High-Q-factor MEMS beam resonator with periodic ripple boundary

Abstract

The invention discloses a high-Q-factor MEMS (Micro Electro Mechanical System) beam resonator with a periodic ripple boundary, and belongs to the field of micro electro mechanical system resonator design. According to the structure, periodic corrugation modulation is introduced into the side wall of a traditional straight beam, a plurality of periodic corrugation units are symmetrically arranged on the boundary of the side wall of a beam body, a corrugation boundary contour curve is defined by a parameterization function, the modulation amplitude and period are adjustable, the beam width is continuously and periodically changed in the length direction, and therefore thermoelastic damping and anchor point loss are reduced at the same time. The corrugated boundaries are integrally and symmetrically arranged and can be directly processed and formed by an SOI (Silicon On Insulator) process, the structure manufacturability is high, and additional assembly is not needed. Under the driving of electrostatic force, the resonator forms more uniform strain distribution and a limited heat flow channel in fundamental mode vibration, so that the energy leakage is obviously weakened, and the resonance quality factor is further improved. The structure is suitable for high-Q-factor demand scenes such as high-sensitivity MEMS sensors, low-noise oscillators and microfilters.

Description

Technical Field

[0001] This invention relates to the field of microelectromechanical systems (MEMS) resonator design, and in particular to a high Q-factor MEMS beam resonator with periodic rippled boundaries. Background Technology

[0002] MEMS beam resonators are widely used in micro / nano devices such as precision sensors, low-noise oscillators, and RF filters, and their performance is mainly determined by the quality factor (Q factor). However, MEMS beams are generally limited by two main energy loss mechanisms in the operating frequency range: TED (Temperature-Delayed Energy) – irreversible heat flow caused by temperature gradients in the compression and tension regions, which is a key energy-consuming mechanism in microscale beam structures; and anchor point loss – the shear force and bending moment at the beam ends during resonance radiate mechanical energy to the substrate, causing energy leakage, which is an important external leakage mechanism limiting the Q value. Traditional strategies to improve the Q value include: reducing TED through slotting, transition regions, and cantilever thinning; and reducing anchor point energy leakage by changing the support structure and introducing phonon bandgap units. However, existing methods generally have the following problems: most designs only optimize for a single loss mechanism; the geometry is complex and difficult to fabricate; and there is limited optimization for the fundamental mode, which limits its application in high-stability applications.

[0003] Therefore, a novel beam resonator structure is needed that is simple in structure, can be directly implemented through planar corrugation design, and can simultaneously suppress TED and anchor point losses. Summary of the Invention

[0004] This invention aims to provide a MEMS beam resonator based on periodic corrugated boundaries. By introducing periodic continuous corrugations at the two side boundaries of the beam, the beam cross-section undergoes periodic changes along its length, thereby simultaneously achieving: perturbing the heat flow path, reducing the temperature gradient to suppress TED (thermal gradient oscillation); and reshaping the strain energy distribution, significantly reducing anchor region energy to suppress anchor point loss. Ultimately, the resonator can still significantly improve the Q factor in the fundamental mode while maintaining excellent manufacturability.

[0005] The present invention mainly includes the following technical solutions: A high Q-factor MEMS beam resonator with periodic rippled boundaries, comprising: Substrate; Anchor point areas at both ends; The vibrating beam located between two anchor points is a double-ended fixed beam with multiple periodic corrugated elements symmetrically arranged on its sidewalls. The corrugated profile curve is defined by the following parametric function: Where x is the coordinate along the beam length, with the origin located at the lower left end of the beam; x∈[0, L] μm; L is the total length of the beam; a is the modulation amplitude coefficient, representing the maximum geometric deviation of each parabolic element; b is the longitudinal scale of a corrugated element, i.e., the period length coefficient, which determines how many corrugations are distributed along the beam length.

[0006] The corrugated elements are continuously distributed along the length of the beam, causing the local width of the beam to change periodically, forming an integral non-uniform cross section.

[0007] The corrugated structure can rearrange the strain energy density under the fundamental mode vibration and significantly reduce the proportion of energy that can be propagated to the anchor point, thereby reducing anchor point loss. Because the periodic change in cross-section causes the heat flow path to be broken and the period to be interrupted, thermoelastic damping can be significantly reduced. The resonator operates in the in-plane bending fundamental mode; Through finite element simulation and parameter optimization, the modulation amplitude coefficient a and period length coefficient b can be designed according to the target frequency range to adapt to different frequency bands.

[0008] Preferably, the local width of the vibrating beam is... It varies periodically along its length, and the corrugated boundary profiles on both sides simultaneously satisfy: The cross-section passing through the midpoint of the beam's length is mirror symmetric. Furthermore, at any position x, the upper and lower boundaries are mirror-symmetric about the centerline of the beam.

[0009] Preferably, the vibrating beam is made of single-crystal silicon material and is prepared by SOI (Silicon-On-Insulator) process, and the thickness of the vibrating beam is preferably 25 μm.

[0010] As a preferred option, parameterized functions It is a non-negative function with continuous curvature.

[0011] Preferably, the total length L of the beam ranges from 400 μm to 1000 μm, and the beam width ranges from 8 μm to 25 μm; more preferably, the thickness T is 25 μm.

[0012] The above-mentioned high Q-factor MEMS beam resonator with periodic rippled boundary is used in MEMS sensors or oscillators. The resonator achieves a high Q-factor through its periodic rippled boundary design and is integrated into a high-sensitivity quality detection sensor or a low-phase-noise oscillator to improve detection sensitivity or reduce signal phase noise.

[0013] The beneficial effects of this invention are: The beam structure of this invention achieves efficient suppression of two major energy dissipation mechanisms—thermoelastic damping (TED) and anchor point loss—at the structural level through a periodic, continuous corrugated boundary defined by a single function. Specifically, the periodically varying cross-section disrupts the uniform heat conduction path, significantly reducing irreversible heat dissipation; simultaneously, the optimized strain energy distribution concentrates more vibrational energy at the beam center, reducing leakage to the substrate through the anchor points. This synergistic effect results in a significant improvement in the Q factor of the resonator in a vacuum environment. Compared with the measured results of a traditional straight beam of uniform width, the structure of this invention, under the same size and manufacturing conditions, can improve the fundamental mode quality factor by approximately 7 times, achieving a measured Q value of 47264, demonstrating significantly enhanced performance. It achieves fundamental mode optimization and obtains high Q without the need for higher-order mode coupling. Furthermore, the continuous curvature of this function avoids stress concentration, improves device reliability and manufacturing yield, and is suitable for various photolithography processes and SOI micromachining. Its modulation amplitude and period parameters have clear adjustability, providing clear engineering design freedom for optimizing performance for different operating frequencies. This makes the resonator structure very suitable for applications such as micro-nano sensors, low phase noise oscillators, and RF filters with extremely high quality factor requirements. Attached Figure Description

[0014] Figure 1 This is the geometric modeling design diagram of the present invention. The sub-diagrams, from left to right, show: a one-dimensional modulation function Y, a two-dimensional beam layout with corrugated edges, and the corresponding three-dimensional structural model.

[0015] Figure 2 This is a three-dimensional finite element simulation model of the periodic ripple MEMS resonator of this invention.

[0016] Figure 3 The figures show a comparison of finite element simulations of the fundamental mode strain energy, TED dissipation density per unit volume, and temperature distribution of the resonator.

[0017] Figure 4 This is a schematic diagram of the fabrication of the experimental device of the present invention. Detailed Implementation

[0018] The present invention will be further described and illustrated below with reference to specific embodiments. The embodiments described are merely examples of the content of this disclosure and do not limit the scope of the invention. The technical features of each embodiment in the present invention can be combined accordingly, provided that there is no mutual conflict.

[0019] This invention provides a high Q-factor MEMS beam resonator based on a periodic corrugated boundary, which aims to significantly suppress thermoelastic damping and anchor point loss by optimizing the beam boundary shape, thereby greatly improving its quality factor (Q factor).

[0020] like Figure 1 and Figure 2 As shown, the resonator is integrally fabricated using bulk silicon micromachining technology based on SOI wafers. The silicon beam thickness and electrode gap are fixed by the process, achieving high geometric accuracy and symmetry without post-assembly. Its main structure includes: Substrate: Serves as the mechanical support for the device.

[0021] Anchor point area: Two anchor point areas are fixed on the substrate to fix the beam and realize electrical connection.

[0022] A vibrating beam with periodically corrugated sidewalls: This is a double-ended fixed beam, with each end rigidly connected to two anchor points. The beam material is (100) oriented monocrystalline silicon, and the device layer thickness is 25 μm. The core feature of the beam is that its sidewall boundaries are not traditionally straight, but are designed as periodically corrugated structures with a specific mathematical form. The whole structure consists of sidewalls modulated by a periodic function and a central working section. The periodic corrugations are continuously distributed along the beam length. Through the joint control of amplitude and period length, the structure forms a more uniform strain distribution and a confined heat flow path in the fundamental mode vibration, thereby simultaneously reducing thermoelastic damping and anchor point energy leakage.

[0023] Figure 1 In the diagram, the blue areas at both ends of the beam represent the deliberately removed corrugated sections. All designs use a uniform thickness of 25 μm, and the dimensions shown here correspond to an 800 × 16 μm beam. The leftmost function curve represents the removal of the same corrugated shape used in the end sections (highlighted in blue), demonstrating the local waveform geometry applied to the beam boundaries.

[0024] The design of the boundary profiles on both sides (upper and lower edges) of the vibrating beam in this invention is strictly defined by a parameterized periodic function: Where x is the coordinate along the beam length, with the origin located at the lower left end of the beam; x∈[0, L] μm; L is the total length of the beam; a is the modulation amplitude coefficient, representing the maximum geometric deviation of each parabolic element; b is the longitudinal scale of a corrugated element, i.e., the period length coefficient, which determines how many corrugations are distributed along the beam length. For the damping characteristics of the bending mode, a finite element simulation combined with parameter scanning is used to optimize the periodic corrugation amplitude and period length, so that the structure obtains the highest Q factor while maintaining manufacturing feasibility. In this embodiment, for a beam 800 μm long and 16 μm wide, a is 0.0026 and b is 4. During this process, in order to accurately verify the suppression effect of the structure of this invention on anchor point loss, a three-dimensional finite element simulation model as shown in Figure 2 was constructed. In this simulation model, a perfect matching layer (PML) is set below the substrate. It should be noted that PML ( Figure 2The blue section is not the physical structure layer in the actual manufacturing of the device, but rather a virtual boundary condition in the simulation calculation. It is used to simulate a semi-infinite bulk silicon environment, capable of absorbing the elastic wave energy transmitted from the vibrating beam through the anchor point to the substrate without reflection, thereby accurately calculating the Q factor caused by the anchor point loss.

[0025] Define the x-axis along the length of the beam (where x=0 and x=L correspond to the left and right ends of the beam, respectively, and L is the total length of the beam, which can be designed within the range of 400μm to 1000μm). The local width w(x) of the beam is given by the following function: All width units are in micrometers (μm). The beam width w(x) is a periodic function of position x with a period of 100 μm. Within one period, the width varies smoothly and continuously between a maximum value of 16 μm and a minimum value of 3 μm, forming a regular wavy profile.

[0026] This function satisfies strict symmetry conditions. First, at any position x, the upper and lower boundaries are strictly mirror-symmetric about the longitudinal centerline of the beam, meaning the width change is synchronous from top to bottom. Second, the entire corrugated profile is strictly mirror-symmetric about the midpoint of the beam's length direction (x = L / 2), meaning the function w(x) is an even function about the midpoint, ensuring the symmetry of the strain energy distribution. The continuous first derivative of this function guarantees a smooth corrugated profile curvature without sharp corners or abrupt changes, effectively avoiding stress concentration and improving the mechanical reliability and process yield of the device.

[0027] The periodically varying cross-section effectively disrupts the uniform and efficient heat conduction path in traditional straight beams. For example... Figure 3 Simulation results show that, compared to traditional straight beams, the corrugated structure results in a more uniform distribution of strain energy, migrating it from the anchorage area towards the beam center, and a significant reduction in the overall average strain energy density. This indicates a decrease in the energy storage ratio in the anchorage area, suppression of energy coupling and radiation at the anchor points, and a corresponding reduction in anchor loss. The corrugated structure effectively weakens the overall temperature gradient of the beam and reduces the thermoelastic energy dissipation density, indicating that thermoelastic damping is simultaneously weakened. In summary, the periodic corrugated boundary design of this invention can simultaneously suppress anchor loss and thermoelastic damping within a single structure, thereby significantly improving the overall quality factor of the resonator.

[0028] The structure of this invention is applicable to SOI-MEMS fabrication platforms. (Refer to...) Figure 4 The resonator can be manufactured and processed using standard SOI (Silicon-on-insulator) technology, and the main steps include: (a) Standardized cleaning of SOI wafers.

[0029] (b) Silicon doping. Thin film preparation: A PSG (phosphosilicate glass) sacrificial layer is prepared on the top silicon layer using chemical vapor deposition as a phosphorus doping pretreatment; doping: The wafer is heated to 1050 °C. ◦ Annealing at C for 3 hours, followed by phosphorus ion implantation, and finally removal of the PSG layer by wet etching.

[0030] (c) Metal stripping. Photolithography: A layer of negative resist is uniformly applied to the wafer surface using a spin coater, followed by exposure and development; Coating: 20nm chromium and 500nm gold are deposited on the photoresist surface using an electron beam evaporation coating process (chromium is used to increase the adhesion between gold and silicon); Stripping: Finally, a stripping process is performed to create the gold electrode.

[0031] (d) Patterning the top silicon layer. Photolithography, resist coating, pre-baking, exposure, development, and post-baking are performed to create a dry etching mask layer; DRIE etching of the top silicon layer is performed, using deep reactive ion etching (DRIE) to etch the top silicon layer down to the oxide layer to create the top silicon pattern, and then the resist is removed.

[0032] (e) Patterning the underlying silicon. Etching pretreatment: First, a top silicon protective layer is pretreated, then a photoresist layer is applied to the back of the wafer, followed by exposure and development; DRIE etching of the underlying silicon: Deep reactive ion etching (DRIE) is used to etch the underlying silicon down to the oxide layer to create the underlying silicon pattern, the photoresist layer is removed, and finally the oxide layer is removed using a wet etching process.

[0033] (f) Structural release. The front protective material is etched away using a dry etching process, and then the exposed oxide layer is removed using vaporized HF.

[0034] In a vacuum environment, resonant excitation is achieved through capacitive drive and detection electrodes, with the resonator operating in the fundamental bending mode. Experiments show that the quality factor of the structure of this invention is about 7 times higher than that of a straight beam of the same size, and the Q value reaches 47264, demonstrating significantly better performance than traditional straight beam structures.

[0035] Thanks to its extremely high Q factor, the MEMS beam resonator of this invention is particularly suitable for micro / nano electromechanical systems with extremely high requirements for frequency stability, phase noise and sensitivity. It can be directly used as a core sensing or frequency reference unit and integrated into devices that require a high Q factor, such as micro sensors, micro oscillators and filters.

[0036] The above-described embodiments are merely illustrative of several implementations of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the invention. Those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention.

Claims

1. A high Q-factor MEMS beam resonator with periodic corrugated boundaries, characterized in that, include: Substrate; Anchor point areas at both ends; The vibrating beam located between two anchor points is a double-ended fixed beam with multiple periodic corrugated elements symmetrically arranged on its sidewalls. The corrugated boundary profile curve is defined by the following parametric function: ； Where x is the coordinate along the beam length direction, x∈[0, L] μm, L is the total length of the beam, a is the modulation amplitude coefficient, and b is the longitudinal dimension of a corrugated unit, i.e. the period length coefficient; The corrugated elements are continuously distributed along the length of the beam, causing the local width of the beam to change periodically.

2. The high Q-factor MEMS beam resonator with periodic corrugated boundaries according to claim 1, characterized in that, The local width of the vibrating beam is It varies periodically along its length, and the corrugated boundary profiles on both sides simultaneously satisfy: The cross-section passing through the midpoint of the beam's length is mirror symmetric. Furthermore, at any position x, the upper and lower boundaries are mirror-symmetric about the centerline of the beam.

3. The high Q-factor MEMS beam resonator with periodic corrugated boundaries according to claim 1, characterized in that, The vibrating beam is made of single-crystal silicon material and is prepared using the SOI process.

4. The high Q-factor MEMS beam resonator with periodic corrugated boundaries according to claim 1, characterized in that, Parameterized functions It is a non-negative function with continuous curvature.

5. The high Q-factor MEMS beam resonator with periodic corrugated boundaries according to claim 1, characterized in that, The total length L of the beam ranges from 400μm to 1000μm, and the modulation amplitude coefficient a and the period length coefficient b are adjustable to adapt to different frequency band designs.

6. The application of the high Q-factor MEMS beam resonator with periodic rippled boundaries as described in any one of claims 1-5 in microelectromechanical system sensors or oscillators, characterized in that, The resonator achieves a high Q factor through its periodic ripple boundary design and is integrated into a high-sensitivity quality detection sensor or a low-phase-noise oscillator to improve detection sensitivity or reduce signal phase noise.

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