Tuning the frequency / temperature response of MEMS resonators using complex eigenmodes

Abstract

A MEMS resonator device is provided. The MEMS resonator device includes a support structure, a composite eigenmode resonator element, an anchor, a drive electrode, and a sense electrode. The composite eigenmode resonator element includes at least two subregions. Each of the at least two single subregions resonates with one eigenmode from a set of at least two eigenmodes and has a temperature coefficient of frequency (TCFN). k ). The respective first and second subregions resonate with first and second eigenmodes from a set of at least two eigenmodes and have specific dopant types, specific doping concentrations, and first and second orientations relative to the crystal axes. An anchor couples the composite eigenmode resonator element to a support structure. A drive electrode drives the composite eigenmode resonator element. A sense electrode senses the composite eigenmode resonator element.

Description

[Technical Field]

[0001] The present disclosure relates to the field of microelectromechanical systems (MEMS) resonators. [Background technology]

[0002] The emergence of the Internet of Things (IoT) has spawned a myriad of sensor-based devices used for remote sensing in wearables, smartphones, industrial and consumer applications. Timing references are ubiquitous in these devices, helping to provide signals used to keep track of time, synchronize events within digital integrated circuits (ICs), and process signals. High-precision microelectromechanical systems (MEMS) resonators can be well-suited for such high-performance electronic applications. Summary of the Invention

[0003] It is an object of the present disclosure to provide improved temperature compensated MEMS devices, such as resonators.

[0004] According to the present disclosure, there is provided a MEMS resonator device, comprising a composite eigenmode resonator element including a support structure and at least two subregions, each of the at least two single subregions configured to resonate with a respective eigenmode from a set of at least two eigenmodes, and each having a respective Nth-order temperature coefficient of frequency (TCFN). k ), wherein the at least two subregions include a first subregion and a second subregion, the single first subregion being configured to resonate with a first eigenmode from a set of at least two eigenmodes, and the single first subregion having a particular dopant type, a particular doping concentration, a first orientation relative to the crystal axes, and a first TCFN. k the single second subregion being configured to resonate with a second eigenmode from a set of at least two eigenmodes, the single second subregion having the particular dopant type, the particular doping concentration, a second orientation relative to the crystal axes, and a second TCFNk (i) the first TCFN; k is the desired Nth-order temperature coefficient of frequency (TCFN) of the composite eigenmode resonator element. desired ) and the second TCFN k is TCFN desired or (ii) the first TCFN is less than k is TCFN desired Less than the second TCFN k is TCFN desired or greater than 100 .mu.m, at least one anchor coupling the composite eigenmode resonator element to a support structure, at least one drive electrode for actuating the composite eigenmode resonator element, and at least one sense electrode for sensing the composite eigenmode resonator element.

[0005] According to the present disclosure, a method for designing a MEMS resonator device is provided, which is based on a desired Nth-order temperature coefficient of frequency (TCFN) of a composite eigenmode resonator element. desired ), wherein the composite eigenmode resonator element includes at least two subregions, each of the at least two subregions configured to resonate with a respective eigenmode from a set of at least two eigenmodes, and a respective temperature coefficient of Nth frequency (TCFN k providing a first subregion of the at least two subregions, the first subregion being configured to resonate with a first eigenmode from a set of at least two eigenmodes, the first subregion having a particular dopant type, a particular doping concentration, and a first orientation relative to the crystal axes, the first subregion being configured to have a first TCFN k providing a second subregion of the at least two subregions, the second subregion being configured to resonate with a second eigenmode from the set of at least two eigenmodes, the second subregion having the particular dopant type, the particular doping concentration, and a second orientation relative to the crystal axes, the second subregion comprising a second TCFN; kand providing, for the composite eigenmode resonator element including the at least two subregions, (TCFN total ) and determine the Nth order temperature coefficient of frequency, TCFN desired TCFN so that it is approximately equal to total Coordinating the TCFN total adjusting the first TCFN k TCFN desired the second TCFN k TCFN desired or (ii) said first TCFN k TCFN desired the second TCFN k TCFN desired the first or second TCFN k This includes changing or adjusting at least one of the above.

[0006] The features, aspects, and advantages of the presently disclosed technology may be better understood by considering the following description, the appended claims, and the accompanying drawings listed below. Those skilled in the relevant art will understand that the features shown in the drawings are for illustrative purposes and that variations, including different and / or additional features and combinations thereof, are possible. [Brief explanation of the drawings]

[0007] [Figure 1A] 1 shows an example cross section of a capacitively transduced composite eigenmode resonator device. [Figure 1B] 1 shows an example cross section of a piezoelectrically transduced composite eigenmode resonator device. [Figure 2] 1 shows the frequency shift as a function of temperature for a composite eigenmode resonator element. [Figure 3A] <100> We show the predictable response of TCF1total at different doping concentrations with different dopant types for the wine-glass / length stretching composite eigenmode of silicon lattice orientation. [Figure 3B]<110> We show the predictable response of TCF1total at different doping concentrations with different dopant types for the wine-glass / length stretching composite eigenmode of silicon lattice orientation. [Figure 3C] <100> We show the predictable response of TCF2total at different doping concentrations with different dopant types for the wine-glass / length stretching composite eigenmode of silicon lattice orientation. [Figure 3D] <110> We show the predictable response of TCF2total at different doping concentrations with different dopant types for the wine-glass / length stretching composite eigenmode of silicon lattice orientation. [Figure 4] 1 shows several examples of composite eigenmode resonator elements. [Figure 4A] Continued from Figure 4. [Figure 5] A method for tuning the temperature-induced frequency response of a composite eigenmode resonator element containing two subregions is shown. [Figure 6A] We present examples of the results of effective mass tuning of the combined plane shear / square stretch eigenmode by adding or removing subregions that resonate with specific eigenmodes. [Figure 6B] We present examples of the results of effective mass tuning of the combined plane shear / square stretch eigenmode by adding or removing subregions that resonate with specific eigenmodes. [Figure 6C] We present examples of the results of effective mass tuning the square stretch / Larmé composite eigenmode by adding or removing subregions that resonate with specific eigenmodes. [Figure 6D] We show examples of the results of effective mass tuning the square-stretch / length-stretch composite eigenmode by adding or removing subregions that resonate with specific eigenmodes. [Figure 6E] We present examples of the results of effective mass tuning of the Lame / Nth-order annular composite eigenmode by adding or removing subregions that resonate with specific eigenmodes. [Figure 6F]We present an example of the results of effective mass tuning of combined plane shear / length stretch eigenmodes by adding or removing subregions that resonate with specific eigenmodes. [Figure 6G] We present an example of the results of effective mass tuning the wineglass / length stretching composite eigenmode by adding or removing subregions that resonate with specific eigenmodes. [Figure 6H] We present examples of the results of effective mass tuning of combined breathing / plane shear eigenmodes by adding or removing subregions that resonate with specific eigenmodes. [Figure 7A] We present examples of the results of effective mass tuning of the Lame / square stretch composite eigenmode by both adding or removing subregions that resonate with a particular eigenmode and by changing the size of the subregion that resonates with that particular eigenmode, without changing the resonant frequency of that particular eigenmode. [Figure 7B] We show an example of the results of effective mass tuning of a wine-glass / length-stretching composite eigenmode by both adding or removing subregions that resonate with a specific eigenmode and by changing the size of the subregion that resonates with that specific eigenmode, without changing the resonant frequency of that specific eigenmode. [Figure 8A] This shows an example of the results of tuning the resonant frequency of TCFNtotal for the square expansion / Lame composite eigenmode. [Figure 8B] This shows an example of the result of adjusting the resonant frequency of TCFNtotal for the square / length extension combined eigenmode. [Figure 8C] This shows an example of the results of tuning the resonant frequency of TCFNtotal for the breathing / plane shear combined eigenmode. [Figure 9A] 10 shows an example of the results of tuning TCF1total by adding cavities to one or more subregions of the composite eigenmode resonator element. [Figure 9B] We present an example of the results of tuning TCF2total by adding cavities to one or more subregions of the composite eigenmode resonator element. DETAILED DESCRIPTION OF THE INVENTION

[0008] I. Overview

[0009] Quartz crystal oscillators have been the foundation of timing and frequency reference applications for the past century, but the rapid development of sensor-based electronics has highlighted certain limitations of this technology, such as power consumption, robustness, size, and CMOS compatibility. Over the past two decades, silicon MEMS resonators have attracted significant attention due to their small size, low cost, and integration compatibility. However, MEMS resonators have yet to replace quartz crystal resonators in many applications.

[0010] A limitation of MEMS resonators that has prevented their widespread adoption is their lack of temperature stability compared to quartz. Silicon MEMS resonators have an inherent temperature-induced first-order frequency drift of approximately -30 ppm / °C, resulting in a temperature stability of approximately 3,750 ppm over the industrial temperature range of -40°C to 85°C. In comparison, AT-cut quartz resonators have a temperature stability of approximately 20 ppm over the industrial temperature range. Several attempts have been made to overcome this temperature-induced frequency drift, including the use of heavily doped silicon substrates and composite materials. Temperature-induced frequency drift compensation using heavily doped silicon substrates can only provide first- or second-order compensation for temperature-induced frequency drift at specific doping concentrations, and may be difficult to source from foundries, making them impractical for certain applications. Furthermore, temperature-induced frequency drift compensation using composite materials can face challenges such as aging, complex processing, and reduced quality factor (Q).

[0011] The change in frequency of a MEMS resonator with respect to temperature is given by:

number

[0012] Although active temperature compensation techniques can be used to minimize temperature-induced frequency drift, such techniques can impose significant burdens on the system in terms of power consumption, circuit size, and circuit complexity. Therefore, it would be beneficial to find a way to passively correct for temperature-induced frequency drift in fabricated silicon MEMS resonators.

[0013] Various techniques have been used to passively compensate for temperature-induced frequency drift. For example, fabricating MEMS resonators with secondary materials such as silicon dioxide (SiO2) is an established technique. SiO2 is known to have a large positive TCF1 (approximately 85 ppm / °C), and due to its compatibility with silicon, it can be incorporated into silicon MEMS resonators to offset the negative TCF1 of silicon. Within the industrial temperature range, temperature-induced frequency drift has been reduced to less than 100 ppm overall using this method. For smaller devices (e.g., thin-film bulk acoustic resonators) operating at higher frequencies within the same temperature range, temperature-induced frequency drift as low as 3 ppm has been demonstrated using this technique. However, the TCF2 effect in oxides is poorly documented and difficult to measure due to its small scale. Furthermore, losses in the resonator coefficient Q associated with thermoelastic damping and / or surface losses must be considered. Furthermore, this technique increases manufacturing complexity and presents reliability issues due to film stress, leading to long-term aging problems.

[0014] Another example of an existing passive compensation technique is to fabricate MEMS resonators using heavily doped silicon substrates. This is an established approach because the TCF1 of a particular eigenmode varies with doping concentration. For example, within the industrial temperature range of -40°C to 85°C, using a heavily doped silicon substrate can reduce the temperature-induced frequency drift of a MEMS resonator operating in the Lamé mode to within the range of 200–400 ppm. However, even if the TCF1 is zero, there can still be significant temperature-induced frequency drift due to higher-order frequency temperature coefficients.

[0015] II. Overview of passive compensation of temperature-induced frequency drift using composite eigenmode resonators

[0016] The composite eigenmode resonator element 104 is a mechanically vibrating resonator element that exhibits at least two distinct eigenmodes at resonance. These eigenmodes occur in distinct regions of the composite eigenmode resonator element 104, further referred to as subregions. A "subregion" therefore refers to an adjacent region that resonates with a particular eigenmode and may be connected to other subregions via contact areas that are significantly smaller than the subregions. The composite eigenmode resonator element 104 may be useful in designing MEMS resonators because it has a predictable and tunable temperature-induced frequency drift response. The Nth-order temperature coefficient of frequency (TCFN) of the composite eigenmode resonator element 104 is total ) is the TCFN (TCFN) of the eigenmodes represented by each subregion alone. k ) can be calculated using a weighted average of the eigenmodes. Here, "eigenmodes exhibited by each subregion alone" refers to the eigenmodes exhibited by each subregion when it is not connected to other subregions via contact regions. Therefore, a specific TCFN can be calculated using a combination of eigenmodes. total It is possible to design composite eigenmode resonator elements 104 with any value. Typically, each single subregion is selected from a set of standard shapes including, but not limited to, (i) a square plate, (ii) a disk, (iii) a bar, or (iv) a toroid. The TCFN of any composite eigenmode resonator element 104 total can be calculated using the following formula:

number

[0017] Effective mass is a quantitative measure of the inertia of a given area relative to resonance and can be calculated using the formula:

number

[0018] 1A illustrates an example cross-section of a capacitively transduced composite eigenmode resonator device 100. The composite eigenmode resonator device includes a support structure 102, a composite eigenmode resonator element 104, at least one anchor 106, at least one drive electrode 108, and at least one sense electrode 110. The composite eigenmode resonator element 104 includes at least two subregions. Each of the at least two independent subregions is configured to resonate with a respective eigenmode from a set of at least two eigenmodes, and a respective TCFN k The at least two subregions include a first subregion 104A and a second subregion 104B. The single first subregion 104 is configured to resonate with a first eigenmode from the set of at least two eigenmodes, and has a particular dopant type, a particular doping concentration, a first orientation relative to the crystal axes, and a first TCFN. kThe second subregion 104B alone is configured to resonate with a second eigenmode from the set of at least two eigenmodes, and has a particular dopant type, a particular doping concentration, a second orientation relative to the crystal axes, and a second TCFN. k (i) The first TCFN k Desired Nth-order temperature coefficient of frequency (TCFN) of the composite eigenmode resonator element 104 desired ) and the second TCFN k <TCFN desired or (ii) the first TCFN. k <TCFN desired And the second TCFN k >TCFN desired At least one anchor 106 couples the composite eigenmode resonator element 104 to the support structure 102. At least one drive electrode 108 drives the composite eigenmode resonator element 104. At least one sense electrode 110 senses the composite eigenmode resonator element 104.

[0019] 1B shows an example cross section of a piezoelectrically transduced composite eigenmode resonator device 100. The addition of piezoelectric electrodes, such as at least one drive electrode 108 and at least one sense electrode 110, allows for the creation of a TCFN. total It is important to note that this has minimal impact on

[0020] In some embodiments, the composite eigenmode resonator device 100 is either (i) a capacitive transduced type or (ii) a piezoelectric transduced type.

[0021] In some embodiments, each of the at least two sub-regions is connected to at least one other sub-region by a contact region, the contact region extending from the point of maximum displacement amplitude.

[0022] In some additional embodiments, each of the at least two subregions is connected to at least one other subregion by a contact region, and the contact region is minimized to minimize modal distortion due to non-ideal coupling of the at least two subregions.

[0023] In some additional embodiments, the contact region further comprises a bar, the volume of the bar being less than 5% of the volume of the composite eigenmode resonator element.

[0024] In some embodiments, the at least two subregions further comprise a third subregion, wherein the sole third subregion resonates with a first eigenmode from the set of at least two eigenmodes and has a particular dopant type, a particular doping concentration, a first orientation relative to the crystal axes, and a first TCFN. k It has.

[0025] In some additional embodiments, the at least two subregions further comprise a fourth subregion and a fifth subregion, wherein the fourth subregion and the fifth subregion are both resonant with a first eigenmode from the set of at least two eigenmodes and are resonant with a particular dopant type, a particular doping concentration, a first orientation relative to the crystal axes, and a first TCFN. k It has.

[0026] In some additional embodiments, the at least two subregions further comprise a sixth subregion, a seventh subregion, an eighth subregion, and a ninth subregion, wherein the sixth subregion, the seventh subregion, the eighth subregion, and the ninth subregion are resonant with a first eigenmode from the set of at least two eigenmodes and are resonant with a particular dopant type, a particular doping concentration, a first orientation relative to the crystal axes, and a first TCFN. k It has.

[0027] In some additional embodiments, the at least two subregions further include a set of 4N+1 subregions, where N is any integer greater than 2. Each subregion of the 4N subregions in a single set of 4N+1 subregions resonates with a first eigenmode from the set of at least two eigenmodes and has a particular dopant type, a particular doping concentration, a first orientation relative to the crystal axes, and a first TCFN. kThe sole remaining subregion (i.e., the +1 subregion in the set of 4N+1 subregions) resonates with a second eigenmode from the set of at least two eigenmodes and has a particular dopant type, a particular doping concentration, a second orientation relative to the crystal axes, and a second TCFN. k The remaining sub-region may be disposed at the center of the set of 4N+1 sub-regions, and the 4N sub-regions within the 4N+1 sub-regions may extend symmetrically from a maximum displacement point of the centrally disposed remaining sub-region.

[0028] In some embodiments, the respective individual resonant frequencies of the eigenmodes from the set of at least two eigenmodes are substantially equal. Throughout this specification and disclosure, "the respective individual resonant frequencies are substantially equal" means that it is sufficient to reduce the difference between the individual resonant frequencies as close to zero as practically possible. In some embodiments, this may include reducing the difference to less than 1 GHz. In other embodiments, this may include reducing the difference to less than 1 MHz. In still other embodiments, this may include reducing the difference to less than 1 kHz. In still other embodiments, this may include reducing the difference between the individual resonant frequencies to be less than 1% of the resonant frequency of the composite eigenmode resonator element 104.

[0029] In some embodiments, the set of at least two eigenmodes includes two or more eigenmodes selected from the group consisting of: (i) Lame eigenmodes, (ii) face-shear eigenmodes, (iii) square-extensional eigenmodes, (iv) width-extensional eigenmodes, (v) length-extensional eigenmodes, (vi) Nth order annulus eigenmodes, (vii) breathing eigenmodes, (viii) wineglass eigenmodes, and (ix) higher-order Lame eigenmodes.

[0030] In some embodiments, the TCFN of at least two distinct subregions k By combining these, TCFN total TCFN desired is approximately equal to

[0031] In some embodiments, the Nth order includes the first order (i.e., TCFN=TCF1), and TCFN desired is approximately equal to 0 ppm / °C. Here, "approximately equal to 0 ppm / °C" does not necessarily mean reducing TCF1 exactly to zero. Reducing TCF1 as close to zero as practically possible is sufficient. In some embodiments, this may involve reducing TCF1 to less than 1 ppm / °C. In other embodiments, this may involve reducing TCF1 to less than 0.1 ppm / °C. In still other embodiments, this may involve reducing TCF1 to less than 0.01 ppm / °C. In some embodiments, the Nth order includes the second order (i.e., TCFN=TCF2), and TCFN desired is 0 ppb / ℃ 2 Here, "0 ppb / ℃ 2 "Approximately equal to" does not necessarily mean reducing TCF2 exactly to zero. It is sufficient to reduce TCF2 as close to zero as practically possible. In some embodiments, this means reducing TCF2 to 1 ppb / °C. 2 In another embodiment, the TCF2 may be reduced to less than 0.1 ppb / °C. 2 In yet another embodiment, the TCF2 can be reduced to less than 0.01 ppb / °C. 2 This may include reducing it to less than

[0032] In some embodiments, the Nth order includes the first order (i.e., TCFN=TCF1), and TCFN desired is equal to a non-zero value that at least partially compensates for the third order frequency temperature coefficient of the composite eigenmode resonator element.

[0033] In some embodiments, each of the at least two sub-regions has a particular dopant type and a particular doping concentration.

[0034] In some embodiments, the shape of the composite eigenmode resonator element further comprises one or more cavities in one or more of the at least two subregions.

[0035] In some embodiments, the difference between the effective mass of the composite eigenmode resonator element 104 and the sum of the effective masses of each of the at least two single subregions is less than 10%.

[0036] In some embodiments, the composite eigenmode resonator device 100 is configured to operate as either (i) an oscillator or (ii) a resonating sensor.

[0037] In some embodiments, the composite eigenmode resonator device 100 includes at least one of single crystal silicon, silicon carbide, polycrystalline silicon, quartz, graphene, and polycrystalline diamond.

[0038] III. Example of a passively compensated multiple eigenmode resonator

[0039] FIG. 2 includes a plot 200 illustrating the frequency shift as a function of temperature for a composite eigenmode resonator element 202, which may be similar to or the same as the composite eigenmode resonator element 104 described above in connection with FIGS. 1A and 1B. Plot 200 compares the composite eigenmode resonator element 202, which includes a first subregion 202A resonating with a plane shear mode and a second subregion 202B resonating with a length stretching mode, with the constituent single eigenmodes (plane shear mode 204 and length stretching mode 206). Plot 200 shows that the frequency shift of the single plane shear mode 204 increases significantly with increasing temperature, while the frequency shift of the single length stretching mode 206 decreases significantly with increasing temperature. The slope of the frequency shift as a function of temperature for the composite eigenmode resonator element 202 is significantly smaller than that of either of the constituent single eigenmodes. Thus, the temperature-induced frequency drift of the multiple eigenmode resonator element 202 is significantly reduced compared to a resonator element having only one of the constituent single eigenmodes.

[0040] The TCFN, described by Equation 2 and shown in plot 200, total It is important to note that the predictable response of may hold regardless of the dopant type, doping concentration, or overall orientation with respect to the crystal axes of the composite eigenmode resonator element 104. Increasing the doping concentration or using a different dopant type may result in a TCFN of the kth single subregion. k There is a possibility that TCFN total can still be predicted using Equation 2. Furthermore, changing the overall orientation of the composite eigenmode resonator element 104 with respect to the crystal axes can affect the TCFN of the kth single subregion. k There is a possibility that TCFN total can still be predicted using Equation 2. Furthermore, TCF1 total (or TCFN total) also exists in undoped silicon, but most eigenmodes have a TCF1 of ~30 ppm / °C, so they do not significantly contribute to temperature-induced frequency drift. It is also important to note that the linearity of Equation 2 has limitations, which are discussed in more detail in the next section.

[0041] Figure 3A shows <100> Wine-glass / length-stretching composite eigenmodes of silicon lattice orientations in TCF1 with different dopant types and doping concentrations total shows a predictable response of

[0042] Figure 3B shows <110> Wine-glass / length-stretching composite eigenmodes of silicon lattice orientations in TCF1 with different dopant types and doping concentrations total shows a predictable response of

[0043] Figure 3C shows <100> Wine-glass / length-stretching composite eigenmodes of silicon lattice orientations in TCF2 with different dopant types and doping concentrations total shows a predictable response of

[0044] Figure 3D shows <110> Wine-glass / length-stretching composite eigenmodes of silicon lattice orientations in TCF2 with different dopant types and doping concentrations total shows a predictable response of

[0045] FIG. 4 illustrates several examples of composite eigenmode resonator elements 104. Image 402 illustrates a square-stretch / plane-shear composite eigenmode resonator element with two subregions. Image 404 illustrates a square-stretch / plane-shear composite eigenmode resonator element with three subregions. Image 406 illustrates a square-stretch / plane-shear composite eigenmode resonator element with five subregions. Image 408 illustrates a square-stretch / plane-shear composite eigenmode resonator element with nine subregions. Image 410 illustrates a square-stretch / Larmé composite eigenmode resonator element with two subregions. Image 412A illustrates a square-stretch / Larmé composite eigenmode resonator element with three subregions, two of which are resonating with Lamé eigenmodes. Image 412B illustrates a square-stretch / Larmé composite eigenmode resonator element with three subregions, two of which are resonating with 1x2 higher-order Lamé eigenmodes. Image 412C shows a square stretch / Larmé composite eigenmode resonator element with three subregions, with two subregions resonating with a 1x3 higher-order Lamé eigenmode. Image 414 shows a square stretch / Larmé composite eigenmode resonator element with five subregions. Image 416 shows a square stretch / Larmé composite eigenmode resonator element with two subregions. Image 418 shows a square stretch / Larmé composite eigenmode resonator element with three subregions. Image 420 shows a square stretch / Larmé composite eigenmode resonator element with five subregions. Image 422 shows a fourth-order annular / Larmé composite eigenmode resonator element with two subregions. Image 424 shows a fourth-order annular / Larmé composite eigenmode resonator element with three subregions. Image 426 shows a fourth-order annular / Larmé composite eigenmode resonator element with five subregions. Image 428 shows a plane-shear / length-stretching composite eigenmode resonator element with two subregions. Image 430 shows a plane-shear / length-stretching composite eigenmode resonator element with three subregions. Image 432 shows a plane-shear / length-stretching composite eigenmode resonator element with five subregions. Image 434 shows a wine-glass / length-stretching composite eigenmode resonator element with two subregions. Image 436 shows a wine-glass / length-stretching composite eigenmode resonator element with three subregions.Image 438 shows a wine-glass / length-stretching composite eigenmode resonator element with five subregions. Image 440 shows a plane-shear / breathing-mode composite eigenmode resonator element with two subregions. Image 442 shows a plane-shear / breathing-mode composite eigenmode resonator element with three subregions. Image 444 shows a plane-shear / breathing-mode composite eigenmode resonator element with five subregions. Image 446 shows a width-stretching / Larmé composite eigenmode resonator element with five subregions, with the central subregion resonating with a third-order Lamé eigenmode. Image 448 shows a width-stretching / Larmé composite eigenmode resonator element with nine subregions, with the central subregion resonating with a fifth-order Lamé eigenmode. Those skilled in the art will recognize that the subregions of the width-stretching / Larmé composite eigenmode resonator elements shown in images 446 and 448 include connecting regions defined by bars.

[0046] IV. Passive Compensation Methods for Complex Eigenmode Resonators

[0047] To design the composite eigenmode resonator element 104 to achieve an accurate temperature-induced frequency response, at least two subregions are required, with each of the at least two single subregions resonating with an eigenmode from a set of at least two eigenmodes, resulting in a TCFN. k Although different exemplary embodiments may utilize a variety of different numbers of subregions resonating with different eigenmodes, for purposes of clarity, many of the examples described herein demonstrate how to passively compensate a composite eigenmode resonator element having five or fewer subregions.

[0048] First, the composite eigenmode resonator element 104 is configured to have TCFNs above and below the desired values. k (e.g., TCF1 desired = 0 ppm / °C or TCF2 desired = 0 ppb / °C) 2 ) to obtain the desired Nth temperature coefficient of frequency (TCFN desired ) is approximately equal to TCFN totalHowever, there are only a limited number of common eigenmodes to choose from (e.g., Lame eigenmodes, plane shear eigenmodes, square stretching eigenmodes, width stretching eigenmodes, length stretching eigenmodes, Nth order annular eigenmodes, breathing eigenmodes, wine-glass eigenmodes, or higher-order Lame eigenmodes), and the TCFN k It is not always possible to select two eigenmodes with the same absolute value but opposite signs. Therefore, simply combining two eigenmodes usually results in a TCFN. total =TCFN desired In most cases, further adjustments will be necessary.

[0049] 5 shows a flowchart 500 of a method for tuning an Nth-order temperature-induced frequency response of a composite eigenmode resonator device 100. In step 502, the method includes selecting a TCFN for a composite eigenmode resonator element 104. In line with the above discussion, the composite eigenmode resonator element 104 is composed of at least two subregions, each of which resonates with an eigenmode from a set of at least two eigenmodes, and the TCFN k It has.

[0050] In step 504, the method includes providing a first subregion of the composite eigenmode resonator element 104. The single first subregion resonates with a first eigenmode from the set of at least two eigenmodes and has a particular dopant type, a particular doping concentration, a first orientation relative to the crystal axes, and a first TCFN. k Then, in step 506, the method includes providing a second subregion of the composite eigenmode resonator element 104. The single second subregion resonates with a second eigenmode from the set of at least two eigenmodes and has a particular dopant type, a particular doping concentration, a second orientation relative to the crystal axes, and a second TCFN. k It has.

[0051] In step 508, the method includes: determining a TCFN of a composite eigenmode resonator element 104 that includes the at least two subregions;total which can be achieved using Equation 2 above (or using a computer program, if desired).

[0052] In step 510, the method total TCFN desired TCFN so that it is approximately equal to total This includes coordinating the TCFN. total Adjusting the first TCFN k TCFN desired The second TCFN is larger than k TCFN desired or (ii) the first TCFN k TCFN desired smaller than the second TCFN k TCFN desired The first or second TCFN k This includes changing at least one of the above.

[0053] In some embodiments, TCFN total TCFN desired TCFN so that it is approximately equal to total Adjusting the first or second TCFN k This may be done by one or a combination of (i) making effective mass adjustments, (ii) making resonant frequency adjustments, or (iii) adding cavities to one or more of the at least two subregions, so that at least one of the subregions is changed. Furthermore, when done in combination, the effective mass adjustments, resonant frequency adjustments, and adding cavities to one or more of the at least two subregions may be applied in any order. The method illustrated in flowchart 500 may be performed using a parametric sweep approach, using currently known and later developed methods, to iteratively design the composite eigenmode resonator element 104.

[0054] As used herein, effective mass tuning refers to modifying the effective mass fraction of a particular eigenmode from a set of at least two eigenmodes on the composite eigenmode resonator element 104 without changing the resonant frequency of the particular eigenmode. The effective mass tuning can be performed by one or a combination of (i) adding or removing a subregion that resonates with the particular eigenmode, or (ii) changing the size of a subregion that resonates with the particular eigenmode, without changing the resonant frequency of the particular eigenmode.

[0055] Adding or removing subregions necessarily changes the number of n subregions in Equation 2. Thus, adding or removing subregions changes the weighted average TCFN. total TCFN desired can be adjusted to be approximately equal to

[0056] Modifying the size of a subregion resonating with a particular eigenmode without changing its resonant frequency can be achieved by utilizing a particular type of eigenmode. For example, changing the width of a subregion resonating with a length-stretching eigenmode or changing the height of a subregion resonating with an in-plane eigenmode (e.g., Lamé, plane shear, etc.) modifies the size (and therefore effective mass) of the subregion. Some eigenmodes can also be distributed along specific axes, such as Lamé eigenmodes and length-stretching eigenmodes in the x and y planes. Doubling the dimensions of a subregion exhibiting a distributable mode causes that subregion to exhibit a corresponding second-order eigenmode without changing its resonant frequency.

[0057] In some cases, TCFN total TCFN desired Effective mass tuning can be advantageous over other methods when tuning the effective mass to approximately equal the effective mass. For example, it is possible to perform effective mass tuning while maintaining the maximum displacement amplitude across a set of at least two eigenmodes. This allows the fabricated MEMS resonator device to operate more practically in the real world, as the displacement amplitude improves the signal-to-noise ratio.

[0058] As used herein, tuning a resonant frequency means scaling the dimensions of a particular subregion resonating with a particular eigenmode so that only the resonant frequency of the particular eigenmode changes, but not the mode shape (e.g., scaling a square subregion that represents a square extension mode, but not making it rectangular). Changing the resonant frequency of a particular eigenmode creates a difference in resonant frequency between the modified eigenmode and the other eigenmodes. This weakens the mechanical coupling between the modified eigenmode and the other eigenmodes in the overall composite eigenmode. The weaker coupling allows one of the eigenmodes to dominate over the others in terms of displacement amplitude at the new resonant frequency. This effectively increases the effective mass of the dominant eigenmode relative to the other eigenmodes, reducing the TCFN of the resonator. total TCFN with dominant eigenmode k shift towards

[0059] TCFN total TCFN desired Tuning the composite eigenmode to be approximately equal to c can also be achieved by adding cavities to specific subregions that are resonant with specific eigenmodes. Adding cavities to one or more of the subregions slightly "distorts" the composite eigenmode. As used herein, "distort" means introducing stray vibrations, causing a deviation from the composite eigenmode present in an equivalent resonator element without a cavity. Adding cavities to one or more of the at least two subregions modulates the elastic constants (c) of silicon for the composite eigenmode. 11 , c 12 , and c 44 ) contributions, the composite eigenmode is distorted, which leads to the TCFN of the composite eigenmode. total shifts.

[0060] Figure 6A shows an example of the results of effective mass tuning by adding or removing subregions that resonate with a particular eigenmode. In this example, the TCF1 of the combined plane shear / square stretch eigenmode totalIn this example, the resonator element is <100> Oriented in the silicon lattice direction, 2.00x10 19 atoms / cm 3 The indicated value of the square expansion (SE) effective mass (SE Effective Mass) is determined according to the following formula: SE Effective Mass = meffSE / meff total The resonator element 602 has a single subregion that resonates with a single plane shear eigenmode, such that the element 602 does not contain a square stretch eigenmode and the SE effective mass=0%. When the SE effective mass=0%, the resulting single plane shear eigenmode is TCF1. total =-28.21 ppm / °C. As shown by resonator elements 604, 606, 608, and 610, one or more square elastic sub-regions may be used to reduce the TCF1 of the composite eigenmode resonator element 104. total the TCF1 of the single square stretching eigenmode (represented by resonator element 612) total As the number of SE subregions increases, the TCF1 total increases and approaches the case of SE effective mass = 100% (i.e., resonator element 612), so that the single square stretching eigenmode is total =-8.42 ppm / ℃.

[0061] Figure 6B shows the TCF1 of the plane shear / square stretch combined eigenmode shown in Figure 6A. total 6 includes a plot 614 that plots an example of the results of effective mass tuning of the plane shear / square stretch composite eigenmode resonator element. In plot 614, the plane shear / square stretch composite eigenmode resonator element includes a single plane shear mode subregion located at the maximum displacement point of the square stretch subregion. In this example, the composite eigenmode resonator element is <100> Oriented in the silicon lattice direction, 2.00x10 19 atoms / cm 3 The effective mass of the electrons is meff. SE / meff totalAs shown in plot 614, the SE effective mass is varied by adjusting the number of square stretchable sub-regions. As further shown in FIG. 6B, plot 616 shows the TCF2 resulting from varying the SE effective mass of the composite eigenmode resonator element by adjusting the number of square stretchable sub-regions in the same or similar manner as related to plot 614. total This shows the impact on

[0062] Figure 6C shows the square-stretched / Larmé composite eigenmode of TCF1. total In plot 618, the square stretch / Larmé composite eigenmode resonator element includes a single square stretch mode subregion positioned at the maximum displacement point of the Lamé subregion. In this example, the composite eigenmode resonator element is <110> Oriented in the silicon lattice direction, 2.00x10 19 atoms / cm 3 The effective mass of the electrons is meff. SE / meff total As shown in plot 618, the SE effective mass is varied by adjusting the number of Lamé subregions. As further shown in FIG. 6C, plot 620 illustrates the TCF2 resulting from varying the SE effective mass of the composite eigenmode resonator element by adjusting the number of Lamé subregions in the same or similar manner as related to plot 618. total This shows the impact on

[0063] Figure 6D shows the square-stretch / length-stretch combined eigenmode of TCF1. total In plot 622, the square stretch / length stretch composite eigenmode resonator element includes a single square stretch mode subregion positioned at the maximum displacement point of the length stretch subregion. In this example, the composite eigenmode resonator element <100> Oriented in the silicon lattice direction, 6.00x10 19 atoms / cm 3 The effective mass of the electrons is meff. SE / meff totalAs shown in plot 622, the SE effective mass of the composite resonator element is varied by adjusting the number of length-stretching mode subregions. As further shown in FIG. 6D, plot 624 illustrates the TCF2 resulting from varying the SE effective mass of the composite eigenmode resonator element by adjusting the number of length-stretching subregions in the same or similar manner as related to plot 622. total This shows the impact on

[0064] Figure 6E shows the Lame / Nth-order annular composite eigenmode of TCF1. total 6 includes a plot 626 illustrating an example of effective mass tuning of the Lame / Nth-order annular composite eigenmode resonator element. In plot 626, the Lame / Nth-order annular composite eigenmode resonator element includes a single Lame mode subregion located at the maximum displacement point of the Nth-order annular. The Lame subregion is <100> Oriented in the silicon lattice direction, 7.5x10 19 atoms / cm 3 It is doped with N to a concentration of . Lamé effective mass = meff Lame / meff total As shown in plot 626, the Lamé effective mass of the composite resonator element is varied by adjusting the number of Nth-order annular mode subregions. As further shown in FIG. 6E, plot 628 illustrates the TCF2 resulting from varying the Lamé effective mass of the composite eigenmode resonator element by adjusting the number of Nth-order annular subregions in the same or similar manner as related to plot 626. total This shows the impact on

[0065] Figure 6F shows the combined plane shear / length stretch eigenmode TCF1 total 6 includes a plot 630 illustrating an example of effective mass tuning of the length stretching / plane shear subregion. In plot 630, the plane shear / length stretching composite eigenmode resonator element includes a single plane shear subregion positioned at the maximum displacement point of the length stretching subregion. In this example, the composite eigenmode resonator element is <100> Oriented in the silicon lattice direction, 2.4x10 20 atoms / cm 3 The surface shear (FS) effective mass is meff. SE / meff totalAs shown in plot 630, the FS effective mass of the composite resonator element is varied by adjusting the number of length-stretching mode subregions. As further shown in FIG. 6F, plot 632 illustrates the TCF2 resulting from varying the FS effective mass of the composite eigenmode resonator element by adjusting the number of length-stretching subregions in the same or similar manner as related to plot 630. total This shows the impact on

[0066] Figure 6G shows the wine-glass / length-stretching combined eigenmode of TCF1. total In plot 634, the wine-glass / length-stretching composite eigenmode resonator element includes a single wine-glass subregion positioned at the maximum displacement point of the length-stretching subregion. In this example, the composite eigenmode resonator element <110> Oriented in the silicon lattice direction, 2.4 × 10 20 atoms / cm 3 The effective mass of the wine glass is meff. wineglass / meff total As shown in plot 634, the wine-glass effective mass of the composite resonator element is varied by adjusting the number of length-stretching mode subregions. As further shown in FIG. 6G, plot 636 illustrates the TCF2 resulting from varying the wine-glass effective mass of the composite eigenmode resonator element by adjusting the number of length-stretching subregions in the same or similar manner as related to plot 634. total This shows the impact on

[0067] Figure 6H shows TCF1 total 6 includes a plot 638 illustrating an example of effective mass tuning of a combined breathing / plane shear eigenmode. In plot 638, the combined breathing / plane shear eigenmode resonator element includes a single breathing mode subregion located at the maximum displacement point of the plane shear subregion. In this example, the combined eigenmode resonator element is <110> Oriented in the silicon lattice direction, 2.4 × 10 20 atoms / cm 3 The effective mass of the breathing mode is meff. breathing / meff total As shown in plot 638, the breathing-mode effective mass of the composite resonator element is varied by adjusting the number of plane shear subregions. As further shown in FIG. 6H, plot 640 illustrates the TCF2 resulting from varying the breathing-effective mass of the composite eigenmode resonator element by adjusting the number of plane shear subregions in the same or similar manner as related to plot 638. total This shows the impact on

[0068] Figure 7A shows an example of the results of effective mass tuning by modifying the size of the subregion resonating with a particular eigenmode without changing the resonant frequency of that particular eigenmode. In this example, the TCF1 of the Lame / square stretching composite eigenmode total and TCF2 total In this example, the composite eigenmode resonator element is <110> Oriented in the silicon lattice direction, 2.00x10 19 atoms / cm 3 Again, the indicated value of the square-expansion (SE) effective mass is determined according to the following formula: SE effective mass = meff SE / meff total Here, element 708 shows a square stretch / Larmé composite eigenmode resonator element having three subregions, the first subregion resonating with the square stretch eigenmode and the two subregions extending from opposite corners of the first subregion resonating with the Lamé eigenmode. Element 708 is shown in Fig. 1 for a simulated TCF1. total =-27.04 ppm / ℃, simulated TCF2 total =-28.99ppb / ℃ 2 Moving on to element 706, we can see that the sides of the two subregions that resonate with the Lamé eigenmodes have doubled in length. Thus, the two subregions exhibit higher-order Lamé eigenmodes that resonate at the same resonant frequency, but with the addition of effective mass, the TCFN of element 706 totaltowards the pure Lame eigenmode (element 702). Similarly, element 704 further increases the side length of the two subregions resonating with the Lame eigenmode, shifting the temperature coefficient of frequency from the simulated TCF1 total =-23.17 ppm / ℃ and simulated TCF2 total =-39.13ppb / ℃ 2 This is further shifting the focus to

[0069] Figure 7B shows an example of the results of effective mass tuning the wine-glass / length-stretch composite eigenmode by both (i) adding or removing subregions that resonate with the particular eigenmode and (ii) changing the size of the subregion that resonates with the particular eigenmode, without changing the resonant frequency of the particular eigenmode. In this example, the composite eigenmode resonator element is <110> Oriented in the silicon lattice direction, 2.4 × 10 20 atoms / cm 3 The effective mass of the wine glass is given by the formula: wineglass / meff total The resonator element 702 has a single subregion that resonates at a single wineglass eigenmode, with the wineglass effective mass=100%. When the wineglass effective mass=100%, the resulting single wineglass eigenmode is TCF1 total =1.18 ppm / °C. As shown by resonator elements 704, 706, 708, and 710, one or more length-stretching (LE) subregions can be used to reduce the TCF1 of a single length-stretching eigenmode (represented by resonator element 712). total Towards TCF1 total To press the LE subregion, it may be placed at the maximum displacement point of the wine glass subregion. As the number of LE subregions increases, the TCF1 total increases until the effective mass of the wine glass becomes 0%, and as a result, the single length-stretching eigenmode is TCF1 total =-8.21 ppm / °C. Furthermore, comparing element 704 and element 706, it can be seen that the TCFN can be improved by modifying the size of the sub-region that resonates with the length-shortening eigenmode without changing the resonant frequency.total It can be seen that the element 704 is TCF1 total =-1.11 ppm / °C, but by increasing the width of the subregion that resonates with the length-stretching eigenmode, element 706 total =-2.59 ppm / ℃.

[0070] 8A to 8C show TCF1 total and TCF2 total Figure 8A shows the effect of tuning the resonant frequency of TCF1. total 8. In plot 802, the square stretched / Lame composite eigenmode resonator element includes a single SE mode subregion and a single Lame mode subregion stretched from a corner of the SE mode subregion, each of which is <110> Oriented in the silicon lattice direction, 1.7 × 10 20 atoms / cm 3 Here again, the SE effective mass = meff SE / meff total As shown in plot 802, the SE effective mass of the composite resonator element is proportional to the side length l of the Lame mode subregion. Lame The specific values ​​are shown in Figure 8A.

[0071] Figure 8B shows TCF1 total 8 includes a plot 804 illustrating an example of resonant frequency tuning by scaling the dimensions of a square-stretched / length-stretched (SE / LE) composite eigenmode to tune . In plot 804, the SE / LE composite eigenmode resonator element includes a single SE mode subregion and a single LE mode subregion extending from a corner of the SE mode subregion, each of which has a <100> Oriented in the silicon lattice direction, 6.0 × 10 19 atoms / cm 3 Here again, the SE effective mass = meff SE / meff total As shown in plot 804, the SE effective mass of the composite resonator element is proportional to the side length l of the SE mode subregion.SE 8B. As further shown in FIG. 8B, plot 806 varies with the side length l of the SE mode subregion in a manner similar or analogous to that related to plot 804. SE The TCF2 resulting from varying the SE effective mass of the composite resonator element by adjusting total This shows the impact on

[0072] Figure 8C shows TCF1 total 8 includes a plot 808 illustrating an example of resonant frequency tuning by scaling the dimensions of the combined breathing / plane shear eigenmode to tune . In plot 808, the combined breathing / plane shear eigenmode resonator element includes a single breathing mode subregion and a single plane shear mode subregion located at the maximum displacement point of the breathing mode subregion, each of which has <110> Oriented in the silicon lattice direction, 2.4 × 10 20 atoms / cm 3 The effective mass of the breathing mode is meff. breathing / meff total As shown in plot 808, the breathing mode effective mass of the composite resonator element is the radius r of the breathing mode subregion. breathing 8C, plot 810 is plotted against the radius r of the breathing mode subregion in the same or similar manner as related to plot 808. breathing The TCF2 resulting from varying the breathing mode effective mass of the composite resonator element by adjusting total This shows the impact on

[0073] Figure 9A shows the TCF1 total In the example shown, the composite eigenmode is a square-stretched (SE) / Lame composite eigenmode. Plot 902 shows the effect of adjusting TCF1 total9B shows an example of the effect of adding a cavity to the SE / Lame composite eigenmode to tune the effective mass. Image 904 shows the SE / Lame composite eigenmode without a cavity (corresponding to the dotted line in plot 902). Image 906 shows the SE / Lame composite eigenmode with a cavity having a side length of 20 μm (corresponding to the dashed line in plot 902). Image 908 shows the SE / Lame composite eigenmode with a cavity having a side length of 25 μm (corresponding to the solid line in plot 902). As shown in plot 902, adding a cavity to the SE / Lame composite eigenmode can increase the sensitivity of the effective mass tuning, as evidenced by the increasing slope of the line in plot 902 as the cavity size increases. While the example shown in FIG. 9B uses four square-shaped cavities arranged in a grid around the center of the resonator element, other examples may use different numbers, sizes, shapes, and arrangements of cavities, similarly affecting the TCF1. total 9B includes a composite SE / Lame eigenmode having a single SE mode and a single Lame mode, it should be understood that the cavity may similarly be implemented in connection with various other composite eigenmode configurations, such as any of the composite eigenmode configurations described herein.

[0074] Figure 9B shows the TCF2 structure by adding cavities to one or more of the at least two subregions. total 9A. That is, plot 910 shows the TCF2 resulting from adding a cavity to the combined SE / Lame eigenmode in a manner similar or analogous to that described above in connection with FIG. 9A. total 1 shows an exemplary effect on

[0075] In some embodiments, the exemplary methods described herein further include providing a third subregion of the at least two subregions, the sole third subregion resonating with the first eigenmode and having a particular dopant type, a particular doping concentration, a first orientation relative to the crystal axes, and a first TCFN. k It has.

[0076] In some further embodiments, the example methods described herein further include providing a fourth subregion and a fifth subregion of the at least two subregions, wherein the fourth subregion and the fifth subregion together resonate with the first eigenmode and have a particular dopant type, a particular doping concentration, a first orientation relative to the crystal axes, and a first TCFN. k It has.

[0077] In some further embodiments, the exemplary methods described herein further include providing a sixth subregion, a seventh subregion, an eighth subregion, and a ninth subregion of the at least two subregions, wherein the sixth subregion, the seventh subregion, the eighth subregion, and the ninth subregion are resonant with a first eigenmode and have a particular dopant type, a particular doping concentration, a first orientation relative to the crystal axes, and a first TCFN. k It has.

[0078] In some embodiments, the respective single resonant frequencies of each eigenmode from the set of at least two eigenmodes are substantially equal.

[0079] In some embodiments, the set of at least two eigenmodes includes at least two eigenmodes selected from the group consisting of: (i) Lame eigenmodes, (ii) plane shear eigenmodes, (iii) square stretch eigenmodes, (iv) width stretch eigenmodes, (v) length stretch eigenmodes, (vi) Nth order annular eigenmodes, (vii) breathing eigenmodes, (viii) wine-glass eigenmodes, and (ix) higher order Lame eigenmodes.

[0080] In some embodiments, the TCFN of at least two distinct subregions k By combining these, TCFN total TCFN desired is approximately equal to

[0081] In some embodiments, the Nth order includes the first order, and the TCFN desiredis approximately equal to 0 ppm / ℃.

[0082] In some embodiments, the Nth order includes a second order, and the TCFN desired is 0 ppb / ℃ 2 is approximately equal to

[0083] In some embodiments, the Nth order includes the first order, and the TCFN desired is a non-zero value that at least partially compensates for the third-order frequency temperature coefficient of the composite eigenmode resonator element.

[0084] In some embodiments, each of the at least two sub-regions has a particular dopant type and a particular doping concentration.

[0085] V. Nonlinear Effects Observed in Complex Eigenmode Resonators

[0086] It is important to note that there are limitations to the linearity of the theoretical predictions made by Equation 2. These limitations arise from the complex eigenmodes present in the complex eigenmode resonator element 104 and the TCFN in Equation 2. k This arises due to modal distortion with the eigenmodes exhibited by each of the n single subregions used. Modal distortion can be quantified by looking at the effective mass of the entire composite eigenmode resonator element 104 compared to the sum of the effective masses of the n single subregion eigenmodes. In the ideal case, when the two are equal, all eigenmodes of the device's n subregions are fully excited, resulting in a TCFN. total The values ​​are in perfect agreement with the result of Equation 2. However, in the presence of modal distortion, the effective mass of the actual composite eigenmode resonator element 104 is lower than the sum of its parts (n sub-regions).

[0087] In the fabricated composite eigenmode resonator device 100, there will always be modal distortions due in part to (i) non-ideal contact regions between one or more subregions, and (ii) weakened coupling between at least two subregions. The non-ideal contact regions cause modal distortions of the composite eigenmode compared to the eigenmodes exhibited by each subregion alone. As these modal distortions increase, the composite eigenmode exhibited by the composite eigenmode resonator element 104 moves away from the combination of the single eigenmodes and toward the TCFN. total This results in a single composite eigenmode that is difficult to predict analytically. A large contact area between two subregions results in large modal distortion. Therefore, it may be desirable to minimize the contact area between the subregions. In some embodiments, the contact area may be defined by the intersection line between the two subregions. Schematic diagram 912 shows an example of the intersection line 914 at the contact area of ​​a square-stretched (SE) / Larmé composite eigenmode. In some embodiments, minimizing the contact area between the two subregions may include making the intersection line less than 10 μm. In other embodiments, this may include making the intersection line at the contact area of ​​the two subregions less than 5 μm. In yet other embodiments, this may include making the intersection line at the contact area of ​​the two subregions less than 1 μm. In some embodiments, the contact area may be defined by a bar between the two subregions. In some embodiments, minimizing the contact area between the two subregions may include making the volume of the bar between the two subregions less than 5% of the total volume of the composite eigenmode resonator element. In other embodiments, minimizing the contact area between two subregions may include making the volume of a bar between the two subregions less than 1% of the total volume of the composite eigenmode resonator element. Utilizing a bar between two subregions may be advantageous over other methods when there is no convenient maximum displacement point to create a sufficiently small contact area that minimizes modal distortion due to non-ideal coupling. Image 446 shows an example of this scenario, where a sufficiently small contact area between the width stretch eigenmode and the third-order Lame eigenmode is not practically feasible.

[0088] Furthermore, weak coupling between the eigenmodes of a subregion induces modal distortion. Tuning the resonant frequency by scaling the dimensions of one or more subregions increases the difference in resonant frequencies between the single eigenmodes, thereby weakening the coupling. As the difference in resonant frequencies between the single eigenmodes increases, one eigenmode becomes more dominant over the others. As shown in Equation 3, the displacement amplitude of a subregion is directly linked to its effective mass. Therefore, as the percentage of the total effective mass of the composite eigenmode resonator element 104 represented by the dominant eigenmode increases, modal distortion becomes more pronounced. In extreme cases, if the resonant frequencies between the single eigenmodes are far enough apart, the eigenmodes may become completely decoupled, and the intended composite eigenmode may no longer be exhibited by the resonator element. In this extreme case, one or more subregions may act as anchors at the maximum displacement point of the dominant eigenmode. This can cause significant distortion from the pure eigenmode, resulting in TCFN. total causes unpredictable changes.

[0089] 8A to 8C show that the increase in modal distortion is total and simulated TCFN total Figure 1 shows the effect on the match of the frequency temperature coefficients. When the dimensions of one or more subregions are scaled such that the resonant frequencies of one or more subregions are no longer equal, the modal distortion increases, and as a result, the difference between the simulated and analytically calculated frequency temperature coefficients increases.

Claims

1. a support structure; A composite eigenmode resonator element comprising at least two subregions, each of the at least two single subregions configured to resonate with a respective eigenmode from a set of at least two eigenmodes, and each having a respective Nth-order temperature coefficient of frequency (TCFN). ｋ ), wherein the at least two subregions include a first subregion and a second subregion, the single first subregion being configured to resonate with a first eigenmode from a set of at least two eigenmodes, and the single first subregion being configured to resonate with a specific dopant type, a specific doping concentration, a first orientation relative to a crystal axis, and a first TCFN. ｋ the single second subregion being configured to resonate with a second eigenmode from a set of at least two eigenmodes, the single second subregion having the specific dopant type, the specific doping concentration, a second orientation relative to the crystal axes, and a second TCFN. ｋ (i) the first TCFN; ｋ is the desired Nth-order temperature coefficient of frequency (TCFN) of the composite eigenmode resonator element. ｄｅｓｉｒｅｄ ) and said second TCFN ｋ is TCFN ｄｅｓｉｒｅｄ or (ii) said first TCFN is less than ｋ is TCFN ｄｅｓｉｒｅｄ and the second TCFN is less than ｋ is TCFN ｄｅｓｉｒｅｄ a composite eigenmode resonator element, at least one anchor coupling the composite eigenmode resonator element to the support structure; at least one drive electrode for driving the composite eigenmode resonator element; at least one sensing electrode for sensing the composite eigenmode resonator element; 1. A MEMS resonator device comprising:

2. The device of claim 1 , wherein the device is either (i) a capacitive transduction type or (ii) a piezoelectric transduction type.

3. The device of claim 1 , wherein each of the at least two sub-regions is connected to at least one other sub-region by a contact region, the contact region extending from a point of maximum displacement amplitude.

4. 4. The device of claim 3, wherein each of the at least two subregions is connected to at least one other subregion by a contact area, the contact area being minimized to minimize modal distortion due to non-ideal coupling of the at least two subregions.

5. The device of claim 4 , wherein the contact region further comprises a bar, the volume of the bar being less than 5% of the volume of the composite eigenmode resonator element.

6. The at least two subregions further include a third subregion, the sole third subregion resonating with the first eigenmode and having the specific dopant type, the specific doping concentration, the first orientation relative to the crystal axes, and the first TCFN. ｋ The device of claim 1 , comprising:

7. The at least two subregions further include a fourth subregion and a fifth subregion, wherein the fourth subregion and the fifth subregion are both resonant with the first eigenmode and have the specific dopant type, the specific doping concentration, the first orientation with respect to the crystal axes, and the first TCFN. ｋ 7. The device of claim 6, wherein:

8. The at least two subregions further comprise a sixth subregion, a seventh subregion, an eighth subregion, and a ninth subregion, and the sixth subregion, the seventh subregion, the eighth subregion, and the ninth subregion are resonant with the first eigenmode and have the specific dopant type, the specific doping concentration, the first orientation with respect to the crystal axes, and the first TCFN. ｋ 8. The device of claim 7, comprising:

9. The device of claim 1 , wherein the respective single resonant frequencies of the respective eigenmodes from the set of at least two eigenmodes are approximately equal.

10. 2. The device of claim 1, wherein the set of at least two eigenmodes comprises two or more eigenmodes selected from the group consisting of: (i) Lame eigenmodes, (ii) face-shear eigenmodes, (iii) square-extensional eigenmodes, (iv) width-extensional eigenmodes, (v) length-extensional eigenmodes, (vi) Nth order annular eigenmodes, (vii) breathing eigenmodes, (viii) wineglass eigenmodes, and (ix) higher-order Lame eigenmodes.

11. The TCFNs of the at least two independent sub-regions ｋ and the N-th order temperature coefficient of frequency (TCFN ｔｏｔａｌ ) to TCFN ｄｅｓｉｒｅｄ The device of claim 1 , wherein the

12. Nth order includes 1st order, TCFN ｄｅｓｉｒｅｄ The device of claim 1 , wherein is approximately equal to 0 ppm / ° C.

13. Nth order includes second order, TCFN ｄｅｓｉｒｅｄ is 0 ppb / ℃ ２ The device of claim 1 , wherein the

14. Nth order includes 1st order, TCFN ｄｅｓｉｒｅｄ 10. The device of claim 1, wherein t is a non-zero value that at least partially compensates for a third-order frequency temperature coefficient of the composite eigenmode resonator element.

15. The device of claim 1 , wherein each of the at least two subregions has the particular dopant type and the particular doping concentration.

16. The device of claim 1 , wherein the composite eigenmode resonator element further comprises one or more cavities in one or more of the at least two subregions.

17. The device of claim 1 , wherein the effective mass of the composite eigenmode resonator element differs from the sum of the effective masses of each of the at least two independent subregions by less than 10%.

18. The device of claim 1 , wherein the device is configured to operate as either (i) an oscillator or (ii) a resonating sensor.

19. 10. The device of claim 1, wherein the device comprises at least one of monocrystalline silicon, silicon carbide, polycrystalline silicon, quartz, graphene, and polycrystalline diamond.

20. 1. A method for designing a MEMS resonator device, comprising: The desired Nth-order temperature coefficient of frequency (TCFN) of the composite eigenmode resonator element ｄｅｓｉｒｅｄ ), wherein the composite eigenmode resonator element includes at least two subregions, each of the at least two subregions configured to resonate with a respective eigenmode from a set of at least two eigenmodes, and a respective Nth-order temperature coefficient of frequency (TCFN ｋ ), providing a first subregion of the at least two subregions, the first subregion being configured to resonate with a first eigenmode from a set of at least two eigenmodes, the first subregion having a particular dopant type, a particular doping concentration, and a first orientation relative to a crystal axis, the first subregion comprising a first TCFN; ｋ having, providing, providing a second subregion of the at least two subregions, the sole second subregion configured to resonate with a second eigenmode from the set of at least two eigenmodes, the sole second subregion having the particular dopant type, the particular doping concentration, and a second orientation relative to the crystal axes, the second subregion comprising a second TCFN; ｋ having, providing, For the composite eigenmode resonator element including the at least two subregions, (TCFN ｔｏｔａｌ ) determining the Nth order frequency temperature coefficient of TCFN ｄｅｓｉｒｅｄ TCFN so that it is approximately equal to ｔｏｔａｌ and coordinating the TCFN ｔｏｔａｌ adjusting the first TCFN ｋ TCFN ｄｅｓｉｒｅｄ and the second TCFN ｋ TCFN ｄｅｓｉｒｅｄ or (ii) said first TCFN ｋ TCFN ｄｅｓｉｒｅｄ and the second TCFN ｋ TCFN ｄｅｓｉｒｅｄ the first or second TCFN so as to be greater than ｋ adjusting, including changing at least one of A method comprising:

21. TCFN ｄｅｓｉｒｅｄ TCFN so that it is approximately equal to ｔｏｔａｌ adjusting the first or second TCFN ｋ 21. The method of claim 20, comprising one or more of: (i) performing effective mass tuning; (ii) performing resonant frequency tuning; or (iii) adding cavities to one or more of the at least two sub-regions, such that at least one of

22. Further comprising providing a third subregion of the at least two subregions, the third subregion resonating with the first eigenmode and having the specific dopant type, the specific doping concentration, the first orientation relative to the crystal axes, and the first TCFN. ｋ 21. The method of claim 20, comprising:

23. and providing a fourth subregion and a fifth subregion of the at least two subregions, the fourth subregion and the fifth subregion both resonating with the first eigenmode and having the specific dopant type, the specific doping concentration, the first orientation relative to the crystal axes, and the first TCFN. ｋ 23. The method of claim 22, comprising:

24. providing a sixth subregion, a seventh subregion, an eighth subregion, and a ninth subregion of the at least two subregions, wherein the sixth subregion, the seventh subregion, the eighth subregion, and the ninth subregion are resonant with the first eigenmode and are of the specific dopant type, the specific doping concentration, the first orientation relative to the crystal axes, and the first TCFN. ｋ 24. The method of claim 23, comprising:

25. 21. The method of claim 20, wherein the respective single resonant frequencies of the respective eigenmodes from the set of at least two eigenmodes are substantially equal.

26. 21. The method of claim 20, wherein the set of at least two eigenmodes comprises two or more eigenmodes selected from the group consisting of: (i) Lame eigenmodes, (ii) plane shear eigenmodes, (iii) square stretch eigenmodes, (iv) width stretch eigenmodes, (v) length stretch eigenmodes, (vi) Nth order annular eigenmodes, (vii) breathing eigenmodes, (viii) wine-glass eigenmodes, and (ix) higher order Lame eigenmodes.

27. TCFNs of said at least two independent sub-regions ｋ By combining these, the TCFN ｔｏｔａｌ The TCFN ｄｅｓｉｒｅｄ 21. The method of claim 20, wherein the value is approximately equal to

28. Nth order includes 1st order, TCFN ｄｅｓｉｒｅｄ 21. The method of claim 20, wherein is approximately equal to 0 ppm / °C.

29. Nth order includes second order, TCFN ｄｅｓｉｒｅｄ is 0 ppb / ℃ ２ 21. The method of claim 20, wherein the .times. ...

30. Nth order includes 1st order, TCFN ｄｅｓｉｒｅｄ 21. The method of claim 20, wherein ∑ i = ...

31. 21. The method of claim 20, wherein each of the at least two sub-regions has the particular dopant type and the particular doping concentration.

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