenges.

[0005] It is an object of certain embodiments of the present invention to provide MEMS resonators with low equivalent series resistance (ESR), or at least to provide an alternative to existing technology. Another object of certain embodiments is to provide increased design freedom regarding the layout of the resonator elements on a die. Another object of certain embodiments is to provide MEMS resonators with low temperature variation of resonant frequency and low ESR.

[0006] According to a first aspect of the present invention, there is provided a MEMS (microelectromechanical system) resonator assembly, comprising: Support structure and; a resonator element suspended on the support structure; an actuator for exciting the resonator element into a resonant mode; Equipped with the resonator element vibrates at a resonant frequency f0 and comprises at least one bulk acoustic resonator; If the equivalent series resistance of the MEMS resonator assembly is ESR and the area of ​​the resonator element is A, then ESR*A*f0 is 12Ωmm 2 MHz to 83Ωmm 2 It is in the MHz range.

[0007] In some embodiments, the bulk acoustic resonator is a Length-Extensional mode resonator (LE resonator), an assembly of multiple interconnected Length-Extensional mode resonators, a Width-Extensional resonator, a Square-Extensional resonator, or a Lame resonator.

[0008] In some embodiments, ESR*A*f0 is 16Ωmm 2 MHz or above or 25Ωmm 2 MHz or higher.

[0009] In some embodiments, the area A of the resonator element is less than 0.001 mm in the plane of the die or wafer on which the resonator element is formed. 2 From 1mm 2 The range is.

[0010] In some embodiments, the ratio of equivalent series capacitance (motional capacitance) to shunt capacitance (shunt capacitance) is less than 0.005.

[0011] In some embodiments, the figure of merit is greater than ten.

[0012] In some embodiments, the bulk acoustic resonator is an in-plane resonator.

[0013] In some embodiments, the actuator is a piezoelectric actuator or an electrostatic actuator.

[0014] In some embodiments, more than 50% of the mass of the resonator element is made from single crystal silicon.

[0015] In some embodiments, the main direction of vibration of the bulk acoustic resonator is the direction of vibration of the monocrystalline silicon layer within the resonator element. <100> substantially parallel to the crystal direction, and the single crystal silicon layer has an average phosphorus dopant concentration of 2×10 19 cm -3 Greater than.

[0016] In some embodiments, the resonator element comprises a layer of piezoelectric AlN, Sc-doped AlN, ZnO, LiNbO3, or LiTaO3.

[0017] In some embodiments, the resonator element has a piezoelectric layer with a thickness in the range of 0.5 μm to 4 μm, such as 1 μm to 2 μm.

[0018] In some embodiments, the average phosphorus dopant concentration of the single crystal silicon in the resonator element is 2×10 19 cm -3 and the ratio of the thickness of the piezoelectric layer to the thickness of the single crystal silicon layer, or the ratio of the thickness of the piezoelectric layer to the sum of the thicknesses of the two single crystal silicon layers in the resonator element, is greater than 0.07.

[0019] In some embodiments, the resonant frequency f0 is in the range of 7 MHz to 160 MHz, such as in the range of 15 MHz to 110 MHz.

[0020] In some embodiments, the curve of resonant frequency f0 versus temperature has two turn-around points in the temperature range from -30°C to 85°C.

[0021] In some embodiments, the variation of the resonant frequency f0 in the temperature range of -30°C to 85°C is within ±30 ppm, such as within ±15 ppm, of the resonant frequency at a temperature of 25°C.

[0022] In some embodiments, the number of silicon oxide layers in the resonator element is zero or one.

[0023] In some embodiments, a cavity separates the resonator element from the support structure.

[0024] In some embodiments, the resonator element comprises two bulk acoustic resonators and a material portion mechanically connecting the two bulk acoustic resonators, the two bulk acoustic resonators vibrating at the resonant frequency f0 in phase or 180 degrees out of phase with each other.

[0025] In some embodiments, the distance between the two bulk acoustic resonators is less than 50 μm, such as less than 20 μm.

[0026] In some embodiments, said part of material mechanically connecting said two bulk acoustic resonators is a (substantially) rigid interconnection element.

[0027] In some embodiments, the material portion mechanically connecting a first bulk acoustic resonator and a second bulk acoustic resonator of the two bulk acoustic resonators is a flexural mode resonator.

[0028] In some embodiments, a first bulk acoustic resonator and a second bulk acoustic resonator of the two bulk acoustic resonators are fixed to the same support structure, the support structure being formed in the plane of the resonator element along a line halfway between the first bulk acoustic resonator and the second bulk acoustic resonator.

[0029] In some embodiments, the resonant frequency f0 is substantially equal to a fundamental frequency or an Nth overtone frequency of the first bulk acoustic resonator and a fundamental frequency or an Nth overtone frequency of the second bulk acoustic resonator.

[0030] In some embodiments, the first and second bulk acoustic resonators resonate in a direction of vibration, and the bending mode resonator is a beam-type resonator, the longest dimension of which is perpendicular to the direction of vibration.

[0031] In some embodiments, the Nth overtone frequency of the flexural mode resonator is substantially equal to the resonant frequency f0.

[0032] In some embodiments, the flexural resonator is an in-plane flexural beam resonator.

[0033] In some embodiments, the bending mode resonator is mechanically connected to the first and second bulk acoustic resonators at an anti-nodal point of the first bulk acoustic resonator and an anti-nodal point of the second bulk acoustic resonator.

[0034] In some embodiments, the flexural resonator is mechanically connected to the first bulk acoustic resonator at one or more connection points on a first side of the flexural resonator and to the second bulk acoustic resonator at one or more connection points on a second side of the flexural resonator opposite the first side.

[0035] In some embodiments, the flexural resonator is mechanically connected to the first bulk acoustic resonator at one or more connection points on one side of the flexural resonator and is mechanically connected to the second bulk acoustic resonator at one or more connection points on the same side of the flexural resonator.

[0036]

[0008] In some embodiments, the resonant mode shape of the bending mode resonator has two types of antinodes: a first type of antinode has a positive displacement along the vibration axis, at least one of which is a connection point for mechanical connection to a first one of the two bulk acoustic resonators; A second type of antinode has a negative displacement along the vibration axis, at least one of which is a connection point for mechanical connection to a second of the two bulk acoustic resonators.

[0037] In some embodiments, the first bulk acoustic resonator resonates in a first vibration direction and the second bulk acoustic resonator resonates in a second vibration direction orthogonal to the first vibration direction.

[0038] In some embodiments, the bending mode resonator has a portion in the form of a rectangular frame.

[0039] In some embodiments, the flexural mode resonator located between a first bulk acoustic resonator and a second bulk acoustic resonator of the two bulk acoustic resonators mechanically connects the first bulk acoustic resonator and the second bulk acoustic resonator via an antinode of the flexural mode resonator.

[0040] In some embodiments, the first and second bulk acoustic resonators comprise a plurality of interconnected resonator element beams, adjacent resonator element beams being connected by at least one mechanical connection (or coupling) element.

[0041] In some embodiments, the resonating beams of the first and second bulk acoustic resonators and the elongated resonating element (or beam) of the flexural mode resonator are each formed of a single crystal silicon layer within the resonating element. <100> Aligned along the crystal direction, <100> If there is any deviation from the crystal orientation, the deviation is at most 15 degrees.

[0042] In some embodiments, the MEMS resonator assembly comprises two bulk acoustic resonators having a plurality of interconnected resonator element beams configured to resonate in an in-plane extensional (LE) resonant mode, and at least one flexural mode resonator mechanically connected to the two bulk acoustic resonators, the at least one flexural mode resonator being connected to the two bulk acoustic resonators via connecting elements located at some of its antinodes, such that the two bulk acoustic resonators resonate in phase.

[0043] In some embodiments, the MEMS resonator assembly comprises two bulk acoustic resonators, each comprised of one or more extensional beams arranged side-by-side with their longitudinal lengths parallel and coupled across their widths (to form an interconnected extensional resonator assembly).

[0044] In some embodiments, the two or more bulk acoustic resonators are arranged side-by-side with parallel longitudinal axes such that the two distal ends of each bulk acoustic resonator periodically move toward and away from each other during in-phase vibration.

[0045] In some embodiments, the in-plane flexural mode resonator is disposed between a pair of stretch mode resonators, with its longitudinal direction perpendicular to the direction of vibration of the stretch mode resonators.

[0046] In some embodiments, assuming free boundary conditions, the flexural mode resonator has substantially the same resonant frequency as the stretching mode resonator. This frequency may be a fundamental resonant frequency or an Nth overtone. In some embodiments, the in-plane flexural vibration is characterized by N+1 nodes and N+2 antinodes. In some embodiments, frequency matching between the flexural mode resonators and the stretching mode resonators is achieved by designing appropriate beam lengths and beam widths.

[0047] In some embodiments, the flexural mode resonator is mechanically coupled to the stretching mode resonator via connection elements arranged at at least one connection point on either side of the flexural mode resonator.

[0048] In some embodiments, the connection points of the bending mode resonator are at antinodes of the bending mode resonator and at antinodes of the (connected) bulk acoustic resonator.

[0049] In some embodiments, the distance between a stretch mode resonator and its neighboring stretch mode resonator is equal to the width of said flexural mode resonator plus twice the length of the connecting element.

[0050] In some embodiments, the stretching mode resonator comprises a monocrystalline silicon layer within the resonator element. <100> In addition, in some embodiments, the flexural coupling elements (flexural mode resonators) have resonators that are (substantially) aligned in a crystal direction (such as <0100>). <100> They are (substantially) aligned in a crystal direction (such as 0010).

[0051] Although various concepts and embodiments have been introduced, they are not presented to limit the scope of the invention. These and the following embodiments are merely used to explain specific aspects and steps that can be used in implementing the present invention. It should be understood that some embodiments can be applied to other embodiments. The embodiments introduced can be appropriately combined. [Brief description of the drawings]

[0052] The invention will now be described, by way of example only, with reference to the accompanying drawings, in which: [Figure 1] 1 illustrates a MEMS resonator assembly according to an embodiment. [Figure 2A] 1 illustrates a cross-section of a MEMS resonator assembly formed on a cavity SOI substrate according to an embodiment. [Figure 2B]1 illustrates a cross section of a MEMS resonator assembly that forms part of a wafer level packaged component according to an embodiment. [Figure 3A] 1 shows an embodiment of a MEMS resonator assembly consisting of two interconnected LE resonator element assemblies and one in-plane flexural resonator designed to vibrate at the third overtone. [Figure 3B] 3B illustrates the maximum displacement of a vibration cycle for the embodiment of FIG. 3A. [Figure 3C] 13 shows the mode shape of the fundamental mode of an interconnected length extension mode resonator assembly upon contraction. [Figure 3D] 13 shows the mode shape of the fundamental mode of an interconnected length extension mode resonator assembly during extension. [Figure 4] 3A and 3B illustrate the meaning of some parameters relevant to the embodiment of FIG. [Figure 5A] An embodiment is shown consisting of three interconnected LE resonator element assemblies and two in-plane flexural resonators designed to vibrate at the third overtone. [Figure 5B] 5B illustrates the maximum displacement of a vibration cycle for the embodiment of FIG. 5A. [Figure 6A] An embodiment is shown that consists of two interconnected LE resonator element assemblies and one in-plane flexural resonator designed to vibrate at the first overtone. [Figure 6B] 6B shows the maximum displacement of a vibration cycle for the embodiment of FIG. 6A. [Figure 7A] An embodiment is shown that consists of two interconnected LE resonator element assemblies and one in-plane flexural resonator designed to vibrate at the seventh overtone. [Figure 7B] 7B shows the maximum displacement of a vibration cycle for the embodiment of FIG. 7A. [Figure 8A] 1 shows another embodiment consisting of two interconnected LE resonator element assemblies and one in-plane flexural resonator designed to vibrate at the seventh overtone. [Figure 8B]8B illustrates the maximum displacement of a vibration cycle for the embodiment of FIG. 8A. [Figure 9A] 1 illustrates an exemplary embodiment of a MEMS resonator assembly having a piezoelectric actuator. [Figure 9B] 9B illustrates an example cross-section of the MEMS resonator assembly of FIG. 9A along line AA′. [Figure 9C] 9B illustrates an example cross-section of the MEMS resonator assembly of FIG. 9A along line BB'. [Figure 10A] 1 shows an embodiment using electrostatic actuation. [Figure 10B] 10B shows an example of a cross-section of the MEMS resonator assembly of FIG. 10A taken along line AA'. [Figure 10C] 10B illustrates an example of a cross-section of the MEMS resonator assembly of FIG. 10A with etch holes, according to an embodiment. [Figure 10D] 10B illustrates an example cross-section of the MEMS resonator assembly of FIG. 10A along line BB'. [Figure 11A] An embodiment is shown that is composed of two WE resonators and one in-plane flexural resonator and designed to vibrate at the third overtone. [Figure 11B] 11B shows the maximum displacement of a vibration cycle for the embodiment of FIG. 11A. [Figure 12A] An embodiment is shown that is composed of two SE resonators and one in-plane flexural resonator and designed to vibrate at the third overtone. [Figure 12B] 12B illustrates the maximum inward displacement of a vibration cycle for the embodiment of FIG. 12A. [Figure 13] 1 shows a MEMS resonator assembly having four SE resonators arranged in a 2×2 array. [Figure 14A] 1 shows another MEMS resonator assembly having two LE resonant elements and a flexure resonator designed to vibrate at the third overtone. [Figure 14B] 14B shows the maximum displacement of a vibration cycle for the embodiment of FIG. 14A. [Figure 15A]1 shows a MEMS resonator assembly having two LE resonating elements and two flexure resonators coupled to opposite distal ends of the LE resonating elements. [Figure 15B] 15B illustrates the maximum contraction of the LE resonating element during a vibration cycle for the embodiment of FIG. 15A. [Figure 16A] 1 shows a MEMS resonator assembly having four LE resonating elements and three flexure resonators coupled to the distal ends of the LE resonating elements. [Figure 16B] 16B illustrates the maximum contraction of the LE resonating element during a vibration cycle for the embodiment of FIG. 16A. [Figure 17] 1 illustrates a MEMS resonator assembly having two interconnected LE resonator element assemblies and a flexural mode resonator coupled to a distal end of the interconnected LE resonator element assemblies. [Figure 18] Two WE resonators are shown tethered to a central support structure. [Figure 19] 13 shows yet another embodiment having a central support structure. [Figure 20] 1 illustrates a MEMS resonator assembly having three interconnected LE resonator element assemblies and a flexural mode resonator coupled to a distal end of the interconnected LE resonator element assemblies. [Figure 21A] 1 shows an extensional mode resonator with its principal vibration axes perpendicular to each other. [Figure 21B] 21B shows the shape of the flexural resonator of the embodiment of FIG. 21A when resonating. [Figure 21C] 13 shows another embodiment of an extensional mode resonator with its principal vibration axes perpendicular to each other. [Figure 21D] FIG. 21D shows the shape of the flexural resonator of the embodiment of FIG. 21C when resonating. [Figure 21E] 1 shows an embodiment of a stretching mode resonator with 180 degrees out of phase, with the principal axes of vibration perpendicular to each other. [Figure 21F] FIG. 21F shows the shape of the flexural resonator of the embodiment of FIG. 21E when resonating. [Figure 22A] 1 illustrates a MEMS resonator assembly having multiple Lame mode resonators coupled by a flexure resonator. [Figure 22B] 1 illustrates a MEMS resonator assembly having multiple Lame mode resonators coupled by a substantially rigid connecting element. [Figure 22C] 1 illustrates a MEMS resonator assembly having two SE resonators that are 180 degrees out of phase with each other and are coupled by a substantially rigid connecting element. [Figure 22D] 1 shows a MEMS resonator assembly having two WE resonators coupled by two substantially rigid connecting elements. [Figure 23A] 1 illustrates a particular MEMS resonator layout according to an embodiment. [Figure 23B] 1 illustrates a particular MEMS resonator layout according to an embodiment. [Figure 23C] 1 illustrates a particular MEMS resonator layout according to an embodiment. [Figure 23D] 1 illustrates a particular MEMS resonator layout according to an embodiment. [Figure 23E] 1 illustrates a particular MEMS resonator layout according to an embodiment. [Figure 23F] 1 illustrates a particular MEMS resonator layout according to an embodiment. [Figure 24] The equivalent circuit model is shown below. [Figure 25A] 23B shows measured electrical impedance of the MEMS resonator assembly of FIG. 23A. [Figure 25B] 23E shows measured electrical impedance of the MEMS resonator assembly of FIG. 23D. [Figure 25C] 23F shows measured electrical impedance of the MEMS resonator assembly of FIG. 23E. [Figure 26] 1 illustrates how the product of ESR x resonant frequency x resonator element area depends on resonant frequency, according to an embodiment. [Figure 27] 4 illustrates the temperature dependence of quality factor in an embodiment. [Figure 28] 4 illustrates the temperature dependence of resonant frequency in an embodiment. Detailed explanation

[0053] In the following description, like reference numbers refer to like elements.

[0054] In certain embodiments disclosed herein, a MEMS resonator assembly comprises at least one bulk acoustic resonator, and the equivalent series resistance (ESR) of the MEMS resonator assembly falls within a particular range depending on the area of ​​the resonator element and the resonant frequency.

[0055] In some embodiments, the resonator element of the MEMS resonator assembly comprises two bulk acoustic resonators and a material portion mechanically connecting the two bulk acoustic resonators, the two bulk acoustic resonators vibrating in phase or 180 degrees out of phase with each other at the resonant frequency f0. The ESR of the MEMS resonator assembly is within a certain range depending on the area of ​​the resonator elements and the resonant frequency. In some embodiments, the material portion is a (substantially) rigid coupling element. In some embodiments, the material portion comprises a flexural mode resonator and two connecting elements, a first one of which connects the flexural resonator to the first bulk acoustic resonator and a second one of which connects the flexural resonator to the second bulk acoustic resonator.

[0056] The bulk acoustic resonators that make up the MEMS resonator assembly include, for example, Length-Extensional (LE) mode resonators, Width-Extensional (WE) mode resonators, Squire-Extensional (SE) mode resonators, and interconnected Length-Extensional mode resonators and Lame mode resonators. Synchronization of two or more bulk acoustic resonators can improve the properties of the resonator assembly. In particular, it can reduce the ESR (equivalent series resistance) and improve the power handling capability.

[0057] In some embodiments, a MEMS resonator assembly includes at least two extensional mode resonators coupled with one or more flexural resonators such that each of the at least two extensional mode resonators resonates in a collective resonant mode, either in phase with the other extensional mode resonator or out of phase with the other resonator by 180 degrees. Examples of extensional mode resonators that may be components of these MEMS resonator assemblies include length extensional mode resonators, width extensional mode resonators, square extensional mode resonators, and interconnected length extensional mode resonators.

[0058] FIG. 1 shows a MEMS resonator assembly 100 according to an embodiment, such as assembled on a die. The resonator assembly 100 comprises a resonator element 101, a support structure 102, and a suspension element 140 that fixes the resonator element 101 to the support structure 102. Four suspension elements are depicted in FIG. 1. The resonator element 101 comprises (at least) two stretching mode resonators (110a, 110b) and (one or more) flexural mode resonators 120. Furthermore, the resonator element 101 comprises (one or more) connection elements 130a connecting the flexural resonator 120 and the stretching mode resonator 110a, and (one or more) connection elements 130b connecting the flexural resonator 120 and the stretching mode resonator 110b. In the example of FIG. 1, there is one connection element for the stretching mode resonator 110a and two connection elements for the stretching mode resonator 110b.

[0059] In some embodiments, at least one of the stretching mode resonators 110a, 110b of the resonator element 101 comprises a piezoelectric thin film actuator for exciting the stretching mode resonator into a resonant mode. When the stretching mode resonator is excited into a resonant mode, the entire resonator element is in collective resonance due to the mechanical coupling of the stretching mode resonators 110a, 110b. In some embodiments, more than 50% of the mass of the resonator element 101 is made of single crystal silicon. In some embodiments, the MEMS resonator assembly 100 comprises an electrostatic actuator for exciting at least one of the stretching mode resonators 110a, 110b into a resonant mode, thereby exciting the entire resonator element into collective resonance due to the mechanical coupling of the stretching mode resonators 110a, 110b.

[0060] In some embodiments, the MEMS resonator assembly is formed on a silicon-on-insulator (SOI) substrate or a cavity-SOI substrate, the latter of which is shown in cross section in FIG. 2A. The support structure 102 and the resonator element 101, consisting of the stretching mode resonators 110a, 110b and the flexure resonator 120, are formed in the so-called device layer of the cavity-SOI substrate. The cavity-SOI substrate comprises a handle layer 105 (typically made of monocrystalline silicon) and an insulator layer 106 (typically made of silicon oxide). A recess is formed in the handle layer to form a cavity 107 between the resonator element 101 and the handle layer 105. A vertical trench 190 passing through the material layer forming the resonator element 101 separates the stretching mode resonators 110a and 110b from the support structure 102 and the flexure resonator 120. The support structure 102 is rigidly attached to the handle layer 105 via the insulator layer 106. The suspension and connecting elements are not shown in FIG. 2A.

[0061] In some embodiments, the MEMS resonator assembly is encapsulated in a low pressure cavity in a ceramic package. In some embodiments, the MEMS resonator assembly is part of a wafer level package component, as shown in FIG. 2B. A cap 108 is attached to the support structure 102, using wafer level packaging techniques to create a hermetically sealed low pressure environment for the resonator element 101. A recess 109 is formed in the cap 108, providing a cavity between the resonator element 101 and the cap 108. Electrical connections are not shown in FIGS. 2A and 2B.

[0062] Further features of the structure and operation of the MEMS resonator assembly 100 of FIG. 1 are described below with reference to FIGS. 3A-3D. The structure shown shows a resonator element 101 and suspension elements 141a, 142a, 143b, 144b connecting the resonator element 101 to a support structure (not shown). The shape of the MEMS resonator assembly in a rest position is shown in FIG. 3A and at maximum displacement of a vibration cycle in FIG. 3B. The extension mode resonators 110a, 110b shown in FIGS. 3A and 3B are interconnected length extension (LE) resonator assemblies, which are composed of elongated beam-like LE mode resonating elements 151. In this example, both resonators 110a, 110b are composed of five LE resonating elements 151. Adjacent LE resonating elements 151 are separated by trenches 153, but are connected at distal end portions 152. The portions 152 connecting two adjacent LE resonating elements are referred to herein as interconnecting elements. The purpose of the interconnection element is to make the multiple beam-type LE resonating elements 151 ganged together and resonating in the same length-stretching resonant mode at the same frequency. These two interconnected LE resonating elements vibrate together in the stretching mode resonance. The dashed and dotted lines in Figs. 3B-3D show the rest positions of the distal ends (or outer edges) of the interconnected LE resonating element assemblies. At the maximum contraction position of the resonant motion shown in Fig. 3B (and 3C), the distal ends of the interconnected LE resonators have moved in the x-direction towards their (central) axis of symmetry, as shown by the four arrows next to the four dashed and dotted lines in Fig. 3B. At the other maximum displacement position, the extended position of the resonant motion, which is 180 degrees out of phase with the position shown in Fig. 3B, the distal ends of the interconnected LE resonators have moved in the x-direction outward from their central axis of symmetry, as shown in Fig. 3D. In the maximum contraction position shown in FIG. 3B, the geometry of the flexure resonator 120 is such that the material portions proximate the connecting element 131a connecting the flexure resonator 120 to the stretch-mode resonator 110a and the material portions proximate the connecting elements 132b, 133b connecting the flexure resonator 120 to the stretch-mode resonator 110b move in opposite directions along the x-axis.The material portion of the flexural resonator 120 closer to the connecting element 131a follows the movement of the distal end of the LE resonant element of the stretch mode resonator 110a, and the material portion of the flexural resonator 120 closer to the connecting elements 132b, 133b follows the movement of the distal end of the interconnected LE resonator of the stretch mode resonator 110b.

[0063] In a preferred embodiment, the flexure resonator 120 has a resonant frequency substantially equal to the resonant frequency of the stretching mode resonators 110a and 110b. The mode shape of the flexure resonator 120 is such that the connection elements 131a, 132b, 133b are substantially located at the antinodes of the eigenmode. For illustrative purposes, the magnitude of the displacements depicted in FIG. 3B has been exaggerated to highlight the resonant mode shape compared to the rest position of the flexure resonator 120 shown in dashed-dotted lines. In the embodiment shown in FIG. 3B, no connection elements are located at the two antinodes at either end of the flexure resonator 120.

[0064] 3C and 3D show further features of an interconnected LE resonator element assembly (e.g., assembly 110a) under fundamental resonant mode. FIG. 3C shows (in an exaggerated manner) how stretching mode vibration along the x-axis of the interconnected LE resonator element assembly causes a central material portion near the nodal point of the resonating LE beam 151 to expand in orthogonal directions (y and z directions) when the LE beam 151 contracts (FIG. 3C) and contract when the LE beam 151 elongates (FIG. 3D). In other figures herein, small lateral contractions and expansions orthogonal to the primary vibration direction of the stretching mode resonance are ignored as being insignificant to the operation of the embodiment.

[0065] The inventors have found that when the stretching mode resonators 110a and 110b are coupled to a flexural resonator 120 vibrating at resonance, the entire resonator element resonates in a collective resonance mode, minimizing energy loss due to coupling. flex-IP can be estimated as follows:

[0066] TIFF2024545463000002.tif22170

[0067] where E is the Young's modulus of the flexural resonator material, ρ is the density, βn is the eigenvalue of the Nth overtone, W is the beam width, and L is the beam length, where βn depends on the boundary conditions. Typical boundary conditions include free, fixed, and pinned. After the overtone order is selected, the beam width W and length L in an embodiment can be dimensioned using equation (1) such that the flexural resonant frequency matches the resonant frequency of the stretching mode resonator to be coupled. For a piezoelectrically coupled MEMS resonator assembly, there may be multiple material layers, such as single crystal layers, piezoelectric layers, and top electrode layers, with each layer having its own material parameters and geometric dimensions. Therefore, a more general form of equation (1) may be used.

[0068] FIG. 4 shows the meaning of the parameters in equation (1) related to the flexure resonators and connecting elements of FIGS. 3A and 3B. The width W and length L of the resonator 120 are dimensioned such that the third overtone frequency is equal to the resonant frequency of the interconnected LE resonator element assemblies 110a and 110b. The location and dimensions of the connecting elements as well as the node positions have some influence on the exact value of the resonant frequency. The widths and lengths of the connecting elements 131a, 132b, and 133b are shown in FIG. 4 as WC1 and LC1, WC2 and LC2, and WC3 and LC3, respectively. FEM simulations may be used to find the optimal locations of the connecting elements as well as the optimal values ​​of the width W and length L of the flexure resonators. As a general rule, proper coupling has only a small effect on the resonant mode shapes of the coupled stretching mode resonators and the flexure resonators.

[0069] The table below lists the dimensions of certain exemplary flexural resonators (length L, width W) and connecting elements (length Lc, width WC) that can be used to mechanically couple stretching mode resonators vibrating in a collective resonant mode at frequency f0.

[0070] TIFF2024545463000003.tif85170

[0071] FIG. 5A shows an embodiment consisting of three interconnected LE resonator element assemblies 210a, 210b, 210c and two in-plane flexural mode resonators 221, 222 designed to vibrate at the third overtone. The flexural resonator 222 is a mirror image of the flexural resonator 221 along its length, making the entire structure symmetrical about both the x-axis and the y-axis. The entire structure shown is a suspended structure. The outermost interconnected LE resonator element assemblies 210a and 210c are connected to the support structure 120 (not shown) via suspension elements 241a, 242a, 243c, 244c. The central interconnected LE resonator element assembly 210b is connected to the outer interconnected LE resonator element assemblies 210a, 210c via the flexural mode resonators 221, 222 and their attached connection elements. The width W and length of each of the flexural resonators 221, 222 are such that their third overtone resonances are substantially equal to each other and to the resonant frequency of the extensional mode resonators (here the interconnected LE resonating element assemblies 210a, 210b, 210c).

[0072] FIG. 5B illustrates the maximum displacement of a vibration cycle for the same embodiment depicted in FIG. 5A. The dashed and dotted lines represent the initial rest position of the interconnected LE resonator element assemblies 210a, 210b, 210c. The flexural mode resonators 221, 222 vibrate at the third overtone. The connecting elements located at the anti-nodal positions of the flexural resonators 221, 222 effectively couple the flexural vibration to the stretching mode vibration. The magnitude of the displacement is exaggerated for illustrative purposes. The entire MEMS resonator assembly illustrated in FIG. 5B vibrates in a collective resonant mode in which the motions of the stretching mode resonators 210a, 210b, 210c are substantially in phase with each other.

[0073] In some embodiments, as shown in FIG. 6A, a MEMS resonator assembly 300 is composed of two interconnected LE resonator element assemblies 310a, 310b and one (in-plane) flexural mode resonator 320 designed to vibrate at its first overtone. The entire structure shown is a suspension structure and is tethered to a support structure by suspension elements 341a, 342a, 343b, 344b. Connecting elements (331a, 332b, 333b) are located at all three antinodes of the flexural mode resonator 320. These connecting elements couple the antinodes of the flexural mode resonator 320 to adjacent sidewalls of the interconnected LE resonator element assemblies 310a, 310b.

[0074] FIG. 6B illustrates the maximum displacement of a vibration cycle for the embodiment of FIG. 6A. The dashed and dotted lines represent the rest positions of the interconnected LE resonator element assemblies 310a, 310b. At the first overtone resonance of the flexure resonator 320, the anti-node position of the connecting element 331a moves in an opposite direction relative to the anti-node positions of the connecting elements 332b, 333c. The width W and length L of the resonator 320 are set such that its first overtone resonant frequency is substantially equal to the resonant frequency of the LE resonator element assemblies 310a, 310b. The entire MEMS resonator assembly 300 of FIG. 6A, 6B vibrates in a collective resonant mode in which the motions of the stretching mode resonators (here the interconnected LE resonator element assemblies 310a, 310b) are substantially in phase with each other.

[0075] 7A shows an embodiment consisting of two interconnected LE resonator element assemblies 410a, 410b and one (in-plane) flexural mode resonator 420 designed to vibrate at the seventh overtone. The entire structure shown is a suspended structure, suspended to a support structure by suspension elements 441a, 442a, 443b, 444b. Seven connecting elements 431a, 432a, 433a, 434b, 435b, 436b, 437b couple the antinodes of the flexural mode resonator 420 to adjacent sidewalls of the interconnected LE resonator element assemblies 410a, 410b.

[0076] FIG. 7B illustrates the maximum displacement of the vibration cycle of the embodiment of FIG. 7A. The dashed and dotted lines represent the rest positions of the interconnected LE resonator element assemblies 410a, 410b. At the seventh overtone resonance of the flexure resonator 420, the antinode positions of the connecting elements 431a, 432a, 433a move in the opposite direction to the antinode positions of the connecting elements 434b, 435b, 436b, 437b. The width W and length L of the resonator 420 are set such that its seventh overtone resonant frequency is substantially equal to the resonant frequency of the interconnected LE resonator element assemblies 410a, 410b. The entire MEMS resonator assembly 400 of FIGS. 7A, 7B vibrates in a collective resonant mode in which the motions of the stretching mode resonators (here, the interconnected LE resonator element assemblies 410a, 410b) are substantially in phase with each other. In this embodiment, no connecting elements are placed at the two antinodes at both ends of the resonator 420.

[0077] 8A shows an embodiment consisting of two interconnected LE resonator element assemblies 510a, 510b and one (in-plane) flexural mode resonator 520 designed to vibrate at the seventh overtone. The entire structure shown is a suspended structure, suspended to a support structure by suspension elements 541a, 542a, 543b, 544b. Four connecting elements 531a, 532a, 533b, 534b couple the antinodes of the flexural mode resonator 520 to adjacent sidewalls of the interconnected LE resonator element assemblies 510a, 510b.

[0078] FIG. 8B illustrates the maximum displacement of the vibration cycle of the embodiment of FIG. 8A. The dashed and dotted lines represent the rest positions of the interconnected LE resonator element assemblies 510a, 510b. At the seventh overtone resonance of the flexure resonator 520, the antinode positions of the connecting elements 531a, 532a move in the opposite direction relative to the antinode positions of the connecting elements 533b, 534b. The width W and length L of the resonator 520 are set such that its seventh overtone resonant frequency is substantially equal to the resonant frequency of the interconnected LE resonator element assemblies 510a, 510b. The entire MEMS resonator assembly 500 of FIGS. 8A, 8B vibrates in a collective resonant mode in which the motions of the stretching mode resonators (here, the interconnected LE resonator element assemblies 510a, 510b) are substantially in phase with each other. In this embodiment, only four of the nine antinodes of the resonator 520 have connecting elements placed on them.

[0079] The width of the flexural resonators and the length of the connecting elements define how closely the spacing of the stretching mode resonators (in the embodiment of Figs. 8a, 8B, the interconnected LE resonator element assemblies) can be made. In some embodiments, these dimensions, together with the corresponding lengths of the flexural resonators according to equation (1), are as small as the technology allows. In advantageous embodiments, the distance between the first and second stretching mode resonators is less than 50 μm. In preferred embodiments, the distance between the first and second stretching mode resonators is less than 20 μm.

[0080] The dimensions of the connecting element define the coupling strength between the stretching mode resonator and the flexural resonator. In some embodiments, the length of the connecting element is made as small as technology allows, and its width is selected as a trade-off between coupling strength and perturbation of the flexural mode shape. In an advantageous embodiment, the connecting element is placed at the antinode of the flexural resonator.

[0081] Not all antinodes of the flexural resonator are necessarily connected to the stretching mode resonator through connecting elements. In some embodiments, such as in Figures 3A-3B, 5A-5B, 7A-7B, and 8A-8B, the antinodes at both ends of the flexural resonator are not connected in order to maintain the free boundary condition.

[0082] In some embodiments (e.g., as in Figures 5A-5B), the number of stretching mode resonators is 3 or more and the number of flexural resonators is 2 or more. In some embodiments, the flexural resonators and their associated connecting elements are identical to each other or are mirror images of each other.

[0083] In some embodiments (e.g., Figures 5A-5B), one of the stretching mode resonators does not have a suspension element connecting to a support structure, but has connecting elements connecting to two adjacent stretching mode resonators via flexure resonators.

[0084] In some embodiments, at least one of the stretching mode resonators that are components of the MEMS resonator assembly vibrates at an overtone frequency. For example, the mutually coupled LE mode resonators shown in Figures 3A and 3B (or the resonators shown in Figures 5A-5B, 6A-6B, 7A-7B, 8A-8B) may vibrate at a third overtone frequency or other odd overtone (or fundamental) frequency, as long as the resonant frequencies of the flexural resonators and the mutually coupled LE mode resonators are substantially equal (which is the frequency of the collective resonant mode). (Please ignore the changes in scale of the x-axis and y-axis.) In some other exemplary embodiments, one of the stretching mode resonators vibrates in a fundamental mode and another stretching mode resonator vibrates in an overtone mode, and these two frequencies and the resonant frequency of the flexural mode resonator coupled to these two stretching mode resonators are substantially equal to each other.

[0085] In some embodiments, the resonator element 101 of the MEMS resonator assembly 100 comprises a piezoelectric actuator for exciting the resonator element into a resonant mode. An exemplary embodiment of such a MEMS resonator assembly is shown in Figures 9A-9C. The resonator assembly 600 comprises a resonator element 601, a support structure 602 (102), and suspension elements 641a, 642a, 643b, 644b that secure the resonator element 601 to the support structure 602. The resonator element 601 comprises two stretching mode resonators (610a, 610b), a flexural mode resonator 620, a connection element 631a that connects the flexural mode resonator 620 to the stretching mode resonator 610a, and connection elements 632b, 633b that connect the flexural mode resonator 620 to the stretching mode resonator 610b. 9A are interconnected LE resonator element assemblies each having five elongated beam-like LE mode resonator elements 671. Each LE resonator element 671 is connected at its two distal ends (i.e., both ends) to adjacent LE resonator elements via a substantially rigid material portion (interconnection element) 672.

[0086] FIG. 9B illustrates a cross-section of the MEMS resonator assembly of FIG. 9A taken along line AA′. The interconnected LE resonator element assemblies 610a, 610b are actuated using a piezoelectric thin film actuator having a layer of piezoelectric material 652 and a top electrode layer 653. The layer of piezoelectric material 652 is formed on the single crystal silicon layer 651 in certain areas of the interconnected LE resonator element assemblies 610a, 610b. A top electrode layer 653 is also formed on the piezoelectric layer 652 in certain areas of the LE resonator element assemblies 610a, 610b. In some embodiments, such as the exemplary embodiment of FIGS. 9A-9B, the entire top surface of the single crystal silicon layer 651 of the LE resonator element assemblies 610a, 610b is covered by the piezoelectric layer 652 and the top electrode layer 653. 9A-9B, the flexure resonator 620 and connecting elements 631a, 632b, 633b include a piezoelectric layer 652 but no top electrode layer 653. However, in some embodiments, the flexure resonator 620 and connecting elements 631a, 632b, 633b also include a piezoelectric layer 652 and top electrode layer 653. The primary driving force for resonating the flexure resonator 620 comes from the piezoelectric actuation of the LE resonating element assemblies 610a, 610b and is mechanically mediated by the connecting elements 631a, 632b, 633b.

[0087] In some embodiments, the single crystal silicon layer 651 is doped with phosphorus dopant, with an average phosphorus dopant concentration of at least 2×10 19 cm -3 In some embodiments, the thickness of the single crystal silicon layer 651 is in the range of 2 μm to 40 μm, for example, 5 μm to 20 μm. In some embodiments, the crystal orientation of the single crystal silicon layer 651 is <100> In some embodiments, the main direction of vibration of the two stretching mode resonators (610a, 610b) is in the plane of the respective layers. <100> The average phosphorus dopant concentration of the single crystal silicon layer in the resonator element (including the upper electrode layer when the upper electrode is made of single crystal silicon as described later) is 2×10 19 cm -3The principal direction is along the x-axis in the embodiment shown in Figures 3B, 5B, 6B, 7B and 8B.

[0088] The material of the piezoelectric layer 652 may be, for example, AlN, Sc-doped AlN, ZnO, LiNbO3, or LiTaO3. In some embodiments, the thickness of the piezoelectric layer is in the range of 500 nm to 4 μm, such as 1 μm to 2 μm.

[0089] In some embodiments, the material of the top electrode layer 653 comprises a metal such as gold, molybdenum, aluminum or tungsten, a metal alloy, degenerately doped polycrystalline silicon, degenerately doped single crystal silicon, another semiconductor material, or any other suitable material that is electrically conductive. In some embodiments, the thickness of the top electrode layer is in the range of 50 nm to 1000 nm, such as 150 nm to 400 nm. In some embodiments, the material of the top electrode layer 653 is made of degenerately doped single crystal silicon and the thickness of the layer 653 is in the range of 2 μm to 40 μm, such as 5 μm to 20 μm, which may significantly improve the quality factor of the resonance by making the material layer stack more symmetrical, thereby reducing energy leakage to the support structure.

[0090] 9B may have other layers, such as one or two silicon oxide layers. In some embodiments, there is also a silicon oxide layer on the underside of the single crystal layer 651 facing the cavity 107. In some embodiments, there may be a silicon oxide layer between the single crystal layer 651 and the piezoelectric layer 652, between the piezoelectric layer 652 and the top electrode layer 653, or on top of the top electrode layer 653. These silicon oxide layers may be used to modify the temperature dependence of the resonant frequency.

[0091] Figure 9C shows a cross section of the MEMS resonator assembly of Figure 9A along line BB'. A contact pad 661 (on the support structure 602) on the (electrically isolated) piezoelectric layer 652 on the single crystal layer 651 is provided for galvanic connection to the top electrode 653 of the stretch mode resonator 610b via a conductor on one of the suspension elements of the resonator 610b (element 643b in Figures 9A,C). As shown by the layout of the MEMS resonator assembly 600 shown in the cross sections of Figures 9A and 9C, the contact pad 661 is also galvanically connected to the top electrodes 653 of the stretch mode resonators 610a and 610b via conductors formed on the suspension elements 641a and 643b, which are electrically isolated from the doped single crystal silicon layer 651 by the piezoelectric layer 652. Another contact pad 662 on the support structure 602 is provided to make a galvanic connection to the single crystal silicon layer 651, which is used as a bottom electrode in the exemplary embodiment of Figures 9A-9C. To provide a galvanic contact between the contact pad 662 and the single crystal silicon bottom electrode 651, an opening 663 is formed in the piezoelectric layer 652 and the material of the top electrode 653 is deposited on the opening 663. In the embodiment shown in Figures 9A-9C, the single crystal silicon layer 651 is degenerately doped with a dopant such as phosphorus or arsenic. Thus, the single crystal silicon portions in the stretch mode resonators 610a, 610b and the support structure 602 are galvanically connected via the single crystal silicon material portions in the suspension elements (641a, 642a, 643b, 644b) and via the single crystal silicon material portions in the flexure resonator 620 and the connecting elements 631a, 632b, 633b. The stretch mode resonator 610b (and 610a) can be excited into stretch mode resonance by applying an AC voltage at the resonant frequency between contact pads 661 and 662, thereby generating an AC electric field across the piezoelectric layer 652. Electrical interconnects for piezoelectric actuation of the MEMS resonator assembly 600 can be achieved in several alternative ways, such as using Through-Silicon-Via (TSB) or wafer-level packaging.

[0092] In some embodiments, a MEMS resonator assembly includes two stretch mode resonators and a flex mode resonator mechanically connecting the two stretch mode resonators. A first of the two stretch mode resonators includes a piezoelectric actuator, while a second of the two stretch mode resonators does not include a piezoelectric actuator. An exemplary embodiment of such a MEMS resonator assembly may be the same assembly as the assembly 600 of FIGS. 9A-C, except for the difference that there is no conductive path on the suspension element 641a to the top electrode 653 layer on the stretch mode resonator 610a. That is, only the stretch mode resonator 610b includes a piezoelectric actuator that can be used to excite the corresponding stretch mode resonator, and thus the resonator element, into a resonant mode. Although the ESR of the entire resonator assembly increases due to the reduction in electromechanical coupling, the power handling of the resonator assembly may be improved due to the sharing of kinetic energy between the mechanically connected stretch mode resonators vibrating in a collective resonant mode.

[0093] In some embodiments, the MEMS resonator assembly includes an electrostatic actuator for exciting the resonator elements into a resonant mode. An exemplary embodiment of such a MEMS resonator 700 formed on a silicon on insulator (SOI) substrate is shown in Figures 10A-10D. The SOI substrate has a single crystal silicon device layer 751, a silicon oxide layer 706, and a single crystal silicon handle layer 705 (shown in Figures 10B-10D). The device layer 751 is doped with an impurity to make it conductive. In some embodiments, the single crystal silicon layer 751 is doped with phosphorus dopant, with an average phosphorus dopant concentration of at least 2 x 10 19 cm -3The MEMS resonator assembly 700 of FIG. 10A includes a resonator element 701, a support structure 702 (represented by support structure portions 781-784), four suspension elements 741a, 742a, 743b, and 744b that anchor the resonator element 701 to the support structure 702, and four fixed electrodes 771-774 used for electrostatic actuation. The resonator element 701 includes two stretching mode resonators (710a and 710b), a flexural mode resonator 720, and three connection elements. One of the three connection elements connects the flexural mode resonator 720 to the stretching mode resonator 710a, and two of the three connection elements connect the flexural mode resonator 720 to the stretching mode resonator 710b. The stretching mode resonators 710a and 710b shown in FIG. 10A are interconnected LE resonator element assemblies, each having five elongated beam-shaped LE mode resonator elements 711. Each LE resonator element 711 is connected at its two distal ends (i.e. both ends) to adjacent LE resonator elements via a substantially rigid material portion (interconnection element) 712. The resonator element 701, the suspension elements 741a, 742a, 743b, 744b, and the fixed electrodes 771-774 are formed in a single crystal silicon device layer 751, as shown in FIG. 10B, which is a cross section AA' of FIG. 10A. Two electrodes 771, 772 are formed facing the outermost sidewalls of the interconnected LE resonator element assemblies. A narrow gap 791 (or 792) separates the electrodes 771 (or 772) from the opposing sidewalls of the interconnected LE resonator element assemblies 710a (or 710b). The width of the gap is typically less than 500 nm, and preferably less than 100 nm. There may also be an electrode facing the innermost sidewall of the interconnected LE resonator element assemblies, as shown in FIG. 10A. Narrow gaps separate electrodes 773, 774, formed in the device layer of the SOI substrate, from opposing sidewalls of the interconnected LE resonator element assemblies 710a, 710b (such as gaps 793, 794, which separate electrodes 774 from interconnected LE resonator element assemblies 710a, 710b, respectively).

[0094] The resonator element 701 is separated from the handle layer 705 by a cavity 707. The cavity 707 may be formed by removing the silicon oxide from the cavity using, for example, a hydrofluoric (HF) acid vapor etch. To release the resonator element, it may be necessary to etch holes 798 (see FIG. 10C) through the device layer using deep reactive ion etching (DRIE) to provide a flow path for the HF vapor to the silicon oxide layer 706. FIG. 10C is the same AA′ cross section as FIG. 10B, but showing the etch holes 798.

[0095] A cross section taken along line BB' of Figure 10A is shown in Figure 10D. Suspension elements 743b and 744b secure interconnected LE resonator element assembly 710b to support structures 783 (702) and 784 (702), respectively, formed in device layer 751 of the SOI substrate.

[0096] The MEMS resonator element 701 can be excited into a collective stretching mode resonance as shown in FIG. 3B by electrostatic actuation. To achieve this, the (fixed) electrodes 771-774 may be galvanically connected to the same terminal (e.g., X1). These galvanic connections may be achieved by etching trenches through the device layer 751 of the SOI substrate to pattern conductive structures and / or by forming conductive structures in the package of the MEMS resonator assembly 700 (e.g., ceramic package or wafer level package). The support structures 781-784 that are galvanically connected to the resonator elements are galvanically connected to another terminal (e.g., X2). For example, by using a conductor formed in the device layer 751 and / or by a conductive structure in the package of the MEMS resonator assembly 700. A DC bias voltage may then be connected to one of the terminals (e.g., X1) through a large impedance to create sufficient electromechanical coupling to the resonator element 701, while the other terminal (e.g., X2) is left at zero DC potential. A stretching mode resonance can then be excited by applying an AC voltage at the resonant frequency between terminals X1 and X2, causing the MEMS resonating element 701 to vibrate at the collective stretching mode resonance shown in FIG. 3B.

[0097] FIG. 11A shows an embodiment consisting of two width-stretch (WE) mode resonators 810a, 810b and an (in-plane) flexural mode resonator 820 designed to vibrate at the third overtone. The entire structure shown is a suspended structure. The WE mode resonators 810a, 810b are tethered to a support structure (not shown) via suspension elements 841a, 842a, 843b, 844b. In the exemplary embodiment shown in FIG. 11A, the suspension elements have a meander-shaped structure, which is useful for example to relieve stress in the y-direction. At the three antinodes of the central part of the flexural resonator 820, connection elements are arranged that couple these antinodes to the adjacent side walls of the WE mode resonators 810a, 810b. The width W and length L of the flexural resonator 820 are such that its third overtone resonance is substantially equal to the resonant frequency of the stretching mode resonators (here WE mode resonators) 810a, 810b.

[0098] FIG. 11B shows the same embodiment as in FIG. 11A at maximum displacement of the vibration cycle. The dashed and dotted lines represent the initial rest position of the WE mode resonators 810a, 810b. The flexural mode resonator 820 is vibrating at the third overtone. The magnitude of the displacement is exaggerated for illustrative purposes. The entire MEMS resonator assembly shown in FIG. 11B vibrates in a collective resonant mode in which the motions of the stretching mode resonators 810a, 810b are substantially in phase with each other.

[0099] In some embodiments, as shown in FIG. 12A, the MEMS resonator assembly is composed of two square extension (SE) mode resonators 910a, 910b and a flexural mode resonator 920 designed to vibrate at the third overtone in the xy plane. The entire structure shown is a suspended structure. The SE mode resonators 910a, 910b are tethered to a support structure (not shown) via suspension elements 941a, 942a, 943b, 944b. In the exemplary embodiment shown in FIG. 12A, the meander-shaped structure of the suspension elements is mechanically compliant to allow for alternating contraction and expansion of the square extension mode resonators 910a, 910b during mechanical resonant motion. That is, it is capable of transmitting forces through elastic deformation. Three antinodes in the central portion of the flexural mode resonator 920 are disposed with connection elements that couple these antinodes to adjacent sidewalls of the SE mode resonators 910a, 910b. The width W and length L of the flexure resonator 920 are such that its third overtone resonance is substantially equal to the resonant frequency of the extension mode resonators (here, square extension (SE) mode resonators) 910a, 910b.

[0100] In other embodiments having an SE mode resonator, the suspension element of the SE resonator may be connected at a different location than in the exemplary embodiment of Figures 12A-12B. For example, the suspension element may be located at the center (central pillar) of the SE resonator, which is the nodal point of the square stretch vibration mode. The suspension element may be formed by a material portion in the cavity of a cavity SOI structure between the device layer and the handle layer. The suspension element may comprise monocrystalline silicon, polycrystalline silicon, and / or silicon oxide.

[0101] Figure 12B shows the same embodiment as Figure 12A at maximum inward displacement of the vibration cycle. The dashed and dotted lines represent the initial rest positions of the SE mode resonators 910a, 910b. The flexural mode resonator 920 is vibrating at the third overtone. The magnitude of the displacement is exaggerated for illustrative purposes. The entire MEMS resonator assembly shown in Figure 12B vibrates in a collective resonant mode in which the motion of the stretching mode resonators 910a, 910b is substantially in phase with each other.

[0102] In some embodiments, as shown in FIG. 13, a MEMS resonator assembly includes a plurality of SE mode resonators (here, four SE resonators 1010a, 1010b, 1010c, 1010d arranged in a 2×2 array configuration) and a plurality of flexural mode resonators (here, four flexural resonators 1021-1024). In the exemplary embodiment of FIG. 13, each flexural mode resonator is connected to two adjacent SE resonators. Each of the flexural mode resonators 1021-1024 has a width W and a length L such that its third overtone resonant frequency is substantially equal to the resonant frequency of the stretching mode (SE) resonators 1010a, 1010b, 1010c, 1010d. The three antinodes in the central portions of the bending mode resonators 1021-1024 have connection elements disposed thereon that couple the antinodes to adjacent sidewalls of the SE mode resonators 1010a, 1010b, 1010c, 1010d. The 2×2 array shown in FIG. 13 is a suspension structure in which meander-shaped suspension elements 1041a, 1042b, 1043c, 1044d tether the resonator elements to a support structure (not shown). In some embodiments, the four SE resonators are suspended by suspension elements that mechanically connect the central nodes of each SE resonator to the handle layer, as described above. The MEMS resonator assembly 1000 vibrates in a collective resonant mode in which the motions of the stretching mode resonators 1010a, 1010b, 1010c, 1010d are substantially in phase with each other.

[0103] FIG. 14A shows a MEMS resonator assembly 1100 with two extensional (LE) mode resonators 1110a, 1110b and an (in-plane) flexural mode resonator 1120 designed to vibrate at the third overtone. The entire structure shown is a suspended structure. The LE mode resonators 1110a, 1110b are tethered to a support structure (not shown) via suspension elements 1141a, 1142a, 1143b, 1144b. In the exemplary embodiment shown in FIG. 14A, the suspension elements have a meander-shaped structure, useful for example to relieve stress in the x-direction. At the two antinodes 1131a, 1132b of the flexural resonator 1120, connection elements are arranged that couple these antinodes to the distal end sidewalls of the LE mode resonators 1110a, 1110b. The width W and length L of the flexural resonator 1120 are set so that the third overtone resonance thereof is substantially equal to the resonance frequency of the extension mode resonators (here, LE mode resonators) 1110a and 1110b.

[0104] Figure 14B shows the same embodiment as Figure 14A at maximum displacement of the vibration cycle (maximum contraction of the LE mode resonators 1110a, 1110b). The dashed and dotted lines represent the initial rest positions of the distal ends of the LE mode resonators 1110a, 1110b. The bending mode resonator 1120 is vibrating at the third overtone. The magnitude of the displacement has been exaggerated for illustrative purposes.

[0105] In the maximum contraction position shown in FIG. 14B, the distal ends of the LE resonators 1110a, 1110b have moved in the y direction toward their (central) axis of symmetry, as indicated by the four arrows next to the two dashed-dotted lines in FIG. 14B. In the maximum extension position of vibration, which is 180 degrees out of phase with respect to maximum contraction, the distal ends of the LE resonators 1110a, 1110b extend outward in the y direction from their central axis of symmetry. In the maximum contraction position shown in FIG. 14B, the shape of the flexure resonator 1120 is such that the portion of material close to the connecting element 1131a connecting the flexure resonator 1120 to the stretch mode resonator 1110a and the portion of material close to the connecting element 1132b connecting the flexure resonator 1120 to the stretch mode resonator 1110b move in the same direction along the y axis to follow the movement of the adjacent distal ends of the LE resonator elements 1110a, 1110b. The flexure resonator 1120 has an anti-node location midway between the anti-node connections to the connecting elements 1131a, 1132b, and two anti-node locations at either end. In the embodiment shown in Figure 14B, these three anti-node locations have no connecting elements attached and move in opposite directions during vibration relative to the two anti-node locations connected to the connecting elements 1131a and 1132b. The entire MEMS resonator assembly shown in Figure 14B vibrates in a collective resonant mode in which the motion of the stretch mode resonators 1110a, 1110b are substantially in phase with each other.

[0106] FIG. 15A shows a MEMS resonator assembly 1200 comprising two extension-and-retraction (LE) mode resonators 1210a, 1210b and two (in-plane) flexural mode resonators 1221, 1222. The two LE resonators are arranged side by side along the x-axis with their longitudinal directions aligned with the y-axis. The distal ends of the LE resonating elements 1210a, 1210b in the positive y-axis direction are coupled via the flexural mode resonator 1221, and the distal ends in the negative y-axis direction are coupled via the flexural mode resonator 1222. The entire structure shown is a suspended structure. The LE mode resonators 1210a, 1210b are tethered to a support structure 120 (not shown) via suspension elements 1241a, 1242a, 1243b, 1244b. At the two antinodes of the flexure resonator 1221, connecting elements 1231a, 1232b are arranged which couple the antinodes to adjacent distal end sidewalls of the LE mode resonators 1210a, 1210b. At the two antinodes of the flexure resonator 1222, connecting elements 1233a, 1234b are arranged which couple the antinodes to adjacent distal end sidewalls of the LE mode resonators 1210a, 1210b. The width W and length L of the flexure resonators 1221, 1222 are such that their third overtone resonance is substantially equal to the resonance frequency of the stretching mode resonators (here, LE mode resonators) 1210a, 1210b.

[0107] Figure 15B illustrates the same embodiment depicted in Figure 15A, the LE mode resonators 1210a, 1210b at maximum contraction during their vibration cycle. The dashed and dotted lines represent the initial rest positions of the distal ends of the LE mode resonators 1210a, 1210b. The magnitude of the displacements has been exaggerated for illustrative purposes. The entire MEMS resonator assembly 1200 vibrates in a collective resonant mode in which the motions of the stretch mode resonators 1210a, 1210b are substantially in phase with each other.

[0108] 16A shows a MEMS resonator assembly 1300 comprising four length-stretch (LE) mode resonators 1310a, 1310b, 1310c, 1310d and three (in-plane) flexural mode resonators 1321, 1322, 1323 coupled to the distal ends of the LE resonator elements. The LE mode resonators are arranged in a line along the x-axis with their longitudinal axes pointing in the y-axis direction. The width W and length L of the flexural mode resonators 1321, 1322, 1323 are such that their third overtone resonance is substantially equal to the resonant frequency of the stretching mode resonators (here LE mode resonators) 1310a, 1310b, 1310c, 1310d. The outermost anti-node positions of the flexure resonators 1321, 1322, 1323 are connected to connecting elements (1331a, 1332b, 1333b, 1334c, 1335c, 1336d). The other ends of these connecting elements are connected to the adjacent distal end sidewalls of the LE mode resonators. The flexure resonator 1321 (or 1323) is connected to the distal ends of the LE resonator elements 1310a and 1310b (or 1310c and 1310d) in the positive y-direction relative to the centerline of the LE resonator elements. The flexure resonator 1322 is connected to the distal ends of the LE resonator elements 1310b and 1310c in the negative y-direction relative to the centerline of the LE resonator elements.

[0109] FIG 16B illustrates the same embodiment depicted in FIG 16A for the LE mode resonators 1310a, 1310b, 1310c, 1310d at maximum contraction during a vibration cycle. The dashed and dotted lines represent the initial rest positions of the distal ends of the LE mode resonators 1310a, 1310b, 1310c, 1310d. The magnitude of the displacements has been exaggerated for illustrative purposes. The entire MEMS resonator assembly 1300 vibrates in a collective resonant mode in which the motions of the stretch mode resonators 1310a, 1310b, 1310c, 1310d are substantially in phase with each other.

[0110] In addition to the advantages mentioned above, the embodiment shown in Figures 14A-14b, 15A-15b, and 16A-16B has further advantages. First, the electrical resistance of AC current passing through the MEMS resonator assembly is low. Each LE resonator component of the MEMS resonator assembly has low electrical resistance to the support structure and low ohmic losses, so ohmic losses in the resonator elements are minimized. Second, because the suspension elements for each LE resonator are in close proximity to each other, thermo-mechanical stresses in the resonator elements are low. This improves the thermal and long-term stability of the resonant frequency.

[0111] FIG. 17 shows a MEMS resonator assembly 1400 comprising two interconnected LE resonator element assemblies 1410a, 1410b and an (in-plane) flexural mode resonator 1420 coupled to the distal end of the interconnected LE resonator element assemblies. The two interconnected LE resonator element assemblies are arranged in a row along the x-axis, with the longitudinal axis of each LE resonator element assembly coinciding with the y-direction. The width W and length L of the flexural mode resonator 1420 are such that its third overtone frequency is substantially equal to the resonant frequency of the stretching mode resonators (here the interconnected LE resonator element assemblies) 1410a, 1410b. The two antinodes of the flexural resonator 1420 are provided with connecting elements 1431a, 1432b that couple the antinodes to adjacent distal end sidewalls of the interconnected LE resonator element assemblies 1410a, 1410b. The interconnected LE resonator element assemblies 1410a, 1410b are tethered to the same support structure 1402 at the center of the two interconnected LE resonator element assemblies via suspension elements 1441a, 1442b. The entire MEMS resonator assembly 1400 vibrates in a collective resonant mode in which the motions of the stretching mode resonators 1410a, 1410b are substantially in phase with respect to each other.

[0112] The support structure 1402 and the suspension elements 1441a, 1442b being symmetrical about a centerline parallel to the y-axis have several advantages. First, such a central support structure reduces the leakage of vibration energy through the suspension elements to the support structure, since there is no vibrational motion along the x-direction during vibration. In contrast, in the embodiment shown in Figures 3C-3D, small alternating contractions and expansions at the centerline of the LE resonating element 151 induce some (albeit small) movement at the connection points of both the two suspension elements 141a, 142a and the support structure. Thus, a MEMS resonator with a central support structure increases the quality factor of the resonance and reduces the ESR (equivalent series resistance). Second, the central support structure reduces the thermomechanical stress experienced by the resonator element, since the connection points of the suspension elements 1441a and 1442b and the support structure 1402 are closer to each other. Third, the central support structure, along with its lower thermomechanical stress, tends to improve the thermal and long-term stability of the resonant frequency.

[0113] There are other embodiments with a central support structure. The term central support structure is used herein to mean a support structure (102) located along a line that is central in the x-direction or y-direction or central in both the x-direction and y-direction of the stretching mode resonators of the MEMS resonator assembly, in the plane of the resonator elements and / or below or above the plane (like the central support structure of the SE resonator described above). In some embodiments, two LE resonator elements connected via flexure resonators are both fixed to the same central support structure. An example of such an embodiment is similar to the MEMS resonator assembly of FIG. 17, but with only one LE resonator element in each interconnected LE resonator element assembly. Another embodiment with a central support structure is shown in FIG. 18. In this embodiment, suspension elements 1541a, 1542a, 1543b, 1544b connect width stretch mode resonators 1510a, 1510b, which are connected via flexure resonator 1520, to a central support structure 1502, 1503 disposed along a line located between width stretch mode resonators 1510a, 1510b. Otherwise, the embodiment of Figure 18 is similar to that described with reference to Figures 11A-11B.

[0114] FIG. 19 shows yet another embodiment with a central support structure. In this embodiment, there are two interconnected LE resonator element assemblies 1610a, 1610b coupled via a flexural mode resonator 1620. The resonator elements of the MEMS resonator assembly 1600 are fixed to three support structures 1602, 1603, 1604, one of which (1603) is a central support structure. Apart from the support structures and associated suspension elements, the MEMS resonator assembly 1600 is similar to the assembly 1400 of FIG. 17. The use of multiple support structures in the MEMS resonator assembly 1600 reduces AC ohmic losses in the suspension structures due to lower electrical resistance to the support structures (due to shorter distances). Another advantage of multiple support structures (such as a central support structure) is reduced sensitivity to g-forces in the z-direction.

[0115] In some embodiments, the central support structure is 0.04 mm (in the xy plane of the resonator element). 2 For example, the x dimension by y dimension of the central support structure may be 0.19 mm by 0.19 mm or 0.1 mm by 0.39 mm.

[0116] In an embodiment using piezoelectric actuation, galvanic connections to the single crystal layer (including the openings in the piezoelectric layer and the contact pads illustrated by 663, 662 in FIG. 9C) and the top electrode (including the contact pad illustrated by 661 in FIG. 9C) are made in the central support structure. The two contact pads 661 and 662 can be further connected to a through silicon via (TSV) structure passing through the cap wafer 108 in a wafer level packaged resonator (see FIG. 2B) or to bond wires in a ceramic package (see discussion related to FIG. 2B).

[0117] FIG. 20 shows a MEMS resonator assembly 1700 comprising three interconnected LE resonator element assemblies 1710a, 1710b, 1710c and an (in-plane) flexural mode resonator 1720 coupled to the distal end of the interconnected LE resonator element assemblies. The LE resonator element assemblies are arranged in two rows along the x-axis with their longitudinal axes aligned in the y-direction. The assemblies 1710a and 1710b are in the lower row (i.e., the row in the negative y-axis direction from the flexural resonator 1720) and the assembly 1710c is in the upper row (i.e., the row in the positive y-axis direction from the flexural resonator 1720). The width W and length L of the flexural mode resonator 1720 are such that its third overtone resonant frequency is substantially equal to the resonant frequency of the stretching mode resonators (here the interconnected LE resonator element assemblies) 1710a, 1710b, 1710c. Located at three antinodes in the central portion of the flexural resonator 1720 are connecting elements 1731a, 1732b, 1733c that couple the antinodes to adjacent distal end sidewalls of the interconnected LE resonator element assemblies 1710a, 1710b, 1710c, respectively. The interconnected LE resonator element assemblies 1710a, 1710b, 1710c are tethered to a support structure (not shown) via suspension elements 1741a, 1742a, 1743b, 1744b, 1745c, 1746c. The entire MEMS resonator assembly 1700 vibrates in a collective resonant mode in which the motions of the stretching mode resonators 1710a, 1710b, 1710c are substantially in phase with each other.

[0118] In some embodiments, the principal vibration axes of the multiple bending mode resonators are perpendicular to each other. An example of such an embodiment is shown in FIGS. 21A-21B. The MEMS resonator assembly 1800 of FIG. 21A comprises two interconnected LE resonator element assemblies 1810a, 1810b. The principal vibration axes of the two LE resonator element assemblies are perpendicular to each other (1810a in the y direction, 1810b in the x direction). The layout of the assembly 1800 is symmetrical about the y axis. The first bending mode resonator 1821 is coupled to the left distal end of the interconnected LE resonator element assembly 1810b (in the negative x direction) using a connecting element 1831b, and is coupled to the top distal end of the interconnected LE resonator 1810a (in the positive y direction) using a connecting element 1831a. A second bending mode resonator 1822 is coupled to the right distal end (positive x-axis) of interconnected LE resonator element assembly 1810b using connecting element 1832b and to the top distal end (positive y-axis) of interconnected LE resonator 1810a using connecting element 1832a. Both bending mode resonators 1821, 1822 consist of two rectangular frames (in this case two identical square shaped frames) and a substantially rigid connecting element between the two frames.

[0119] The shape of the flexural resonator 1822 at resonance is shown in FIG. 21B. The opposing sides of the two frames 1822a, 1822b alternately move either inward or outward during resonant vibration, and the adjacent sides of the two frames move in different directions (either inward or outward) relative to each other. The two sides of the frames 1822a and 1822b, which are connected to a substantially rigid connecting element 1822c, move in the same direction in the y direction, but in opposite directions (either inward or outward) relative to the center of each frame. Thus, the resonant motion of the flexural mode resonator 1822 is such that when the connecting element 1832a moves in the positive y direction, the connecting element 1832b moves in the positive x direction, as shown in FIG. 21B. The rectangular frames 1822a and 1822b are dimensioned such that their resonant frequencies are substantially equal to each other and to the resonant frequencies of the interconnected LE resonating element assemblies 1810a and 1810b. The mode shape of the bending mode resonator 1822 at resonance is such that the magnitude of the motion of the connecting element 1832a in the positive y-axis direction is substantially equal to the magnitude of the motion of the connecting element 1832b in the positive x-axis direction. Thus, the effect of the bending mode resonator 1822 is that the entire MEMS resonator assembly 1800 vibrates in a collective resonant mode in which the motions of the stretching mode resonators (here, the interconnected LE resonator element assemblies) 1810a, 1810b, which have mutually orthogonal principal vibration directions, are substantially in phase with each other. This effect is similar to that of the bending mode resonator 1821. The resonator elements of the MEMS resonator assembly 1800 are connected by three suspension elements, of which two suspension elements 1841a, 1842a are connected to the stretching mode resonator 1810a, and the suspension element 1843b is connected to the stretching mode resonator 1810b. The entire structure shown in FIG. 21A is a suspended structure.

[0120] Another embodiment with a similar technical effect is shown in Figs. 21C-21D. The MEMS resonator assembly 1900 of Fig. 21C comprises two interconnected LE resonator element assemblies 1910a, 1910b. The principal vibration axes of these two LE resonator element assemblies are perpendicular to each other (1910a in the y-direction, 1910b in the x-direction). The layout of the assembly 1900 is symmetrical about the y-axis. The first bending mode resonator 1921 is coupled to the left distal end of the interconnected LE resonator element assembly 1910b (in the negative x-axis direction) using connecting elements 1931b and 1931c, and to the top distal end of the interconnected LE resonator 1910a (in the positive y-axis direction) using connecting element 1931a. The second flexural mode resonator 1922 is coupled to the right distal end (positive x-axis) of the interconnected LE resonator element assembly 1910b using connection elements 1932b and 1932c, and to the top distal end (positive y-axis) of the interconnected LE resonator element 1910a using connection element 1932a. Both flexural mode resonators 1921, 1922 are composed of a flexural beam resonator and a rectangular frame (square shaped frame in this example) with a substantially rigid link between the flexural beam resonator and the rectangular frame. The resonant shape of the flexural resonator 1922 is shown in FIG. 21D. The opposing sides of the frame 1922b move alternately inward or outward during resonant vibration, and the adjacent sides of the two frames move in different directions (either inward or outward) relative to each other. The flexure beam resonator 1922a has its longitudinal axis along the y-direction and is connected to the frame 1922b at a distance from the resonator 1922a in the positive x-direction using a substantially rigid connecting element 1922c. In the exemplary embodiment of FIG. 21D, the first overtone frequency of the flexure beam resonator 1922a and the resonant frequency of the rectangular frame 1922b are substantially equal to the resonant frequency of the interconnected LE resonant element assemblies 1910a and 1910b. Thus, the resonant motion of the flexure mode resonator 1922 is such that when the connecting element 1932a moves in the negative y-direction, the connecting elements 1932b and 1932c move in the negative x-direction, as shown in FIG. 21D.Thus, the effect of the bending mode resonator 1922 is that the entire MEMS resonator assembly 1900 vibrates in a collective resonant mode in which the motions of the stretching mode resonators (here interconnected LE resonator element assemblies) 1910a, 1910b, which have mutually orthogonal principal vibration directions, are substantially in phase with each other. This effect is similar for the bending mode resonator 1921. The resonator elements of the MEMS resonator assembly 1900 are connected by three suspension elements, two of which, 1941a, 1942a, are connected to the stretching mode resonator 1910a, and suspension element 1943b is connected to the stretching mode resonator 1910b. The entire structure shown in FIG. 21C is a suspended structure.

[0121] The embodiment in which the flexural resonator connects stretching mode resonators whose principal vibration axes are perpendicular to each other can be used to minimize the product of the equivalent series resistance and the area required for the resonator element (ESR*A). In some embodiments, the optimal ESR*A is obtained with a resonator element having a shape close to a square (such as the exemplary embodiments of Figures 21A and 21C). In some embodiments, the area A of the resonator element is less than 0.001 mm2. 2 from 1mm 2 The range is.

[0122] The MEMS resonator assembly 2900 of FIG. 21E comprises two interconnected LE resonator element assemblies 2910a, 2910b. The principal directions of vibration of these two LE resonator element assemblies are perpendicular to each other (2910a in the y direction, 2910b in the x direction). The layout of the assembly 2900 is symmetrical about the y axis. The first bending mode resonator 2921 is coupled to the left distal end of the interconnected LE resonator element assembly 2910b (in the negative x direction) using a connecting element 2931b and to the top distal end of the interconnected LE resonator element 2910a (in the positive y direction) using a connecting element 2931a. The second flexural mode resonator 2922 is coupled to the right distal end (positive x-axis direction) of the interconnected LE resonating element assembly 2910b using a connecting element 2932b, and to the top distal end (positive y-axis direction) of the interconnected LE resonating element 2910a using a connecting element 2932a. Both flexural mode resonators 2921, 2922 are constructed with rectangular frames (square frames in this case). The shape of the flexural resonator 2922 at resonance is shown in FIG. 21F. During resonant vibration, the opposing sides of the frame 2922 move alternately either inward or outward, and the adjacent sides of the two frames move in different directions (either inward or outward) relative to each other. In the exemplary embodiment of FIG. 21F, the resonant frequency of the flexural resonator 2922 is substantially equal to the resonant frequencies of the interconnected LE resonating element assemblies 2910a and 2910b. Thus, the resonant motion of the bending mode resonator 2922 is such that when the connecting element 2932a moves in the negative y-axis direction, the connecting element 2932b moves in the positive x-axis direction, as shown in FIG. 21F. Thus, the effect of the bending mode resonator 2922 is that the entire MEMS resonator assembly 2900 vibrates in a collective resonant mode in which the motions of the stretching mode resonators (here, the interconnected LE resonating element assemblies) 2910a, 2910b, which have mutually orthogonal principal vibration directions, are substantially 180 degrees out of phase with each other. As a result, when the stretching mode resonator 2910a contracts the most, the stretching mode resonator 2910b expands the most. This effect is the same for the bending mode resonator 2921.The resonator elements of the MEMS resonator assembly 2900 are connected by three suspension elements, two of which, 2941a and 2942a, are connected to the stretch mode resonator 2910a, and suspension element 2943b is connected to the stretch mode resonator 2910b. The entire structure shown in FIG. 21E is a suspended structure.

[0123] In some embodiments, the MEMS resonator assembly comprises Lame mode resonators. The MEMS resonator assembly 2100 shown in FIG. 22 has four Lame mode resonators 2110a, 2110b, 2110c, 2110d arranged in a 2×2 array, and four flexural mode resonators 2121-2124, each connected to two adjacent Lame mode resonators. The Lame resonators have a square plate shape in the xy plane. The flexural mode resonators 2121-2124 have the shape of a rectangular frame (in this case, a square frame). The shape of the resonator assembly 2100 is exaggerated to show the mode shape consisting of the fundamental Lame mode and the fundamental flexural mode of the resonators 2121-2124 in collective resonance. In the fundamental Lame mode, the central sidewalls of adjacent square plates move (stretch or shrink) 180 degrees out of phase with each other, as shown in FIG. 22. The opposing side walls of two adjacent Lame resonators are coupled via a flexure resonator connected to the (anti)center points of the two side walls. The opposing side walls of the frame-shaped flexure resonators 2121-2124 move alternately either inward or outward during resonant vibration, and the adjacent side walls of the two frames move in different directions (either inward or outward) relative to each other (similar to the resonator 2922 of FIG. 21F). The dimensions of the beams forming the rectangular frames 2121-2124 are selected such that the resonant frequencies of the flexure resonators 2121-2124 are substantially equal to the resonant frequencies of the Lame mode resonators 2110a, 2110b, 2110c, 2110d. The effect of the flexural resonator coupler is therefore to synchronize the Lame resonators, so that the MEMS resonator assembly 2100 oscillates in a collective resonant mode, in particular in a collective resonant mode in which the motions of the stretching mode resonators 2110a, 2110b, 2110c, 2110d are substantially in phase with each other. The entire structure shown in Fig. 22 is a suspended structure. The corner points are the nodal points of the fundamental Lame mode, and therefore the suspension elements are connected to these points (in the embodiment shown, only eight suspension elements 2141a, 2142a, 2143b, 2144b, 2145c, 2146c, 2147d, 2148d are connected to the corners of the resonators).

[0124] 22B illustrates an embodiment 2200 having two square-shaped Lame mode resonators 2210a, 2210b and a (substantially) rigid mechanical coupling element 2252 connecting the center points of the opposing sidewalls of the Lame resonators. The opposing sidewalls vibrate 180 degrees out of phase with each other such that when the connection point of the coupling element 2252 to the first resonator 2210a is maximally displaced toward the second resonator 2210b, the connection point of the coupling element 2252 to the second resonator 2210b is maximally displaced away from the first resonator 2210a. (Note that the geometry of the resonator assembly 2200 shown is an exaggerated depiction of collective resonance.) Thus, the effect of the substantially rigid mechanical coupling element 2252 is to synchronize the two Lame resonators such that the MEMS resonator assembly 2200 vibrates in a collective resonance mode. The entire structure shown in FIG. 22B is suspended from the corners of the square shaped Lame resonator by suspension elements 2241a, 2242a, 2243a, 2244a, 2245b, 2246b, 2247b, 2248b to a support structure (not shown).

[0125] Figure 22C shows an embodiment 2300 comprising two square extension (SE) mode resonators 2310a, 2310b and a (substantially) rigid mechanical coupling element 2352 connecting the center points of the opposing side walls of the SE resonators. The opposing side walls vibrate 180 degrees out of phase with each other such that when the connection point of the coupling element 2352 to the first resonator 2310a is maximally displaced inward (away from the second resonator 2310b), the connection point of the coupling element 2352 to the second resonator 2310b is maximally displaced outward (toward the first resonator 2310a). Note that Figure 22C shows an exaggerated view of the shape of the resonator assembly 2300 at this point (when the SE resonator 2310a is maximally contracted and the SE resonator 2310b is maximally extended). The effect of the substantially rigid mechanical coupling element 2352 is to synchronize the SE resonators such that the MEMS resonator assembly 2300 vibrates in a collective resonant mode in which the two SE resonators 2310a, 2310b are 180 degrees out of phase with respect to each other. The entire structure shown in Figure 22C is suspended from the corners of the SE resonators by suspension elements 2341a, 2342a, 2343a, 2344a, 2345b, 2346b, 2347b, 2348b to a support structure (not shown).

[0126] FIG. 22D shows a MEMS resonator assembly 2400 having two width-extension (WE) resonators 2410a, 2410b and two (substantially) rigid mechanical coupling elements 2451, 2452 (having a length LC) formed between the two rectangular plate-like WE resonators and connecting the distal parts of the opposing side walls of the two WE resonators. The main direction of vibration of the WE resonators is along the y-direction. As shown in FIG. 22D, the length of the side along the x-direction (LR) is greater than the length of the side along the y-direction (WR). Thus, the resonators 2410a, 2410b are described as width-extension resonators. The purpose of the substantially rigid mechanical coupling elements 2451 and 2452 is to ensure that the WE resonators 2410a, 2410b resonate in tandem in the same width-extension resonance mode at the same frequency. (The WE resonators 2410a, 2410b may have slightly different resonant frequencies due to manufacturing tolerances if they were not mechanically connected to each other.) The interconnected WE resonators 2410a, 2410b are tethered to the same (central) support structure 2402 at the center of the two WE resonators via suspension elements 2441a, 2442b. The entire MEMS resonator assembly 2400 vibrates in a collective resonant mode in which the motions of the stretching mode resonators 2410a, 2410b are substantially in phase with each other.

[0127] Those skilled in the art will appreciate in light of the above exemplary embodiments that all of the above exemplary embodiments of the MEMS resonator assembly may include an actuator for exciting the resonator into a resonant mode. They will also appreciate that the actuator may be of the piezoelectric or electrostatic type, as described above with reference to Figures 9A-9C and 10A-10D. In some embodiments, a piezoresistive sensor is used to measure the vibrational motion of the resonator element.

[0128] The bulk acoustic resonators that are part of a MEMS resonator assembly (such as 100) do not have to be of the same type. In some embodiments, for example, the MEMS resonator assembly comprises an interconnected length-stretching (LE) resonator assembly and a square-stretching mode (SE) resonator. In another exemplary embodiment, the MEMS resonator assembly comprises an SE mode resonator and a width-stretching (WE) mode resonator. In yet another exemplary embodiment, the MEMS resonator assembly comprises an LE mode resonator and an interconnected LE resonator element assembly. When considering the order of the resonant modes, still other alternative embodiments are provided. In addition to the embodiment in which the bulk acoustic resonators vibrate at the fundamental frequency, there are also embodiments in which at least one of the bulk acoustic resonators vibrates at a third overtone frequency that is substantially equal to the resonant frequency of the collective resonant mode of the MEMS resonator assembly.

[0129] With reference to the disclosed embodiments, the following table summarizes certain characteristics of exemplary MEMS resonator assemblies (shown as legends S1-S6).

[0130] TIFF2024545463000004.tif117170

[0131] Layouts of the exemplary MEMS resonator assemblies in the above table are shown in Figures 23A-23F for embodiments S1-S6, respectively. The area of ​​a resonator element is defined herein as the area of ​​the smallest rectangle that encloses the resonator element (as a whole) (in a plane that contains the die or wafer on which the resonator element is formed). The width and length values ​​of such a rectangle are shown in the layout.

[0132] The embodiment shown in Figures 23A-23C and 23F is an interconnected LE resonator element assembly consisting of elongated beam-like LE resonator elements 151. The LE resonator elements 151 are separated by trenches 153 but are connected to each other at both ends by distal portions 152. The length (along the x-direction) of the interconnected LE resonator elements was selected to obtain the resonant frequencies tabulated above. The number of interconnected LE resonator elements was selected to fit into the space reserved in the y-direction for the resonator assembly.

[0133] The embodiment shown in FIG. 23D comprises two interconnected LE resonator element assemblies, both of which are connected to a flexure resonator using a connecting element (three connecting elements in total). The flexure resonator is dimensioned to vibrate at a third overtone resonance at 40 MHz. (The dimensions L and W of the flexure mode resonator and the dimensions of the connecting elements LC=LC1=LC2=LC3, WC=WC1=WC2=WC3 are given in the table associated with the discussion of FIG. 4.) Meanwhile, the interconnected LE resonator element assemblies vibrate at a fundamental resonant mode at 40 MHz substantially in phase with each other. The embodiment shown in FIG. 23D embodies the embodiment shown in FIGS. 3A and 3B, but with nine LE resonator elements in the assembly instead of five.

[0134] The embodiment shown in FIG. 23E comprises three interconnected LE resonator element assemblies. These assemblies are coupled by two flexure resonators and a connecting element (connecting element between a flexure resonator and its adjacent assembly). The flexure resonators are dimensioned to vibrate at a third overtone resonance at 40 MHz. (The dimensions L and W of the flexure mode resonators and the connecting element dimensions LC=LC1=LC2=LC3, WC=WC1=WC2=WC3 are given in the table associated with the discussion of FIG. 4.) Meanwhile, the interconnected LE resonator element assemblies vibrate at a fundamental resonant mode at 40 MHz substantially in phase with each other. The embodiment shown in FIG. 23E embodies the embodiment shown in FIGS. 5A and 5B, but with nine LE resonators in the assembly instead of five.

[0135] The mask layout shown in the figure was used to etch trenches through the material layers of the piezoelectrically coupled MEMS resonators. The material layers of these resonators have layers according to the diagrams of Figures 9A-9C. The electrical and physical properties of the MEMS resonator assemblies (equivalent circuit parameters, resonant frequency and its temperature dependence, etc.) depend on the dimensions, mechanical and physical properties of the materials forming the resonator elements. The crystal orientation of the monocrystalline silicon layer 651 of the exemplary MEMS resonator assemblies S1-S6 is such that the main direction of vibration of the two interconnected LE resonator element assemblies (x-direction in Figures 23A-23F) is aligned along the x-axis. <100> Substantially parallel to the crystal direction (e.g., <0100>) and the y direction is different <100> The average phosphorus dopant concentration of the single crystal silicon layer 651 in the resonator element is 2×10 19 cm -3 A larger value (e.g., 1.5×10 20 cm -3 ~2.5×10 20 cm -3 ). The thickness of the single crystal silicon layer 651 of each of the exemplary MEMS resonator assemblies S1-S6 is 11 μm. The material of the piezoelectric layer 652 of the exemplary MEMS resonator assemblies S1-S6 is AlN, and the thickness of each AlN layer is 1.6 μm. The thickness of the top electrode layer 653 ranges from 200 nm to 300 nm. There is no silicon oxide layer in the resonator elements of the exemplary MEMS resonator assemblies S1-S6.

[0136] The exemplary MEMS resonator assemblies S1-S6 were investigated by electrical impedance measurements and the results were fitted to the equivalent circuit model shown in FIG. 24 to extract key performance parameters such as ESR values ​​summarized above. The equivalent circuit model is also known as the Butterworth-Van Dyke (BVD) model. This model has a series LC resonator characterized by an equivalent series capacitance C1, an equivalent series inductance L1, and a series resistance R1, where the series resistance R1 represents the energy loss during the mechanical resonant motion. A shunt capacitance C0 is in parallel with the LC resonant circuit. The electrical resistance of the current path is the resistance R of the equivalent circuit model in FIG. 24. e In this disclosure, the term "equivalent series resistance" and its abbreviation "ESR" refer to the resistance ESR=R1+R, which is the real part of the complex impedance of a MEMS resonator assembly measured at the (series) resonant frequency f0. e In the analysis of the complex impedance of the MEMS resonator assemblies S1 to S6, the electrical resistance (R e ) is much smaller than the series resistance R1 and has been neglected. Within this approximation, the equivalent series resistance ESR is equal to the series resistance. The quality factor Q of the measured resonance is Q=(2πf0C1R1) -1 where f0 is the (series) resonant frequency.

[0137] 25A-25C show electrical characteristics of a MEMS resonator according to an embodiment at room temperature and low ambient pressure below 10 mbar. FIG. 25A shows the measured electrical impedance of the MEMS resonator assembly S1 of FIG. 23A, which is an interconnected piezoelectric driven LE resonator element assembly with a resonant frequency close to 24 MHz. The top graph of FIG. 25A shows the impedance magnitude and the bottom graph shows the phase. The 0 dB level on the vertical scale of the top graphs of FIGS. 25A-25C represents an impedance of 1 Ω. Fitting the data to an equivalent circuit model yields the following performance parameters: parallel capacitance C0=2.6 pF, equivalent series capacitance C1=8.2 fF, ESR=39.8 Ω, and quality factor Q=20240.

[0138] FIG. 25B shows the measured electrical impedance of the MEMS resonator assembly S4 shown in FIG. 23D. This assembly has two interconnected piezoelectrically driven LE resonator element assemblies and a resonant frequency close to 40 MHz. The top graph in FIG. 25B shows the impedance magnitude and the bottom graph shows the phase. Fitting this data to an equivalent circuit model yields the following performance parameters: parallel capacitance C0=2.2 pF, equivalent series capacitance C1=5.1 fF, ESR=41.7 Ω, and quality factor Q=18870. The Q factor of the MEMS resonator assembly S4 with two interconnected LE resonator element assemblies is comparable to the Q factors of the MEMS resonator assemblies S1, S2, S3, S6 with only a single interconnected LE resonator element assembly. This indicates that flexure resonators according to embodiments of the present invention can be used to mechanically couple extension mode resonators without causing significant degradation in resonator performance.

[0139] FIG. 25C shows the measured electrical impedance of the MEMS resonator assembly S5 shown in FIG. 23E. This assembly has three interconnected piezoelectrically driven LE resonator element assemblies and a resonant frequency close to 40 MHz. The top graph in FIG. 25C shows the impedance magnitude, and the bottom graph shows the phase. Fitting this data to an equivalent circuit model yields the following performance parameters: parallel capacitance C0=3.2 pF, equivalent series capacitance C1=7.2 fF, ESR=28.4 Ω, and quality factor Q=19570. The Q factor of the MEMS resonator assembly S5 with three interconnected LE resonator element assemblies is comparable to the Q factors of the MEMS resonator assemblies S1, S2, S3, and S6 with only a single interconnected LE resonator element assembly.

[0140] The ESR*A*f0 values ​​of the exemplary MEMS resonator assemblies S1-S6 are in units of Ωmm 2 The ESR*A*f0 of these embodiments is expressed in MHz and shown as a filled circle in Figure 26. 2 MHz to 60Ωmm 2 It is observed that the frequency is in the range of MHz.

[0141] The data points at a frequency of 40 MHz for the exemplary MEMS resonator assemblies S3-S5 have the same ESR*A within ±10%. This is due to the fact that the Q value of the extension mode resonators is not degraded when multiple resonators are coupled by mechanical connection elements with flexure resonators. It is also due to the fact that the area of ​​the mechanical connection elements with flexure resonators is relatively small, and the suspension and mechanical fixation of the assemblies S3-S5 are substantially the same, leading to similar fixation losses. Thus, the ESR of the MEMS resonator assemblies can be reduced by increasing the area of ​​the resonator elements by coupling multiple extension mode resonators using mechanical connection elements with flexure resonators. In the embodiment including the exemplary MEMS resonator assemblies S4 and S5, multiple extension mode resonators are spaced apart from each other along their primary vibration direction (x-direction in Figs. 23D-23E), and there are mechanical connection elements between adjacent extension mode resonators. In alternative embodiments, including those shown in Figures 15A-15B, 16A-16B, 17 and 19, the extension mode resonators are spaced apart from each other in a direction perpendicular to their primary vibration directions.

[0142] In some embodiments, the ESR of a MEMS resonator assembly can be reduced by increasing the area of ​​the resonator elements by coupling multiple extension mode resonators using substantially stiff mechanical connection elements (interconnection elements). For example, referring to the exemplary resonators S1-S6 of Figures 23A-23F, the ESR can be further reduced by increasing the width of the interconnected LE resonator element assembly by adding more LE resonator elements (shown at 671 in Figure 9A) and their interconnection elements (shown at 672 in Figure 9A).

[0143] The area between the two dashed lines in Figure 26, the upper limit g U =83 (unit: Ωmm 2 MHz) and lower limit g L =12 (unit: Ωmm 2 MHz) represents the value ESR*A*f0 of the MEMS resonator assembly according to an embodiment of the present invention.

[0144] The inventors of the present application have determined that the upper limit g U = 83 (unit: Ωmm 2 Several embodiments (of the MEMS resonator assemblies) have been implemented that have ESR*A*f0 values ​​between 1000 MHz and the data points of the exemplary resonators S1-S6. These embodiments include, for example, resonator assemblies in which the connection points between the suspension element and the resonator element are in non-nodal locations. The non-nodal locations of the connection points tend to decrease the Q factor and therefore increase the ESR*A*f0. In other embodiments, the resonator assemblies were operated at ambient pressures where the quality factor decreased to the range of 11000 to 14000 due to increased gas damping, increasing the ESR*A*f0 values. According to the above equation and the electrical performance parameters associated with FIGS. 25A-25C (summarized in the table above), the figure of merit FOM=1 / (2πf0C0R1) of the exemplary MEMS resonator assemblies S1-S6 would clearly be greater than 10, even at ambient pressures sufficient for robust operation of the oscillator circuit driving the MEMS resonator assemblies. Hermetically packaging under an inert gas atmosphere instead of a low pressure environment makes packaging of the MEMS resonators easier and reduces manufacturing costs.

[0145] Lower limit of ESR*A*f0 g L can be determined by further analyzing the various energy loss mechanisms of the MEMS resonator. The Q factor is given by the following formula: 1 / Q = 1 / Q anch + 1 / Q TED + 1 / Q gas It is determined by, where Q anch represents the energy loss due to the anchor (anchoring loss, i.e., energy leaking from the resonator element through the suspension element to the support structure), and Q TED represents thermoelastic dissipation (i.e., dissipation due to heat flow between hot and cold regions within the resonator element), and Q gasrepresents the energy leakage from the resonator element to the surrounding gas environment. Experiments carried out by the inventors in various controlled low pressure environments show that for MEMS resonator assemblies sealed at pressures below 10 mbar, the energy leakage to the surrounding gas environment is negligible. This condition can be routinely achieved using state-of-the-art chip-scale packaging (CSP) or low-pressure wafer-level packaging (WLP) technologies.

[0146] The contributions of anchoring losses and thermoelastic dissipation are 1 / Q TED is proportional to absolute temperature, whereas anchoring loss is independent of temperature, and can be determined by examining the temperature dependence of the quality factor. Figure 27 shows experimental measurements of 1 / Q for a group of 20 resonant elements for an embodiment such as the exemplary sample S1. The solid line in Figure 27 is a best fit to the experimental data in the form of 1 / Q=a+bT. The value of the fitting coefficient is a=4.2x10 -5 , b=3.2x10 -8 Therefore, Q TED =1 / (bT)=3.13x10 7 / T (temperature is expressed in degrees Kelvin (K)). At temperature T = 300K, the heat dissipation is Q TED = 104000. Therefore, the quality factor Q (corresponding to these fitting parameters) = (1 / Q anch +1 / Q TED ) -1 = 19370 is clearly limited by anchoring losses. The anchoring losses can be reduced by optimizing the mechanical connections of the suspension elements to the resonator elements and support structures, for example by using a central support structure as described with reference to FIG. 17 or by using a (more) symmetrical material layer stack as described with reference to FIG. 9B. (Thereby Q anch For a MEMS resonator in the 10 MHz frequency range, 5 Q much higher than 1x10 6It is possible to achieve a Q value of over 1000 kHz. Energy leakage through the suspension elements is negligible compared to thermoelastic dissipation. The improvement in Q value is given by R1 = (2πQf0C1) -1 According to the formula, ESR=R1+R e The ohmic loss of the resonator (R e ) is negligible compared to R1. Therefore, the ESR*A*f0 value of the MEMS resonator assembly in the embodiment of the present invention is anch = 200000 and Q TED = 104000, which corresponds to a quality factor Q of 68000 (lower limit) L =12 (unit: Ωmm 2 It can be concluded that in certain embodiments where the suspension element and its mechanical connection to the resonator element and support structure are not fully optimized, the ESR*A*f0 of the MEMS resonator assembly can be reduced by a factor of Q anch = 100000 and Q TED = 104000, which corresponds to a quality factor Q = 51000 2 In other particular embodiments in which the suspension element and its mechanical connection to the resonator element and support structure are not further optimized, the ESR*A*f0 of the MEMS resonator assembly is equal to or greater than Q anch = 50000 and Q TED = 104000, which corresponds to a quality factor Q = 34000 2 It is equal to or greater than 100 MHz.

[0147] An oscillator is prone to phase noise if the frequency pull range (or frequency tunability) of the resonator is too large. The maximum frequency pull range is proportional to the ratio of the equivalent series capacitance to the parallel capacitance (C1 / C0). Certain embodiments of the present invention exhibit both low ESR and small frequency pull range. In certain embodiments, the ratio C1 / C0 is less than 0.005, preferably less than 0.004, such as in the exemplary MEMS resonator assemblies S1-S6.

[0148] Certain embodiments of the present invention exhibit both low ESR and small temperature variation of resonant frequency. To illustrate this point, FIG. 28 shows the measured resonant frequency of an exemplary MEMS resonator assembly S1 (see FIG. 23A) of a 24 MHz length-extension mode resonator as the temperature is varied from −30° C. to 85° C. The vertical axis of FIG. 28 shows the difference in frequency relative to the frequency of the resonator at a temperature of 30° C. The frequency difference is in parts per million (ppm) relative to the frequency at a temperature of 30° C. The data in FIG. 28 shows that the variation in resonant frequency ranges from −7.5 ppm to +2.5 ppm in the temperature range of −30° C. to 85° C. As can be seen from the data, there are two turnaround points (points where the first derivative of the resonant frequency with respect to temperature is zero) in the resonant frequency vs. temperature curve in the temperature range of −30° C. to 85° C., the first turnaround point is near −15° C. and the second turnaround point is near 55° C. By having two turn points in the temperature range of 30° C. to 85° C., with a lower turn point in the temperature range of −30° C. to 0° C. and an upper turn point in the temperature range of 40° C. to 85° C., it is possible to reduce the overall variation in resonant frequency in the temperature range of −30° C. to 85° C. The value of each turn point temperature can be designed by adjusting the doping level of impurities (such as phosphorus) in the single crystal silicon layer in the resonator element, or by adjusting the relative thicknesses of the single crystal silicon layer, the piezoelectric layer, and the silicon oxide layer (if the embodiment has these material layers).

[0149] In some embodiments, the resonant frequency varies within ±30 ppm from the resonant frequency at 25° C. in the temperature range of −30° C. to 85° C. In more advantageous embodiments, the resonant frequency varies within ±15 ppm from the resonant frequency at 25° C. in the temperature range of −30° C. to 85° C. In some embodiments, there are two turn-around points in the resonant frequency vs. temperature curve in the temperature range of −30° C. to 85° C. In some embodiments, the average phosphorus dopant concentration in the single crystal silicon in the resonator element is 2×10 19 cm -3In certain embodiments, such as the exemplary MEMS resonator assemblies S1-S6, the average phosphorus dopant concentration of the single crystal silicon in the resonator elements is greater than 2×10 19 cm -3 and the ratio of the thickness of the piezoelectric layer to the thickness of the single crystal silicon layer, or the ratio of the thickness of the piezoelectric layer to the sum of the thicknesses of the two single crystal silicon layers in the resonator element, is greater than 0.07. In a particular embodiment, the variation of the resonant frequency with temperature is reduced in a particular temperature range, such as a temperature range of -30°C to 85°C, by adjusting the impurity (such as phosphorus) doping level of the single crystal silicon layer(s) in the resonator element, as well as by either: Adjusting the relative thickness of the single crystal silicon layer (or two single crystal silicon layers) and the piezoelectric layer within the resonator element; Adjusting the relative thicknesses of the single crystal silicon layer (or two single crystal silicon layers), the piezoelectric layer and the silicon oxide layer (or two silicon oxide layers) in the resonator element.

[0150] The present invention is applicable to MEMS resonators over a wide frequency range. In some embodiments, the resonant frequency f0 is in the range of 7 MHz to 160 MHz, for example in the range of 15 MHz to 110 MHz.

[0151] Without limiting the scope and interpretation of the claimed invention, one or more technical advantages of the exemplary embodiments disclosed herein are listed below: A technical advantage is low ESR of the resonator. Another technical advantage is design freedom for the layout of the resonator elements on the die. Further technical advantages include high power handling capability and low sensitivity to thermo-mechanical stresses.

[0152] The above description provides a complete and informative description of the best mode currently contemplated by the inventors for carrying out the present invention, by way of non-limiting examples of specific implementations and embodiments of the present invention. However, as will be apparent to those skilled in the art, the details of the above-described embodiments do not limit the present invention, and may be implemented in other embodiments using equivalent means without departing from the characteristics of the present invention.

[0153] Moreover, features of the embodiments of the present invention disclosed above may be used without the corresponding use of other features. Accordingly, the foregoing description should be considered as merely illustrative of the principles of the present invention, and not in limitation thereof. The scope of the present invention is therefore limited only by the appended claims.

Claims

1. 1. A MEMS resonator assembly comprising: a support structure; a resonator element suspended on the support structure; an actuator for exciting the resonator element into a resonant mode; Equipped with The resonator element has a resonant frequency f ０ and comprising at least one bulk acoustic resonator; If the equivalent series resistance of the MEMS resonator assembly is ESR and the area of ​​the resonator element is A, then ESR*A*f ０ is 12Ωmm 2 MHz to 83Ωmm 2 in the MHz range, MEMS resonator assembly.

2. The MEMS resonator assembly of claim 1 , wherein the bulk acoustic resonator is a stretch length resonator, an interconnected stretch length resonator assembly, a stretch width resonator, a square stretch resonator, or a Lame resonator.

3. ESR*A*f ０ is 16Ωmm 2 MHz or above or 25Ωmm 2 MHz or greater.

4. The area A of the resonator element is 0.001 mm 2 from 1 mm 2 2. The MEMS resonator assembly of claim 1, wherein the MEMS resonator assembly has a capacitance in the range of 0.1 to 0.5 .mu.m.

5. The MEMS resonator assembly of claim 1 , wherein the ratio of equivalent series capacitance to parallel capacitance is less than 0.

005.

6. The MEMS resonator assembly of claim 1 , wherein the figure of merit is greater than 10.

7. The MEMS resonator assembly of claim 1 , wherein the bulk acoustic resonator is an in-plane resonator.

8. The MEMS resonator assembly of claim 1 , wherein the actuator is a piezoelectric actuator or an electrostatic actuator.

9. The MEMS resonator assembly of claim 1 , wherein at least 50% of the mass of the resonator element is composed of single crystal silicon.

10. a main direction of vibration of the bulk acoustic resonator is substantially parallel to a <100> crystallographic direction of a single crystal silicon layer within the resonator element, and the single crystal silicon layer has an average phosphorus dopant concentration of 2×10 19 cm -3 The MEMS resonator assembly of claim 1 , which is larger.

11. The resonator elements are made of piezoelectric AlN, Sc-doped AlN, ZnO, LiNbO 3 , LiTaO 3 The MEMS resonator assembly of claim 1 , comprising any one of the layers:

12. 2. The MEMS resonator assembly of claim 1, wherein the resonator element has a piezoelectric layer thickness in the range of 0.5 μm to 4 μm, such as in the range of 1 μm to 2 μm.

13. The average phosphorus dopant concentration of the single crystal silicon layer in the resonator element is 2×10 19 cm -3 13. The MEMS resonator assembly of claim 12, wherein a ratio of the thickness of the piezoelectric layer to the thickness of the single crystal silicon layer or a ratio of the thickness of the piezoelectric layer to the sum of the thicknesses of two single crystal silicon layers in the resonator element is greater than 0.

07.

14. The resonance frequency f ０ The MEMS resonator assembly of claim 1 , wherein the frequency is in the range of 7 MHz to 160 MHz, for example in the range of 15 MHz to 110 MHz.

15. The resonance frequency f ０ The MEMS resonator assembly of claim 1 , wherein the curve of temperature has two turning points in the temperature range of −30° C. to 85° C.

16. The resonant frequency f in the temperature range from -30°C to 85°C ０ The MEMS resonator assembly of claim 1 , wherein the variation of the resonant frequency is within ±30 ppm, for example, within ±15 ppm, at a temperature of 25° C.

17. The MEMS resonator assembly of claim 1 , wherein the number of silicon oxide layers in the resonator element is 0 or 1.

18. The MEMS resonator assembly of claim 1 , wherein a cavity separates the resonator element from the support structure.

19. The resonator element includes two bulk acoustic resonators and a material portion that mechanically connects the two bulk acoustic resonators, and the two bulk acoustic resonators resonate at the resonant frequency f in phase or 180 degrees out of phase with each other. ０ The MEMS resonator assembly of claim 1 , wherein the MEMS resonator assembly vibrates at a frequency of 1000 kHz.

20. 20. The MEMS resonator assembly of claim 19, wherein the distance between the two bulk acoustic resonators is 50 [mu]m or less, such as 20 [mu]m or less.

21. 20. The MEMS resonator assembly of claim 19, wherein the portion of material mechanically connecting the two bulk acoustic resonators is a substantially rigid interconnection element.

22. 20. The MEMS resonator assembly of claim 19, wherein the material portion mechanically connecting a first bulk acoustic resonator and a second bulk acoustic resonator included in the two bulk acoustic resonators is a flexural mode resonator.

23. 20. The MEMS resonator assembly of claim 19, wherein a first bulk acoustic resonator and a second bulk acoustic resonator of the two bulk acoustic resonators are fixed to the same support structure, the support structure being formed in the plane of the resonator element along a line midway between the first bulk acoustic resonator and the second bulk acoustic resonator.

24. The resonance frequency f ０ 20. The MEMS resonator assembly of claim 19, wherein is substantially equal to a fundamental frequency or an Nth overtone frequency of the first bulk acoustic resonator and a fundamental frequency or an Nth overtone frequency of the second bulk acoustic resonator.

25. 23. The MEMS resonator assembly of claim 22, wherein the first and second bulk acoustic resonators resonate in a direction of vibration, and the bending mode resonator is a beam-type resonator having its longest dimension perpendicular to the direction of vibration.

26. The N-th overtone frequency of the bending mode resonator is the resonant frequency f ０ 23. The MEMS resonator assembly of claim 22, wherein:

27. The MEMS resonator assembly of claim 22, wherein the bending mode resonator is an in-plane bending beam resonator.

28. A MEMS resonator assembly as described in claim 22, wherein the bending mode resonator is mechanically connected to the first and second bulk acoustic resonators at antinodes of the first and second bulk acoustic resonators.

29. A MEMS resonator assembly as described in claim 22, wherein the bending mode resonator is mechanically connected to the first bulk acoustic resonator at one or more connection points on a first side of the bending mode resonator and mechanically connected to the second bulk acoustic resonator at one or more connection points on a second side opposite the first side of the bending mode resonator.

30. A MEMS resonator assembly as described in claim 22, wherein the bending mode resonator is mechanically connected to the first bulk acoustic resonator at one or more connection points on one side of the bending mode resonator and mechanically connected to the second bulk acoustic resonator at one or more connection points on the same side as the one side of the bending mode resonator.

31. 23. The MEMS resonator assembly of claim 22, The resonant mode shape of the bending mode resonator has two types of antinodes, a first type of antinode has a positive displacement along the vibration axis, at least one of which is a connection point for mechanical connection to a first one of the two bulk acoustic resonators; a second type of antinode has a negative displacement along the vibration axis, at least one of which is a connection point for mechanical connection to a second one of the two bulk acoustic resonators; A MEMS resonator assembly.

32. 23. The MEMS resonator assembly of claim 22, wherein the first bulk acoustic resonator resonates in a first vibration direction and the second bulk acoustic resonator resonates in a second vibration direction that is orthogonal to the first vibration direction.

33. 23. The MEMS resonator assembly of claim 22, wherein the bending-mode resonator has a portion in the form of a rectangular frame.

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