Thermoelastic damping in MEMS gyroscopes at high frequencies

MEMS gyroscopes are critical in consumer and automotive electronics, where reliability under harsh conditions (−40 °C to +120 °C, external vibrations) is essential. Historically, robustness focused on frequencies up to tens of kHz, but modern applications demand resilience against large external loads at much higher frequencies, where high-frequency eigenmodes can strongly influence sensor response. The quality factor of these modes is a key determinant of sensitivity. While low-frequency modes at millibar pressures are known to be limited by gas damping, the damping of high-frequency modes in MEMS gyroscopes has not been exhaustively studied. This work investigates whether thermoelastic damping (TED) significantly limits the quality factors of high-frequency modes and assesses whether a finite-element-based approach can accurately predict TED across temperature, enabling predictive design of robust MEMS gyroscopes.

Dominant damping mechanisms in polysilicon MEMS resonators include gas damping, thermoelastic damping (TED), and anchor losses; material losses in silicon are typically negligible due to its linear behavior, and surface losses are more relevant in nanoresonators. Akhiezer damping generally becomes relevant only above ~10 MHz and for very high Q·f products. Classical TED theory originates from Zener’s analytic treatment for beam bending modes, refined by Lifshitz and Roukes. Prior works have used FEM to obtain TED via complex eigenvalue problems or energy-based approaches. Gas damping trends: in the molecular regime Q ∝ p, transitioning to viscous regime where dissipation scales as p^−1/2; at low uncontrolled pressures, gas damping scales with temperature as approximately T0^(−3/2). The gap in the literature is an exhaustive assessment of high-frequency mode damping in industrial MEMS gyroscopes and a computationally efficient method suitable for large-scale models.

Experimental: Two industrial three-axis polysilicon MEMS gyroscope designs (A and B) were characterized via scanning laser Doppler vibrometry (SLDV). Design A (unencapsulated chip) was measured at 1 mbar and 25 °C in vacuum, with out-of-plane excitation via a piezo-shaker using a chirp from 10 kHz to 2 MHz; a 1D SLDV measured out-of-plane response on ~400-point grids, identifying modes by comparison with simulated shapes. Design B (on-wafer, unencapsulated) was excited electrostatically via comb-drive and multiple electrode pairs using a pseudo-random broadband signal from 30 kHz to 1.25 MHz; a 3D SLDV measured in-plane and out-of-plane modes (∼100-point grid for out-of-plane; ∼30-point targeted grids for individual in-plane modes). Temperature and pressure were controlled using a thermal chuck and vacuum chamber. Quality factors (Q) were extracted from resonance linewidths with ~1 Hz frequency resolution. Gas damping was simulated with a Bosch internal molecular-flow tool (COMSOL-based) for comparison. Numerical analysis: Starting from linear thermoelastic continuum equations (momentum balance with thermal expansion, linearized heat equation with Fourier conduction), the authors derived global FEM matrices for mechanics and heat (M, Ku, Kθ; C, KθT, KuT). Assuming harmonic response at frequency ω, the heat equation is eliminated to yield an effective mechanical equation of motion with frequency-dependent damping and stiffness contributions from thermoelastic coupling. A modal reduced-order model (ROM) is formed from mass-normalized mechanical eigenmodes. Intermodal coupling terms and small stiffness shifts from thermoelasticity are neglected, focusing on diagonal modal damping terms. The reciprocal TED quality factor for mode n at its eigenfrequency ωn is evaluated efficiently via a symmetric linear system of size equal to thermal DOFs: QTED,n^−1 = Re{ Φnᵀ (Kθᵀ + i ωn C)^−1 (KθT) Φn } / ωn (as presented in the paper’s Eq. (16) formulation). The FEM assembly and TED computation were implemented in self-written Matlab code. Material temperature dependencies considered for TED include α, κ, and Cv; E, ν, and ρ were treated as weakly temperature dependent and neglected for TED evaluation.

• For design A (out-of-plane modes up to 1.8 MHz, 1 mbar, 25 °C), measured quality factors saturate at higher eigenfrequencies, in contrast to gas-damping simulations that increase approximately linearly with frequency. The low-frequency regime up to ~200 kHz shows good agreement with gas-damping predictions, validating the gas model there and indicating an additional damping mechanism at higher frequencies.
• Gas damping simulations (COMSOL-based molecular flow) match measurements at low frequencies; divergence at higher frequencies motivates inclusion of TED.
• For design B, Q vs pressure at 20 °C for seven modes (a–g) shows linear reciprocal-Q dependence on pressure (Q^−1 = m p + b), confirming molecular flow regime. Pressure dependence becomes very small below 10^−2 mbar; subsequent measurements at 10^−3 mbar ensure negligible gas damping, isolating TED and anchor losses.
• Seven modes in design B, spanning simulated eigenfrequencies from 118.98 kHz (mode a) to 733.46 kHz (mode g), were unambiguously identified via mode shapes. Both in-plane (2 modes) and out-of-plane (5 modes) were measured.
• The proposed FEM-ROM method yields TED quality factors that agree well with measured Q across the investigated high-frequency range and temperatures, supporting TED as a dominant damping source for high-frequency modes in these gyroscopes.
• The approach enables extraction of temperature-dependent thermoelastic material parameters for polycrystalline silicon from vacuum Q(T) measurements.
• Practical measurement details: frequency resolution ~1 Hz; typical operational pressure for gyros ~1 mbar; excitation signals: chirp (10 kHz–2 MHz) and pseudo-random (30 kHz–1.25 MHz).

The study addresses whether thermoelastic damping limits quality factors of high-frequency modes in MEMS gyroscopes. Experimental observations show saturation of Q at high frequencies that cannot be explained by gas damping alone, while simulations incorporating TED reproduce the measurements, demonstrating that TED is a significant, often limiting, mechanism for high-frequency mode damping. By performing pressure sweeps (confirming molecular regime) and measuring at ultra-low pressures (10^−3 mbar), the authors isolate TED (and anchor losses) from gas damping and verify the temperature-dependent behavior predicted by the thermoelastic model. The derived FEM-based ROM method provides an efficient and predictive tool for estimating TED in complex industrial-scale MEMS structures, enabling designers to anticipate high-frequency robustness and optimize structures against vibration-induced performance degradation. The results reinforce the importance of including TED in the design and verification of MEMS gyroscopes operating under demanding environmental conditions.

This work demonstrates that thermoelastic damping is a key contributor to the damping of high-frequency modes in industrial MEMS gyroscopes. The authors develop an efficient FEM-based reduced-order modeling framework that eliminates the heat equation to yield a frequency-dependent mechanical damping operator, enabling accurate TED quality factor prediction via a symmetric linear solve per mode. Measurements on two gyroscope designs, combined with gas damping simulations, show that gas damping dominates at lower frequencies but cannot explain high-frequency behavior, whereas the TED model matches the observed Q, confirming TED’s significance. The method is generic and applicable to a wide range of micro- and nanomechanical structures, supporting predictive MEMS design. Potential future research directions include: extending the framework to quantify and separate anchor losses more precisely; incorporating full intermodal thermoelastic coupling and stiffness shifts; validating and calibrating temperature-dependent material properties over broader temperature ranges and materials; and exploring regimes where Akhiezer damping or surface effects might contribute at higher frequencies or in structures with very high Q·f products.

• Assumes isotropic linear material behavior for polycrystalline silicon; anisotropy and nonlinearities are neglected.
• Other damping mechanisms (e.g., material, surface, Akhiezer below ~10 MHz) are assumed negligible; temperature- and pressure-independent losses are lumped into anchor losses and not separately identified.
• Intermodal thermoelastic coupling and thermoelastic-induced stiffness shifts are neglected in the ROM (diagonal approximation), which may introduce small errors.
• Boundary conditions assume fixed supports and insulating thermal boundaries on free surfaces; real device boundary conditions may deviate.
• Mode identification for design A used semi-automatic matching of out-of-plane modes, noted as prone to errors; design B only allowed excitation/measurement of a limited set of modes due to electrode placement.
• Temperature dependence of E, ν, and ρ is neglected for TED evaluation, potentially introducing minor inaccuracies at temperature extremes.