Tuning frequency-temperature response of MEMS resonators with composite eigenmodes

Abstract

A MEMS resonator apparatus is provided. A 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 sub-regions. Each of the at least two isolated sub-regions resonates with an eigenmode from a set of at least two eigenmodes and has a temperature coefficient of order N (TCFNfc). The isolated first sub-region and the isolated second sub-region resonate with a first eigenmode and a second eigenmode from a set of at least two eigenmodes and have a particular dopant type, a particular doping concentration, and a first orientation and a second orientation relative to the crystal axis. An anchor couples the composite eigenmode resonator element to the support structure. The drive electrode actuates the composite eigenmode resonator element. The sensing electrode senses the composite eigenmode resonator element.

Description

Technical Field

[0001] This disclosure relates to the field of microelectromechanical systems (MEMS) resonators. Background Art

[0002] The emergence of the Internet of Things (IoT) has given rise to numerous sensor-based devices used in wearable devices, smartphones, and remote sensing for industrial and consumer applications. Timing references are ubiquitous in these devices and help provide signals for tracking time, synchronizing events in digital integrated circuits (ICs), and processing signals. High-precision microelectromechanical systems (MEMS) resonators may be desirable for such high-performance electronic applications. Summary of the Invention

[0003] An object of this disclosure is to implement an improved temperature-compensated MEMS device, such as a resonator.

[0004] According to this disclosure, a MEMS resonator device is provided. The MEMS resonator device includes: a support structure; a composite eigenmode resonator element including at least two sub-regions, wherein each of the at least two isolated sub-regions is configured to resonate with a corresponding eigenmode from a set of at least two eigenmodes and has a temperature coefficient of the corresponding Nth-order frequency (TCFN k ), wherein the at least two sub-regions include a first sub-region and a second sub-region, wherein the isolated first sub-region is configured to resonate with a first eigenmode from a set of at least two eigenmodes and has a specific dopant type, a specific doping concentration, a first orientation relative to the crystal axis, and a first TCFN k , wherein the isolated second sub-region is configured to resonate with a second eigenmode from a set of at least two eigenmodes and has a specific dopant type, a specific doping concentration, a second orientation relative to the crystal axis, and a second TCFN k , and wherein (i) the first TCFN k is greater than the temperature coefficient of the desired Nth-order frequency of the composite eigenmode resonator element (TCFN 期望 ) and the second TCFN k is less than TCFN 期望 , or (ii) the first TCFN k is less than TCFN 期望 and the second TCFN k is greater than TCFN 期望 ; at least one anchor that couples the composite eigenmode resonator element to the 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. The method includes: selecting a temperature coefficient of the Nth order frequency (TCFN 期望 ) for a composite eigenmode resonator element, where the composite eigenmode resonator element includes at least two sub-regions, and each of the at least two sub-regions is configured to resonate with a corresponding eigenmode from a set of at least two eigenmodes and is configured to have a temperature coefficient of the Nth order frequency (TCFN k ); providing a first sub-region of the at least two sub-regions, where the isolated first sub-region is configured to resonate with a first eigenmode from the set of at least two eigenmodes and has a specific dopant type, a specific doping concentration, and a first orientation relative to the crystal axis, and where the first sub-region has a first TCFN k ; providing a second sub-region of the at least two sub-regions, where the isolated second sub-region is configured to resonate with a second eigenmode from the set of at least two eigenmodes and has a specific dopant type, a specific doping concentration, and a second orientation relative to the crystal axis, and where the second sub-region has a second TCFN k ; for the composite eigenmode resonator element including the at least two sub-regions, determining the temperature coefficient of the Nth order frequency of (TCFN 总 ); and tuning the TCFN 总 such that it is approximately equal to TCFN 期望 , where tuning the TCFN 总 includes changing at least one of the first or second TCFN k such that (i) the first TCFN k is greater than TCFN 期望 and the second TCFN k is less than TCFN 期望 or (ii) the first TCFN k is less than TCFN 期望 and the second TCFN k is greater than TCFN 期望 . BRIEF DESCRIPTION OF THE DRAWINGS

[0006] The features, aspects, and advantages of the present disclosure technology can be better understood with reference to the following description, the appended claims, and the drawings. Those skilled in the relevant art will understand that the features shown in the drawings are for illustrative purposes, and variations including different and / or additional features and their arrangements are possible.

[0007] Figure 1A An example cross-section of a composite eigenmode resonator device for capacitive transduction is depicted.

[0008] Figure 1BDepicts an example cross-section of a composite intrinsic mode resonator device for piezoelectric transduction.

[0009] Figure 2 Depicts the frequency shift due to the temperature of a composite intrinsic mode resonator element.

[0010] Figure 3A Depicts the predictable response of TCF1 for a wine glass / length extension composite intrinsic mode in a <100> silicon lattice orientation 总 at different doping concentrations and different dopant types.

[0011] Figure 3B Depicts the predictable response of TCF1 for a wine glass / length extension composite intrinsic mode in a <110> silicon lattice orientation 总 at different doping concentrations and different dopant types.

[0012] Figure 3C Depicts the predictable response of TCF2 for a wine glass / length extension composite intrinsic mode in a <100> silicon lattice orientation 总 at different doping concentrations and different dopant types.

[0013] Figure 3D Depicts the predictable response of TCF2 for a wine glass / length extension composite intrinsic mode in a <110> silicon lattice orientation 总 at different doping concentrations and different dopant types.

[0014] Figure 4 Depicts multiple examples of composite intrinsic mode resonator elements.

[0015] Figure 5 Depicts a method for tuning the temperature-induced frequency response of a composite intrinsic mode resonator element that includes two sub-regions.

[0016] Figure 6A Depicts example results of effective mass tuning of a surface shear / square extension composite intrinsic mode by adding or removing sub-regions that resonate with a specific intrinsic mode.

[0017] Figure 6B Depicts example results of effective mass tuning of a surface shear / square extension composite intrinsic mode by adding or removing sub-regions that resonate with a specific intrinsic mode.

[0018] Figure 6C Depicts example results of effective mass tuning of a square extension / Lamé composite eigenmode by adding or removing sub-regions that resonate with a specific intrinsic mode.

[0019] Figure 6D Illustrates exemplary results of effectively mass tuning a square extension / length extension composite eigenmode by adding or removing sub-regions resonating with a specific eigenmode.

[0020] Figure 6E Illustrates exemplary results of effectively mass tuning a Lamé / N - order toroidal composite eigenmode by adding or removing sub-regions resonating with a specific eigenmode.

[0021] Figure 6F Illustrates exemplary results of effectively mass tuning a face shear / length extension composite eigenmode by adding or removing sub-regions resonating with a specific eigenmode.

[0022] Figure 6G Illustrates exemplary results of effectively mass tuning a wine glass / length extension composite eigenmode by adding or removing sub-regions resonating with a specific eigenmode.

[0023] Figure 6H Illustrates exemplary results of effectively mass tuning a breathing / face shear composite eigenmode by adding or removing sub-regions resonating with a specific eigenmode.

[0024] Figure 7A Illustrates exemplary results of effectively mass tuning a Lamé / square extension composite eigenmode by adding or removing sub-regions resonating with a specific eigenmode and modifying the dimensions of the sub-regions resonating with the specific eigenmode without changing the resonance frequency of the specific eigenmode.

[0025] Figure 7B Illustrates exemplary results of effectively mass tuning a wine glass / length extension composite eigenmode by adding or removing sub-regions resonating with a specific eigenmode and modifying the dimensions of the sub-regions resonating with the specific eigenmode without changing the resonance frequency of the specific eigenmode.

[0026] Figure 8A Illustrates the TCFN for tuning the resonance frequency of a square extension / Lamé composite eigenmode total of exemplary results.

[0027] Figure 8B Illustrates the TCFN for tuning the resonance frequency of a square extension / length extension composite eigenmode total of exemplary results.

[0028] Figure 8C Illustrates the TCFN for tuning the resonance frequency of a breathing / face shear composite eigenmode total of exemplary results.

[0029] Figure 9A Illustrates exemplary results of tuning TCF1 by adding a cavity to one or more sub-regions of a composite eigenmode resonator element total of exemplary results.

[0030] Figure 9B Depicts example results of tuning the TCF2 by adding cavities to one or more sub-regions of a composite eigenmode resonator element total as shown. DETAILED DESCRIPTION I. OVERVIEW

[0031] Although quartz crystal oscillators have been the foundation for timing and frequency reference applications for the past century, the rapid development of sensor-based electronic devices has highlighted certain limitations of this technology, such as power consumption, robustness, size, and CMOS compatibility. In the past two decades, MEMS resonators fabricated using silicon have attracted significant attention due to their small size, low cost, and integration compatibility. However, MEMS resonators still cannot replace their quartz counterparts in many applications.

[0032] The limitations of MEMS resonators that impede their widespread adoption are their lack of temperature stability compared to quartz. Silicon MEMS resonators have an inherent first-order temperature-induced 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 contrast, AT-cut quartz resonators have a temperature stability of approximately 20 ppm over the industrial operating temperature range. Several attempts have been made to overcome these temperature-induced frequency drifts, such as methods involving the use of highly doped silicon substrates and composite materials. Temperature-induced frequency drift compensation using highly doped silicon substrates may be impractical for some applications because it may only provide first-order or second-order compensation for temperature-induced frequency drift at specific doping concentrations, which may be difficult to obtain from a foundry. Additionally, temperature-induced frequency drift compensation using composite materials may face issues of aging, increased processing complexity, and reduced quality (Q) factor.

[0033] The frequency change with respect to the temperature of a MEMS resonator is given by the following equation: f(T) = f0[TCF1*(ΔΤ) + TCF2*(ΔΤ) 2 +…] (1) where f o is the resonant frequency of the MEMS resonator at the reference temperature, ΔT is the deviation from the reference temperature, TCF1 is the first-order temperature coefficient of the frequency, and TCF2 is the second-order temperature coefficient of the frequency. For single-crystalline silicon, depending on the dopant type and concentration, the value of TCF2 typically ranges from -25 to -80 ppb / °C 2Within this range, this can result in a temperature-induced frequency drift of about 200 - 400 ppm in the industrial temperature range. Although this second-order temperature-induced frequency drift is relatively small compared to the uncompensated temperature-induced frequency drift of silicon (which is -3750 ppm based on -30 ppm / °C), it is still significantly worse than the typical temperature-induced frequency drift that an AT-cut quartz crystal can provide. Therefore, it is desirable to further reduce the temperature-induced frequency drift of the fabricated silicon MEMS resonator so that it can be widely adopted in the timing market.

[0034] Although active temperature compensation techniques can be used to minimize the temperature-induced frequency drift, such techniques can impose a significant burden on the system in terms of power consumption, circuit size, and circuit complexity. Therefore, it may be beneficial to find ways to passively compensate for the temperature-induced frequency drift in the fabricated silicon MEMS resonator.

[0035] Previously, different techniques have been used to passively compensate for the temperature-induced frequency drift. For example, fabricating MEMS resonators that include a secondary material such as silicon dioxide (SiO2) is a well-established technique. It is known that SiO2 has a large positive TCF1 (about 85 ppm / °C), and due to its compatibility with silicon, SiO2 can be incorporated in the silicon MEMS resonator to balance the negative TCF1 of silicon. In the industrial temperature range, a total temperature-induced frequency drift as low as below 100 ppm has been achieved by this method. In the same temperature range, for smaller devices operating at higher frequencies (e.g., thin film bulk acoustic resonators), temperature-induced frequency drifts as small as 3 ppm have been demonstrated using this technique. However, the TCF2 effect of the oxide must be considered, as well as the loss of the resonator Q factor associated with thermoelastic damping and / or surface losses. The TCF2 effect is not well documented in the literature and is difficult to measure due to its small scale. In addition, this technique increases the manufacturing complexity and has reliability issues due to film stress, which leads to long-term aging problems.

[0036] Another example of existing passive compensation techniques involves fabricating MEMS resonators using highly doped silicon substrates. This is a well-established technique because the TCF1 of a particular eigenmode varies with the doping concentration. For example, in the industrial temperature range of -40°C to 85°C, the temperature-induced frequency drift of a MEMS resonator operating in the Lamé mode can be limited to the range of 200 - 400 ppm using a highly doped silicon substrate. However, even when the TCF1 becomes zero, there may still be a significant amount of temperature-induced frequency drift caused by higher-order temperature coefficients of the frequency. II. Review of Passively Compensating Temperature-Induced Frequency Drift Using Composite Eigenmode Resonators

[0037] The composite eigenmode resonator element 104 is a mechanical vibration resonator element that exhibits at least two different eigenmodes at resonance. These eigenmodes will occur in different regions of the composite eigenmode resonator element 104, which are further referred to as sub-regions. Thus, a "sub-region" is a continuous region that resonates with a specific eigenmode and can be connected to other sub-regions via a contact region that is significantly smaller than the sub-region. The composite eigenmode resonator element 104 may be useful in designing MEMS resonators because they have a predictable and tunable temperature-induced frequency drift response. The Nth-order temperature coefficient of frequency (TCFN 总 ) of the composite eigenmode resonator element 104 can be calculated using a weighted average of the TCFN (TCFN k ) of the eigenmodes exhibited by each respective isolated sub-region. In this context, the "eigenmodes exhibited by each respective isolated sub-region" refers to the eigenmodes exhibited by each respective sub-region when each respective sub-region is not connected to another sub-region via a contact region. Thus, a combination of eigenmodes can be used to design a composite eigenmode resonator element 104 with a specific TCFN total value. Typically, each respective isolated sub-region is selected from a set of standard geometries, including but not limited to (i) a square plate, (ii) a disk, (iii) a bar, or (iv) a ring. The TCFN total of any composite eigenmode resonator element 104 can be calculated using the following equation: where n is the total number of sub-regions, meff k is the effective mass of the kth isolated sub-region, and meff 总 is the effective mass of the composite eigenmode resonator element 104. Analytical or simulation-based methods that are currently known or developed in the future can be used to calculate the TCFN k . It is important to note that the temperature coefficient of any order of the frequency of the composite eigenmode resonator element 104 can be calculated using a weighted average of the TCFN k of the eigenmodes exhibited by each respective isolated sub-region, although calculating the temperature coefficients of the first and second orders of frequency may be the most common.

[0038] The effective mass is a quantitative measure of the inertia of a given region with respect to resonance. It can be calculated using the following expression: where ρ is the density of the sub-region volume V k , d k is the displacement field of the eigenmode in the kth isolated sub-region, and A 总is the maximum displacement amplitude across the entire composite eigenmode resonator element 104. Evidently, the effective mass of a sub-region depends on the maximum displacement amplitude across the entire composite eigenmode resonator element 104. The effective mass is a weighted parameter in Equation 2 as it incorporates the dimensional differences and relative displacement amplitudes of the n sub-regions.

[0039] Figure 1A Depicts an example cross-section of a capacitive transduction 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 sub-regions. Each of the at least two isolated sub-regions is configured to resonate with a corresponding eigenmode from a set of at least two eigenmodes and has a corresponding TCFN k . The at least two sub-regions include a first sub-region 104A and a second sub-region 104B. The isolated first sub-region 104A is configured to resonate with a first eigenmode from a set of at least two eigenmodes and has a specific dopant type, a specific doping concentration, a first orientation relative to the crystal axis, and a first TCFN k . The isolated second sub-region 104B is configured to resonate with a second eigenmode from a set of at least two eigenmodes and has a specific dopant type, a specific doping concentration, a second orientation relative to the crystal axis, and a second TCFN k . (i) The first TCFN k > the temperature coefficient of the desired Nth order frequency of the composite eigenmode resonator element 104 (TCFN 期望 ) and the second TCFN k < TCFN 期望 , or (ii) the first TCFN k < TCFN 期望 and the second TCFN k > TCFN 期望 . At least one anchor 106 couples the composite eigenmode resonator element 104 to the support structure 102. At least one drive electrode 108 actuates the composite eigenmode resonator element 104. At least one sense electrode 110 senses the composite eigenmode resonator element 104.

[0040] Figure 1B Depicts an example cross-section of a piezoelectric transduction composite eigenmode resonator device 100. It is important to note that the addition of piezoelectric electrodes (such as at least one drive electrode 108 and at least one sense electrode 110) will have a minimal impact on the TCFN 总 .

[0041] In some embodiments, the composite eigenmode resonator device 100 is (i) capacitively transduced or (ii) piezoelectrically transduced.

[0042] In some embodiments, each of at least two sub-regions is connected to at least one other sub-region by a contact region that extends from a point of maximum displacement amplitude.

[0043] In some additional embodiments, each of at least two sub-regions is connected to at least one other sub-region by a contact region, and the contact region is minimized such that modal distortion due to non-ideal coupling of the at least two sub-regions is minimized.

[0044] In some additional embodiments, the contact region further includes a rod, and the volume of the rod is less than 5% of the volume of the composite eigenmode resonator element.

[0045] In some embodiments, the at least two sub-regions further include a third sub-region, where the isolated third sub-region resonates with a first eigenmode from a set of at least two eigenmodes and has a specific dopant type, a specific doping concentration, a first orientation relative to a crystal axis, and a first TCFN k .

[0046] In some additional embodiments, the at least two sub-regions further include a fourth sub-region and a fifth sub-region, where both the isolated fourth sub-region and the isolated fifth sub-region resonate with a first eigenmode from a set of at least two eigenmodes and have a specific dopant type, a specific doping concentration, a first orientation relative to a crystal axis, and a first TCFN k .

[0047] In some additional embodiments, the at least two sub-regions further include a sixth sub-region, a seventh sub-region, an eighth sub-region, and a ninth sub-region, where the isolated sixth sub-region, the isolated seventh sub-region, the isolated eighth sub-region, and the isolated ninth sub-region resonate with a first eigenmode from a set of at least two eigenmodes and have a specific dopant type, a specific doping concentration, a first orientation relative to a crystal axis, and a first TCFN k .

[0048] In some additional embodiments, the at least two sub-regions further include a set of 4N + 1 sub-regions, where N is any integer greater than 2. Each of the isolated 4N sub-regions of the set of 4N + 1 sub-regions resonates with a first eigenmode from a set of at least two eigenmodes and has a specific dopant type, a specific doping concentration, a first orientation relative to a crystal axis, and a first TCFN k。The isolated remaining sub-region (i.e., the +1 sub-region in the set of 4N+1 sub-regions) resonates with a second eigenmode from a set of at least two eigenmodes and has a specific dopant type, a specific dopant concentration, a second orientation relative to the crystal axis, and a second TCFN k 。The remaining sub-region can be placed at the center of the set of 4N+1 sub-regions, and 4N of the 4N+1 sub-regions can extend symmetrically from the maximum displacement point of the remaining sub-region placed at the center.

[0049] In some embodiments, the isolated respective resonance frequencies of the eigenmodes from the set of at least two eigenmodes are approximately equal. Throughout this document and the entire disclosure, "the isolated respective resonance frequencies are approximately equal" means that the difference between the isolated resonance frequencies is reduced to as close to zero as possible. In some embodiments, this may involve reducing the difference to less than 1 GHz. In other embodiments, this may involve reducing the difference to less than 1 MHz. Still in other embodiments, this may involve reducing the difference to less than 1 kHz. Still in other embodiments, this may involve reducing the difference between the isolated resonance frequencies to less than 1% of the resonance frequency of the composite eigenmode resonator element 104.

[0050] In some embodiments, the set of at least two eigenmodes includes two or more eigenmodes selected from the group consisting of: (i) Lamé eigenmodes, (ii) face shear eigenmodes, (iii) square extension eigenmodes, (iv) width extension eigenmodes, (v) length extension eigenmodes, (vi) Nth order ring eigenmodes, (vii) breathing eigenmodes, (viii) wine glass eigenmodes, and (ix) higher order Lamé eigenmodes.

[0051] In some embodiments, the TCFN of the isolated at least two sub-regions k are combined such that the TCFN 总 is approximately equal to the TCFN 期望 。

[0052] In some embodiments, the Nth order includes the first order (i.e., TCFN = TCF1), where the TCFN 期望 is approximately equal to 0 ppm / °C. "Approximately equal to 0 ppm / °C" herein does not necessarily involve reducing the TCF1 precisely to zero. Instead, it is sufficient to reduce the TCF1 to as close to zero as possible. In some embodiments, this may involve reducing the TCF1 to less than 1 ppm / °C. In other embodiments, this may involve reducing the TCF1 to less than 0.1 ppm / °C. Still in other embodiments, this may involve reducing the TCF1 to less than 0.01 ppm / °C. In some embodiments, the Nth order includes the second order (i.e., TCFN = TCF2), and the TCFN 期望 is approximately equal to 0 ppb / °C2 In this text, "approximately equal to 0 ppb / °C" 2 does not necessarily involve precisely reducing the TCF2 to zero. Instead, it is sufficient to reduce the TCF2 to as close to zero as possible. In some embodiments, this may involve reducing the TCF2 to less than 1 ppb / °C 2 In other embodiments, this may involve reducing the TCF2 to less than 0.1 ppb / °C 2 Still in other embodiments, this may involve reducing the TCF2 to less than 0.01 ppb / °C 2 .

[0053] In some embodiments, the Nth order includes the first order (i.e., TCFN = TCF1), and TCFN 期望 is equal to a non-zero value that at least partially compensates for the third-order temperature coefficient of the frequency of the composite eigenmode resonator element.

[0054] In some embodiments, each of at least two sub-regions has a specific dopant type and a specific doping concentration.

[0055] In some embodiments, the geometry of the composite eigenmode resonator element further includes one or more cavities in one or more of the at least two sub-regions.

[0056] In some embodiments, the effective mass of the composite eigenmode resonator element 104 has a difference of <10% from the sum of the effective masses of each of the isolated at least two sub-regions.

[0057] In some embodiments, the composite eigenmode resonator device 100 is configured to operate as (i) an oscillator or (ii) a resonant sensor.

[0058] In some embodiments, the composite eigenmode resonator device 100 includes at least one of single-crystalline silicon, silicon carbide, polycrystalline silicon, quartz, graphene, and polycrystalline diamond. III. Examples of Passive Compensation Composite Eigenmode Resonators

[0059] Figure 2 Comprising a graph 200 depicting the frequency shift as a function of the temperature of the composite eigenmode resonator element 202, which composite eigenmode resonator element 202 may be combined with the above Figure 1A and Figure 1Bis similar to or identical with the described composite eigenmode resonator element 104. Graph 200 compares the composite eigenmode resonator element 202, which includes a first sub-region 202A resonating with a face shear mode and a second sub-region 202B resonating with a length extensional mode, with the constituent isolated eigenmodes (face shear mode 204 and length extensional mode 206). As can be seen in Graph 200, the frequency shift of the isolated face shear mode 204 increases significantly with increasing temperature, while the frequency shift of the isolated length extensional mode 206 decreases significantly with increasing temperature. The slope of the frequency shift due to the temperature of the composite eigenmode resonator element 202 is significantly less than that of either of the constituent isolated eigenmodes. Therefore, the frequency drift caused by temperature of the composite eigenmode resonator element 202 will be significantly reduced relative to a resonator element having only one of the constituent isolated eigenmodes.

[0060] It is important to note that the predictable response of the TCFN described by Equation 2 and depicted in Graph 200 总 can hold regardless of the dopant type, dopant concentration, or overall orientation with respect to the crystal axes of the composite eigenmode resonator element 104. Although increasing the dopant concentration or using a different dopant type may shift the TCFN k of the isolated k-th sub-region, Equation 2 can still be used to predict the TCFN 总 . In addition, changing the overall orientation with respect to the crystal axes of the composite eigenmode resonator element 104 may shift the TCFN k of the isolated k-th sub-region, but Equation 2 can still be used to predict the TCFN 总 . Moreover, a predictable response of TCF1 总 (or TCFN 总 ) exists in undoped silicon, but it has little effect on the frequency drift caused by temperature because the TCF1 of most eigenmodes is approximately -30 ppm / °C. It is also important to note that there are limitations to the linearity of Equation 2, which will be discussed in detail in the next section.

[0061] Figure 3A depicts the predictable response of TCF1 总 for the wine glass / length extension composite eigenmode in the <100> silicon lattice orientation at different dopant concentrations and different dopant types.

[0062] Figure 3B depicts the predictable response of TCF1 总 for the wine glass / length extension composite eigenmode in the <110> silicon lattice orientation at different dopant concentrations and different dopant types.

[0063] Figure 3CDepicts the predictable response of TCF2 for the wine glass / length extension composite eigenmode in the <100> silicon lattice orientation. 总 At different doping concentrations and different dopant types.

[0064] Figure 3D Depicts the predictable response of TCF2 for the wine glass / length extension composite eigenmode in the <110> silicon lattice orientation. 总 At different doping concentrations and different dopant types.

[0065] Figure 4Depicts multiple examples of the composite eigenmode resonator element 104. Image 402 depicts a square extension / surface shear composite eigenmode resonator element with two sub-regions. Image 404 depicts a square extension / surface shear composite eigenmode resonator element with three sub-regions. Image 406 depicts a square extension / surface shear composite eigenmode resonator element with five sub-regions. Image 408 depicts a square extension / surface shear composite eigenmode resonator element with nine sub-regions. Image 410 depicts a square extension / Lamé composite eigenmode resonator element with two sub-regions. Image 412A depicts a square extension / Lamé composite eigenmode resonator element with three sub-regions, where two sub-regions resonate with the Lamé eigenmode. Image 412B depicts a square extension / Lamé composite eigenmode resonator element with three sub-regions, where two sub-regions resonate with the 1x2 higher-order Lamé eigenmode. Image 412C depicts a square extension / Lamé composite eigenmode resonator element with three sub-regions, where two sub-regions resonate with the 1x3 higher-order Lamé eigenmode. Image 414 depicts a square extension / Lamé composite eigenmode resonator element with five sub-regions. Image 416 depicts a square extension / length extension composite eigenmode resonator element with two sub-regions. Image 418 depicts a square extension / length extension composite eigenmode resonator element with three sub-regions. Image 420 depicts a square extension / length extension composite eigenmode resonator element with five sub-regions. Image 422 depicts a 4th-order ring / Lamé composite eigenmode resonator element with two sub-regions. Image 424 depicts a 4th-order ring / Lamé composite eigenmode resonator element with three sub-regions. Image 426 depicts a 4th-order ring / Lamé composite eigenmode resonator element with five sub-regions. Image 428 depicts a surface shear / length extension composite eigenmode resonator element with two sub-regions. Image 430 depicts a surface shear / length extension composite eigenmode resonator element with three sub-regions. Image 432 depicts a surface shear / length extension composite eigenmode resonator element with five sub-regions. Image 434 depicts a wineglass / length extension composite eigenmode resonator element with two sub-regions. Image 436 depicts a wineglass / length extension composite eigenmode resonator element with three sub-regions. Image 438 depicts a wineglass / length extension composite eigenmode resonator element with five sub-regions. Image 440 depicts a surface shear / breathing mode composite eigenmode resonator element with two sub-regions. Image 442 depicts a surface shear / breathing mode composite eigenmode resonator element with three sub-regions. Image 444 depicts a surface shear / breathing mode composite eigenmode resonator element with five sub-regions. Image 446 depicts a width extension / Lamé composite eigenmode resonator element with five sub-regions, where the central sub-region resonates with the 3rd-order Lamé eigenmode.Image 448 depicts a width - extended / Lamé composite eigenmode resonator element having nine sub - regions, where the central sub - region resonates with a 5th - order Lamé eigenmode. Those skilled in the art will note that the sub - regions of the width - extended / Lamé composite eigenmode resonator element depicted in Images 446 and 448 include connection regions defined by bars. IV. Method for Passive Compensation of Composite Eigenmode Resonators

[0066] Designing the composite eigenmode resonator element 104 to achieve an accurate temperature - induced frequency response requires at least two sub - regions, where each of the at least two isolated sub - regions resonates with an eigenmode from a set of at least two eigenmodes and has a TCFN k . While in different example embodiments, various different numbers of sub - regions resonating with various eigenmodes can be utilized, for clarity, many of the examples described herein show methods for passive compensation of composite eigenmode resonator elements having five or fewer sub - regions.

[0067] Initially, the composite eigenmode resonator element 104 can be roughly designed to have a temperature coefficient (TCFN 期望 = 0 ppm / °C or TCF2 期望 = 0 ppb / °C 2 ) approximately equal to the desired N - th order frequency by selecting sub - regions that resonate with eigenmodes having TCFN k higher than and lower than the desired value (e.g., TCF1 期望 ). However, there is a limited number of common eigenmodes to choose from (e.g., Lamé, face - shear, square - extension, width - extension, length - extension, N - th order ring, breathing, wine - glass, or higher - order Lamé eigenmodes), and it may not always be possible to select two eigenmodes having the same TCFN 总 absolute value but opposite signs. Thus, simply combining two eigenmodes generally will not result in TCFN k = TCFN 总 . In most cases, further tuning is required.

[0068] Figure 5

[0069] Figure 500 depicts a flow - chart of a method for tuning the N - th order temperature - induced frequency response of the composite eigenmode resonator device 100. At step 502, the method involves selecting a TCFN 期望 for the composite eigenmode resonator element 104. Consistent with the above discussion, the composite eigenmode resonator element 104 includes at least two sub - regions, where each of the at least two isolated sub - regions resonates with an eigenmode from a set of at least two eigenmodes and has a TCFN k .

[0069] At step 504, the method involves providing a first sub-region of the composite eigenmode resonator element 104. The isolated first sub-region resonates with a first eigenmode from a set of at least two eigenmodes and has a specific dopant type, a specific dopant concentration, a first orientation relative to the crystal axis, and a first TCFN k . At step 506, the method involves providing a second sub-region of the composite eigenmode resonator element 104. The isolated second sub-region resonates with a second eigenmode from the set of at least two eigenmodes and has a specific dopant type, a specific dopant concentration, a second orientation relative to the crystal axis, and a second TCFN k .

[0070] At step 508, the method involves determining the TCFN of the composite eigenmode resonator element 104 that includes at least two sub-regions 总 , which can be done using Equation 2 above (or, if desired, with a computer program).

[0071] At step 510, the method involves tuning the TCFN 总 such that it is approximately equal to TCFN 期望 . Tuning the TCFN 总 includes changing at least one of the first or second TCFN k such that (i) the first TCFN k is greater than TCFN 期望 and the second TCFN k is less than TCFN 期望 , or (ii) the first TCFN k is less than TCFN 期望 and the second TCFN k is greater than TCFN 期望 .

[0072] In some embodiments, tuning the TCFN 总 such that it is approximately equal to TCFN 期望 can be done by one or a combination of the following: (i) performing effective mass tuning, (ii) performing resonance frequency tuning, or (iii) adding a cavity to one or more of the at least two sub-regions such that at least one of the first or second TCFN k is changed. Additionally, if combined, effective mass tuning, resonance frequency tuning, and adding a cavity to one or more of the at least two sub-regions can be applied in any order. The method depicted in flowchart 500 can be performed using a parameter sweep approach, using methods known now and developed later, to iteratively design the composite eigenmode resonator element 104.

[0073] In this document, effective mass tuning means modifying the percentage of the effective mass of a particular eigenmode from a set of at least two eigenmodes on the composite eigenmode resonator element 104 without changing the resonance frequency of the particular eigenmode. Effective mass tuning can be accomplished by either (i) adding or removing a sub-region that resonates with the particular eigenmode or (ii) modifying the size of a sub-region that resonates with the particular eigenmode, either alone or in combination, without changing the resonance frequency of the particular eigenmode.

[0074] Adding or removing a sub-region will necessarily change the number of n sub-regions in Equation 2. Therefore, by adding or removing a sub-region, the weighted average TCFN 总 can be tuned to be approximately equal to TCFN 期望 .

[0075] Modifying the size of a sub-region that resonates with a particular eigenmode without changing the resonance frequency of the particular eigenmode can be achieved by exploiting a particular type of eigenmode. For example, changing the width of a sub-region that resonates with a length extension eigenmode or changing the height of a sub-region that resonates with an in-plane eigenmode (e.g., Lamé, face shear, etc.) will modify the size of the sub-region (and thus the effective mass). Some eigenmodes can also be distributed along a particular axis, such as the Lamé eigenmode in the x and y plane directions and the length extension eigenmode in the length direction. Doubling the size of a sub-region that exhibits a distributable mode will cause that sub-region to exhibit the corresponding second-order eigenmode without changing the resonance frequency.

[0076] In some cases, when tuning the TCFN 总 such that it is approximately equal to TCFN 期望 , effective mass tuning may be more advantageous than other methods. For example, effective mass tuning can be achieved while maintaining the maximum displacement amplitude in a set of at least two eigenmodes, which can make the fabricated MEMS resonator device more practical to operate in the real world because the displacement amplitude improves the signal-to-noise ratio.

[0077] As described herein, resonance frequency tuning means scaling the size of a particular sub-region that resonates with a particular eigenmode in such a way that only the resonance frequency of the particular eigenmode changes without changing the mode shape (e.g., making a square sub-region that exhibits a square extension mode larger or smaller, but not rectangular). Changing the resonance frequency of a particular eigenmode will result in a resonance frequency difference between the changed eigenmode and other eigenmodes. This will weaken the mechanical coupling between the changed eigenmode and other eigenmodes in the entire composite eigenmode. The weak coupling will cause one of the eigenmodes to dominate the other eigenmodes in terms of the displacement amplitude at the new resonance frequency. This will effectively increase the effective mass of the dominant eigenmode relative to other eigenmodes and shift the TCFN 总 of the resonator towards the TCFN k of the dominant eigenmode.

[0078] Tuning the TCFN 总 such that it is approximately equal to the TCFN 期望 It can also be achieved by adding a cavity to a specific sub-region that resonates with a specific eigenmode. Adding a cavity to one or more of the sub-regions within the sub-region will cause the composite eigenmode to be slightly "distorted". As used herein, "distortion" means introducing stray vibrations, resulting in a deviation from the composite eigenmode present in an equivalent resonator element without a cavity. Adding a cavity to one or more of at least two sub-regions will distort the composite eigenmode by changing the contribution of the elastic constants of silicon (c 11 , c 12 and c 44 ), thereby shifting the TCFN of the composite eigenmode 总 .

[0079] Figure 6A depicts exemplary results of effective mass tuning via addition or removal of sub-regions that resonate with a specific eigenmode. In this example, tuning the TCF1 of the face-shear / square-extensional composite eigenmode 总 . In this example, the resonator element is oriented in the <100> silicon lattice orientation and is N-doped to a concentration of 2.00×10 19 atoms / cm 3 . The depicted value of the square-extensional (SE) effective mass is determined according to the equation: SE effective mass = meff SE / meff 总 . The resonator element 602 has a single sub-region that resonates in an isolated face-shear eigenmode such that the element 602 does not contain a square-extensional eigenmode and the SE effective mass = 0%. When the SE effective mass = 0%, the resulting isolated face-shear eigenmode has a TCF1 总 = -28.21 ppm / °C. As shown by resonator elements 604, 606, 608, and 610, one or more square-extensional sub-regions can be placed at one or more corners of the face-shear sub-region to push the TCF1 总 of the composite eigenmode resonator element 104 towards the TCF1 总 of the isolated square-extensional eigenmode (represented by resonator element 612). As the number of SE sub-regions increases, the TCF1 总 increases and approaches the case where the SE effective mass = 100% (i.e., resonator element 612), and the resulting isolated square-extensional eigenmode has a TCF1 总 = -8.42 ppm / °C.

[0080] Figure 6B includes graph 614, which plots Figure 6ATCF1 of the face-shear / square-extension composite eigenmode shown in 总 Example results of effective mass tuning. In graph 614, the face-shear / square-extension composite eigenmode resonator element includes a single face-shear mode sub-region placed at the maximum displacement point of the square-extension sub-region. In this example, the composite eigenmode resonator element is oriented in the <100> silicon lattice orientation and N-doped to 2.00x10 19 atoms / cm 3 concentration. SE effective mass = meff SE / meff 总 . As shown in graph 614, the SE effective mass varies by adjusting the number of square-extension sub-regions. As Figure 6B further shown in, graph 616 depicts the effect on TCF2 总 resulting from changing the SE effective mass of the composite eigenmode resonator element by adjusting the number of square-extension sub-regions in the same or similar manner as graph 614.

[0081] Figure 6C Includes graph 618, which depicts an example of effective mass tuning for TCF1 总 of the square-extension / Lamé composite eigenmode. In graph 618, the square-extension / Lamé composite eigenmode resonator element includes a single square-extension mode sub-region placed at the maximum displacement point of the Lamé sub-region. In this example, the composite eigenmode resonator element is oriented in the <100> silicon lattice orientation and N-doped to 2.00x10 19 atoms / cm 3 concentration. SE effective mass = meff SE / meff 总 . As shown in graph 618, the SE effective mass varies by adjusting the number of Lamé sub-regions. As Figure 6C further shown in, graph 620 depicts the effect on TCF2 总 resulting from changing the SE effective mass of the composite eigenmode resonator element by adjusting the number of Lamé sub-regions in the same or similar manner as graph 618.

[0082] Figure 6D Includes graph 622, which depicts an example of effective mass tuning for TCF1 总 of the square-extension / length-extension composite eigenmode. In graph 622, the square-extension / length-extension composite eigenmode resonator element includes a single square-extension mode sub-region placed at the maximum displacement point of the length-extension sub-region. In this example, the composite eigenmode resonator element is oriented in the <100> silicon lattice orientation and N-doped to 6.00x 1019 atoms / cm 3 concentration. SE effective mass = meff SE / meff 总 . As shown in graph 622, the SE effective mass of the composite eigenmode resonator element varies by adjusting the number of length-extension mode sub-regions. As Figure 6B further shown in, graph 624 depicts the effect on TCF2 resulting from changing the SE effective mass of the composite eigenmode resonator element by adjusting the number of length-extension sub-regions in the same or similar manner as graph 622 总 .

[0083] Figure 6E Includes graph 626, which depicts an example of the effective mass tuning of TCF1 for the Lamé / N-order ring composite eigenmode. In graph 626, the Lamé / N-order ring composite eigenmode resonator element includes a single Lamé mode sub-region placed at the maximum displacement point of the N-order ring. The Lamé sub-region is oriented in the <100> silicon lattice orientation, and N is doped to 7.5x 10 总 atoms / cm 19 concentration. Lamé effective mass = meff 3 / meff Lamé / meff 总 . As shown in graph 626, the Lamé effective mass of the composite resonator element varies by adjusting the number of N-order ring mode sub-regions. As Figure 6E further shown in, graph 628 depicts the effect on TCF2 resulting from changing the Lamé effective mass of the composite eigenmode resonator element by adjusting the number of N-order ring sub-regions in the same or similar manner as graph 626 总 .

[0084] Figure 6F Includes graph 630, which depicts an example of the effective mass tuning of TCF1 for the face-shear / length-extension composite eigenmode. In graph 630, the face-shear / length-extension composite eigenmode resonator element includes a single face-shear sub-region placed at the maximum displacement point of the length-extension sub-region. In this example, the composite eigenmode resonator element is oriented in the <100> silicon lattice orientation, and P is doped to 2.4x 10 总 atoms / cm3 concentration. Face-shear (FS) effective mass = meff 20 / meff FS / meff 总 . As shown in graph 630, the FS effective mass of the composite resonator element varies by adjusting the number of length-extension mode sub-regions. As Figure 6FAs further shown in, graph 632 depicts the effect on TCF2 resulting from changing the FS effective mass of the composite eigenmode resonator element by adjusting the number of length-extension sub-regions in the same or a similar manner as graph 630 总 thereof.

[0085] Figure 6G Includes graph 634, which depicts an example of the effective mass tuning of TCF1 总 for the wine glass / length-extension composite eigenmode. In graph 634, the wine glass / length-extension composite eigenmode resonator element includes a single wine glass sub-region placed at the maximum displacement point of the length-extension sub-region. In this example, the composite eigenmode resonator element is oriented in the <110> silicon lattice orientation and P-doped to a concentration of 2.4x 10 20 atoms / cm3. Wine glass effective mass = meff 酒杯 / meff 总 . As shown in graph 634, the wine glass effective mass of the composite resonator element varies by adjusting the number of length-extension mode sub-regions. As Figure 6G further shown in, graph 636 depicts the effect on TCF2 resulting from changing the wine glass effective mass of the composite eigenmode resonator element by adjusting the number of length-extension sub-regions in the same or a similar manner as graph 634 总 thereof.

[0086] Figure 6H Includes graph 638, which depicts an example of the effective mass tuning of TCF1 总 for the breathing / surface shear composite eigenmode. In graph 638, the breathing / surface shear composite eigenmode resonator element includes a single breathing mode sub-region placed at the maximum displacement point of the surface shear sub-region. In this example, the composite eigenmode resonator element is oriented in the <110> silicon lattice orientation and P-doped to 2.4x10 20 atoms / cm3 concentration. Breathing mode effective mass = meff 呼吸 / meff 总 . As shown in graph 638, the breathing mode effective mass of the composite resonator element varies by adjusting the number of surface shear sub-regions. As Figure 6H further shown in, graph 640 depicts the effect on TCF2 resulting from changing the breathing effective mass of the composite eigenmode resonator element by adjusting the number of surface shear sub-regions in the same or a similar manner as graph 638 总 thereof.

[0087] Figure 7ADepicts exemplary results of effective mass tuning by modifying the dimensions of sub-regions resonating with a particular eigenmode without changing the resonance frequency of the particular eigenmode. In this example, tuning the TCF1 of the Lamé / square extension composite eigenmode 总 and TCF2 总 . In this example, the composite eigenmode resonator element is oriented in the <110> silicon lattice orientation and N-doped to a concentration of 2.00x 10 19 atoms / cm 3 . Also, the depicted value of the square extension (SE) effective mass is determined according to the equation: SE effective mass = meff SE / meff 总 . Here, element 708 depicts a square extension / Lamé composite eigenmode resonator element having three sub-regions, where the first sub-region resonates with the square extension eigenmode and two sub-regions extending from opposite corners of the first sub-region resonate with the Lamé eigenmode. Element 708 has simulated TCF1 总 = -27.04 ppm / °C and simulated TCF2 总 = -28.99 ppb / °C 2 . Turning to element 706, it can be seen that the side lengths of the two sub-regions resonating with the Lamé eigenmode have been doubled. Thus, the two sub-regions exhibit a higher order Lamé eigenmode that resonates at the same resonance frequency but with an additional effective mass, which will shift the TCFN of element 706 总 towards the TCFN of the pure Lamé eigenmode (element 702) 总 . Similarly, element 704 has further increased the side lengths of the two sub-regions resonating with the Lamé eigenmode, further shifting the temperature coefficient of the frequency to simulated TCF1 总 = -23.17 ppm / °C and simulated TCF2 总 = -39.13 ppb / °C 2 .

[0088] Figure 7B Depicts exemplary results of effective mass tuning of the wine glass / length extension composite eigenmode by both (i) adding or removing sub-regions resonating with a particular eigenmode and (ii) modifying the dimensions of sub-regions resonating with a particular eigenmode without changing the resonance frequency of the particular eigenmode. In this example, the composite eigenmode resonator element is oriented in the <110> silicon lattice orientation and P-doped to a concentration of 2.4x 10 20 atoms / cm 3 . The depicted value of the wine glass effective mass is determined according to the equation: wine glass effective mass = meff 酒杯 / meff 总。The resonator element 702 has a single sub-region that resonates in an isolated wine-glass eigenmode such that the wine-glass effective mass = 100%. When the wine-glass effective mass = 100%, the resulting isolated wine-glass eigenmode has a TCF1 总 = 1.18 ppm / °C. As shown by resonator elements 704, 706, 708, and 710, one or more length-extension (LE) sub-regions can be placed at the maximum displacement point of the wine-glass sub-region to shift the TCF1 总 towards the TCF1 总 of an isolated length-extension eigenmode (represented by resonator element 712). As the number of LE sub-regions increases, the TCF1 总 increases until the wine-glass effective mass = 0%, and the resulting isolated length-extension eigenmode has a TCF1 总 = -8.21 ppm / °C. Additionally, comparing element 704 with element 706, it can be seen that without changing the resonance frequency, modifying the dimensions of the sub-region resonating with the length-extension eigenmode can tune the TCFN 总 . Element 704 has a TCF1 总 = -1.11 ppm / °C, but by increasing the width of the sub-region resonating with the length-extension eigenmode, element 706 has a TCF1 总 = -2.59 ppm / °C.

[0089] Figures 8A - 8C Depicts the effect of resonance frequency tuning on TCF1 总 and TCF2 总 . Figure 8A Contains graph 802, which depicts an example of resonance frequency tuning by scaling the dimensions of a square-extension / Lamé composite eigenmode to tune the TCF1 总 . In graph 802, the square-extension / Lamé composite eigenmode resonator element includes a single SE mode sub-region and a single Lamé mode sub-region extending from the corner of the SE mode sub-region, where each sub-region is oriented in the <110> silicon lattice orientation and N-doped to a concentration of 1.7x 10 20 atoms / cm 3 . Similarly, the SE effective mass = meff SE / meff 总 . As shown in graph 802, the SE effective mass of the composite resonator element varies by adjusting the side length l Lamé of the Lamé mode sub-region, and the specific value of the side length l Lamé is indicated in Figure 8A .

[0090] Figure 8BIncludes graph 804, which depicts an example of tuning the resonance frequency of TCF1 by scaling the dimensions of the scaled square extension / length extension (SE / LE) composite eigenmode. In graph 804, the SE / LE composite eigenmode resonator element includes a single SE mode sub-region and a single LE mode sub-region extending from the corner of the SE mode sub-region, where each sub-region is oriented in the <100> silicon lattice orientation and N-doped to a concentration of 6.0x10 总 atoms / cm 19 . Similarly, the SE effective mass = meff 3 / meff SE . As shown in graph 804, the SE effective mass of the composite resonator element varies by adjusting the side length l 总 of the SE mode sub-region, and the specific value of this side length l SE is indicated in SE Figure 8B . As further shown in Figure 8B , graph 806 depicts the effect on TCF2 SE resulting from changing the SE effective mass of the composite eigenmode resonator element by adjusting the side length l 总 of the SE mode sub-region in the same or similar manner as graph 804.

[0091] Figure 8C Includes graph 808, which depicts an example of tuning the resonance frequency of TCF1 by scaling the dimensions of the breathing / surface shear composite eigenmode. In graph 808, the breathing / surface shear composite eigenmode resonator element includes a single breathing mode sub-region and a single surface shear mode sub-region placed at the maximum displacement point of the breathing mode sub-region, and each surface shear mode sub-region is oriented in the <110> silicon lattice orientation and P-doped to a concentration of 2.4x10 总 atoms / cm 20 . The breathing mode effective mass = meff 3 / meff 呼吸 . As shown in curve 808, the breathing mode effective mass of the composite resonator element varies by adjusting the radius r 总 of the breathing mode sub-region, and the specific value of this radius r 呼吸 is indicated in 呼吸 Figure 8C . As further shown in Figure 8C , graph 810 depicts the effect on TCF2 呼吸 resulting from changing the breathing mode effective mass of the composite eigenmode resonator element by adjusting the radius r 总 of the breathing mode sub-region in the same or similar manner as graph 808.

[0092] Figure 9A ​​Depicts tuning of TCF1 by adding cavities to one or more of at least two sub-regions 总 The effect. In the depicted example, the composite eigenmode is a square extension (SE) / Lamé composite eigenmode. Graph 902 depicts the example resulting effect of adding cavities to the SE / Lamé composite eigenmode to tune TCF1 总 . Image 904 depicts the SE / Lamé composite eigenmode without cavities (which corresponds to the dotted line in Graph 902). Image 906 depicts the SE / Lamé composite eigenmode with cavities having a side length of 20 μm (which corresponds to the dashed line in Graph 902). Image 908 depicts the SE / Lamé composite eigenmode with cavities having a side length of 25 μm (which corresponds to the solid line in Graph 902). As shown in Graph 902, adding cavities to the SE / Lamé composite eigenmode can increase the sensitivity of effective mass tuning, as evidenced by the slope of the line in Graph 902 that increases as the cavity size gets larger. Although Figure 9B the example depicted in involves four square cavities arranged in a grid pattern around the center of the resonator element, it should be understood that in other examples, the number, size, shape, and arrangement of the cavities can vary, which can similarly alter the effect on TCF1 总 . Additionally, although Figure 9B the example depicted in involves an SE / Lamé composite eigenmode having a single SE mode and a single Lamé mode, it should be understood that cavities can similarly be implemented in conjunction with various other composite eigenmode configurations (such as any of the composite eigenmode configurations described herein).

[0093] Figure 9B Depicts example results of tuning TCF2 by adding cavities to one or more of at least two sub-regions 总 . That is, Graph 910 depicts the exemplary effect on TCF2 Figure 9A resulting from adding cavities to the SE / Lamé composite eigenmode in the same or a similar manner as described above in connection with 总 .

[0094] In some embodiments, the example methods described herein further include providing a third sub-region of at least two sub-regions, wherein the isolated third sub-region resonates with a first eigenmode and has a specific dopant type, a specific dopant concentration, a first orientation relative to the crystal axis, and a first TCFN k .

[0095] In some further embodiments, the example methods described herein further include providing a fourth sub-region and a fifth sub-region in at least two sub-regions, wherein both the isolated fourth sub-region and the isolated fifth sub-region resonate with a first eigenmode and have a specific dopant type, a specific doping concentration, a first orientation with respect to the crystal axis, and a first TCFN k .

[0096] In some further embodiments, the example methods described herein further include providing a sixth sub-region, a seventh sub-region, an eighth sub-region, and a ninth sub-region in at least two sub-regions, wherein the isolated sixth sub-region, the isolated seventh sub-region, the isolated eighth sub-region, and the isolated ninth sub-region resonate with a first eigenmode and have a specific dopant type, a specific doping concentration, a first orientation with respect to the crystal axis, and a first TCFN k .

[0097] In some embodiments, the isolated respective resonance frequencies of the respective eigenmodes from a set of at least two eigenmodes are approximately equal.

[0098] In some embodiments, the set of at least two eigenmodes includes at least two eigenmodes selected from the group consisting of: (i) Lamé eigenmodes, (ii) face shear eigenmodes, (iii) square extension eigenmodes, (iv) width extension eigenmodes, (v) length extension eigenmodes, (vi) N-th order ring eigenmodes, (vii) breathing eigenmodes, (viii) wine glass eigenmodes, and (ix) higher order Lamé eigenmodes.

[0099] In some embodiments, the TCFNs of the isolated at least two sub-regions k are combined such that the TCFN 总 is approximately equal to the TCFN 期望 .

[0100] In some embodiments, N-th order includes first order, and wherein the TCFN 期望 is approximately equal to 0 ppm / °C.

[0101] In some embodiments, N-th order includes second order, and wherein the TCFN 期望 is approximately equal to 0 ppm / °C 2 .

[0102] In some embodiments, N-th order includes first order, and wherein the TCFN 期望 is a non-zero value that at least partially compensates for the third-order temperature coefficient of the frequency of the composite eigenmode resonator element.

[0103] In some embodiments, each of the at least two sub-regions has a specific dopant type and a specific doping concentration. V. Nonlinear Effects in Composite Eigenmode Resonators

[0104] It is important to note that there are limitations to the linearity predicted by theory provided by Equation 2. These limitations are caused by modal distortions in the composite eigenmodes present in the composite eigenmode resonator element 104 relative to the eigenmodes exhibited by each isolated corresponding sub-region, the TCFN of this composite eigenmode resonator element 104 k used in Equation 2. The modal distortion can be quantified by observing the effective mass of the entire composite eigenmode resonator element 104 compared to the sum of the effective masses of the n isolated sub-region eigenmodes. In an ideal case, if the two are equal, all eigenmodes on the n sub-regions of the device are fully excited, and the TCFN 总 value will exactly match the result of Equation 2. However, if there is modal distortion, the effective mass of the actual composite eigenmode resonator element 104 will be lower than the sum of its parts (n sub-regions).

[0105] In the fabricated composite eigenmode resonator device 100, there will always be modal distortion partly due to (i) non-ideal contact regions between one or more sub-regions and (ii) reduced coupling between at least two sub-regions. The non-ideal contact regions will cause modal distortion in the composite eigenmode compared to the eigenmodes exhibited by each corresponding isolated sub-region. As these modal distortions increase, the composite eigenmode exhibited by the composite eigenmode resonator element 104 will deviate from being a combination of isolated eigenmodes and become a complex eigenmode, whose TCFN 总Difficult to predict analytically. A larger contact area between two sub-regions will result in more modal distortion. Thus, it is preferable to minimize the contact area between sub-regions. In some embodiments, the contact area may be defined by the intersection line between two sub-regions. Schematic diagram 912 provides an example of the intersection line 914 in the contact area of the square extension (SE) / Lamé composite eigenmode. In some embodiments, minimizing the contact area between two sub-regions may involve reducing the intersection line to less than 10 μm. In other embodiments, this may involve reducing the intersection line in the contact area of the two sub-regions to less than 5 μm. In other embodiments, this may involve reducing the intersection line in the contact area of the two sub-regions to less than 1 μm. In some embodiments, the contact area may be defined by a bar between two sub-regions. In some embodiments, minimizing the contact area between two sub-regions may involve reducing the volume of the bar between the two sub-regions to less than 5% of the total volume of the composite eigenmode resonator element. In other embodiments, minimizing the contact area between two sub-regions may involve reducing the volume of the bar between the two sub-regions to less than 1% of the total volume of the composite eigenmode resonator element. When there is no convenient maximum displacement point to form a small enough contact area, using a bar between two sub-regions may be more advantageous than other methods, and the small enough contact area minimizes modal distortion caused by non-ideal coupling. Image 446 provides an example of this scenario, where a small enough contact area between the width extension and the 3rd order Lamé eigenmode is actually not feasible.

[0106] In addition, weaker coupling between sub-region eigenmodes will result in modal distortion. When performing resonance frequency tuning by scaling the dimensions of one or more sub-regions, as a result of increasing the difference between the resonance frequencies of isolated eigenmodes, weaker coupling will occur. Increasing the difference between the resonance frequencies of isolated eigenmodes will cause one eigenmode to dominate other eigenmodes. According to Equation 3, the displacement amplitude of a sub-region is directly related to its effective mass. Thus, as the percentage increase in the total effective mass of the composite eigenmode resonator element 104 is represented by the dominant eigenmode, modal distortion will be more prominent. In the extreme case, if the resonance frequencies of isolated eigenmodes are far enough apart, the eigenmodes can be completely decoupled, and the resonator element can no longer exhibit the expected composite eigenmode. In this extreme case, one or more sub-regions may act as an anchor at the maximum displacement point of the dominant eigenmode. This may result in large distortion from the pure eigenmode, which will lead to TCFN 总 Changing unpredictably.

[0107] Figures 8A - 8C Shows the increased modal distortion on the theoretical and simulated TCFN 总Effect on the consistency between. As the sizes of one or more sub-regions are scaled such that the resonant frequencies of the one or more sub-regions are no longer equal, modal distortion increases, and thus the difference between the temperature coefficient of the simulated frequency and the temperature coefficient of the frequency calculated analytically increases.

Claims

1. A MEMS resonator device, the device comprising: A support structure; A composite eigenmode resonator element comprising at least two sub-regions, wherein each of the at least two sub-regions in isolation is configured to resonate with a corresponding eigenmode from a set of at least two eigenmodes and has a corresponding temperature coefficient of frequency N (TCFN) k ), wherein the at least two sub-regions include a first sub-region and a second sub-region, wherein the isolated first sub-region is configured to resonate with a first eigenmode from the set of the at least two eigenmodes and has a specific dopant type, a specific doping concentration, a first orientation relative to a crystal axis, and a first TCFN k , wherein the isolated second sub-region is configured to resonate with a second eigenmode from the set of the at least two eigenmodes and has the specific dopant type, the specific doping concentration, the second orientation relative to the crystal axis, and the second TCFN k , and wherein, (i) the first TCFN k greater than the temperature coefficient of the desired Nth order frequency (TCFN) of the composite eigenmode resonator element. 期望 ) and the second TCFN k Less than TCFN 期望 , or (ii) the first TCFN k Less than TCFN 期望 And the second TCFN k Greater than TCFN 期望 ; At least one anchor that couples the composite eigenmode resonator element to the 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.

2. The device according to claim 1, wherein, The device is (i) capacitively transduced or (ii) piezoelectrically transduced.

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

4. The device according to claim 3, wherein, Each of the at least two sub-regions is connected to at least one other sub-region through a contact region, wherein the contact region is minimized such that modal distortion due to non-ideal coupling of the at least two sub-regions is minimized.

5. The device according to claim 4, wherein The contact region further includes a rod, and wherein the volume of the rod is less than 5% of the volume of the composite eigenmode resonator element.

6. The device according to claim 1, wherein, The at least two sub-regions further include a third sub-region, wherein the isolated third sub-region resonates with the first eigenmode and has the specific dopant type, the specific doping concentration, the first orientation relative to the crystal axis, and the first TCFN k .

7. The device according to claim 6, wherein, The at least two sub-regions further include a fourth sub-region and a fifth sub-region, wherein both the isolated fourth sub-region and the isolated fifth sub-region resonate with the first eigenmode and have the specific dopant type, the specific doping concentration, the first orientation with respect to the crystal axis, and the first TCFN k .

8. The apparatus according to claim 7, wherein, The at least two sub-regions further include a sixth sub-region, a seventh sub-region, an eighth sub-region, and a ninth sub-region, wherein the isolated sixth sub-region, the isolated seventh sub-region, the isolated eighth sub-region, and the isolated ninth sub-region resonate with the first eigenmode and have the specific dopant type, the specific dopant concentration, the first orientation with respect to the crystal axis, and the first TCFN k .

9. The device according to claim 1, wherein Is approximately equal to the isolated respective resonance frequencies of the respective eigenmodes from the set of at least two eigenmodes.

10. The device according to claim 1, wherein, The set of at least two eigenmodes includes two or more eigenmodes selected from the group consisting of (i) Lamé eigenmodes, (ii) face shear eigenmodes, (iii) square extension eigenmodes, (iv) width extension eigenmodes, (v) length extension eigenmodes, (vi) N-th order ring eigenmodes, (vii) breathing eigenmodes, (viii) wine glass eigenmodes, and (ix) higher order Lamé eigenmodes.

11. The device according to claim 1, wherein, Combine the TCFN of the at least two isolated sub-regions k such that the Nth order temperature coefficient (TCFN 总 ) of the frequency of the composite eigenmode resonator element is approximately equal to the TCFN 期望 .

12. The apparatus according to claim 1, wherein The N-th order includes the first order, and wherein, TCFN 期望 is approximately equal to 0 ppm / °C.

13. The device according to claim 1, wherein, The Nth order includes the second order, and wherein, TCFN 期望 is approximately equal to 0 ppm / °C 2 .

14. The device according to claim 1, wherein The Nth order includes the first order, and wherein, TCFN 期望 is a non-zero value that at least partially compensates for a third-order temperature coefficient of the frequency of the composite eigenmode resonator element.

15. The device according to claim 1, wherein, Each of the at least two sub-regions has the specific dopant type and the specific doping concentration.

16. The device according to claim 1, wherein, The composite eigenmode resonator element further includes one or more cavities in one or more of the at least two sub-regions.

17. The device according to claim 1, wherein The effective mass of the composite eigenmode resonator element has a difference of <10% from the sum of the effective masses of each of the at least two isolated sub-regions.

18. The apparatus according to claim 1, wherein, The device is configured to operate as (i) an oscillator or (ii) a resonant sensor.

19. The device according to claim 1, wherein, The device includes at least one of single-crystalline silicon, silicon carbide, polysilicon, quartz, graphene, and polycrystalline diamond.

20. A method for designing a MEMS resonator device, the method comprising: Selecting a desired temperature coefficient of the Nth order frequency (TCFN 期望 ) for a composite eigenmode resonator element, wherein the composite eigenmode resonator element includes at least two sub-regions, and each of the at least two sub-regions is configured to resonate with a corresponding eigenmode from a set of at least two eigenmodes and is configured to have a corresponding temperature coefficient of the Nth order frequency (TCFN k ); Provide a first sub-region of the at least two sub-regions, wherein the isolated first sub-region is configured to resonate with a first eigenmode from a set of the at least two eigenmodes, and has a specific dopant type, a specific doping concentration, and a first orientation relative to a crystal axis, and wherein the first sub-region has a first TCFN k ; Provide a second sub-region among the at least two sub-regions, wherein the isolated second sub-region is configured to resonate with a second eigenmode from the set of the at least two eigenmodes, and has the specific dopant type, the specific doping concentration, and a second orientation with respect to the crystal axis, and wherein the second sub-region has a second TCFN k ; For the composite eigenmode resonator element including the at least two sub-regions, determine the temperature coefficient of the N-th order frequency of (TCFN 总 ); and Adjust the TCFN 总 to make it approximately equal to the TCFN 期望 , Among them, the tuning TCFN 总 includes changing at least one of the first or second TCFN k such that (i) the first TCFN k is greater than TCFN 期望 and the second TCFN k is less than TCFN 期望 , or (ii) the first TCFN k is less than TCFN 期望 and the second TCFN k is greater than TCFN 期望 .

21. The method according to claim 20, wherein, Tuning the TCFN 总 such that it is approximately equal to the TCFN 期望 comprising one or more of the following: (i) performing effective mass tuning, (ii) performing resonant mode tuning, or (iii) adding a cavity to one or more of the at least two sub-regions such that the first or second TCFN is changed k at least one of which 22. The method according to claim 20, further comprising providing a third sub-region among the at least two sub-regions, wherein, The isolated third sub-region resonates with the first eigenmode and has the specific dopant type, the specific dopant concentration, the first orientation with respect to the crystal axis, and the first TCFN k .

23. The method according to claim 22, further comprising providing a fourth sub-region and a fifth sub-region in the at least two sub-regions, wherein, Both the isolated fourth sub-region and the isolated fifth sub-region resonate with the first eigenmode and have the specific dopant type, the specific dopant concentration, the first orientation with respect to the crystal axis, and the first TCFN k .

24. The method according to claim 23, further comprising providing a sixth sub-region, a seventh sub-region, an eighth sub-region, and a ninth sub-region in the at least two sub-regions, wherein, The isolated sixth sub-region, the isolated seventh sub-region, the isolated eighth sub-region, and the isolated ninth sub-region resonate with the first eigenmode and have the specific dopant type, the specific dopant concentration, the first orientation with respect to the crystal axis, and the first TCFN k .

25. The method according to claim 20, wherein, Is approximately equal to the isolated respective resonance frequencies of the respective eigenmodes from the set of at least two eigenmodes.

26. The method according to claim 20, wherein, The set of at least two eigenmodes includes two or more eigenmodes selected from the group consisting of (i) Lamé eigenmodes, (ii) face shear eigenmodes, (iii) square extension eigenmodes, (iv) width extension eigenmodes, (v) length extension eigenmodes, (vi) N-th order ring eigenmodes, (vii) breathing eigenmodes, (viii) wine glass eigenmodes, and (ix) higher order Lamé eigenmodes.

27. The method according to claim 20, wherein, Combine the TCFN of the at least two isolated sub-regions k so that the TCFN 总 is approximately equal to the TCFN 期望 .

28. The method according to claim 20, wherein The Nth order includes the first order, and wherein, TCFN 期望 is approximately equal to 0 ppm / °C.

29. The method according to claim 20, wherein The Nth order includes the second order, and wherein, TCFN 期望 is approximately equal to 0 ppm / °C 2 .

30. The method according to claim 20, wherein, The Nth order includes the first order, and wherein, TCFN 期望 is a non-zero value that at least partially compensates for the third-order temperature coefficient of the frequency of the composite eigenmode resonator element.

31. The device according to claim 20, wherein, Each of the at least two sub-regions has the specific dopant type and the specific doping concentration.

Images