Estimating Temperature-Dependent Glass Substrate CTE via Frequency Measurements in a MEMS Gyroscope

Estimating Temperature-Dependent Glass Substrate CTE via Frequency Measurements in a MEMS Gyroscope Mehran Hosseini-Pishrobat1 and Erdinc Tatar1, 2 Emails: [email protected] [email protected] 1Department of Electrical and Electronics Engineering 2National Nanotechnology Research Center Bilkent University, Ankara, Turkey Abstract—We present a novel physics-based optimal parameter estimation framework to extract the coefficient of thermal expansion (CTE) of the glass substrate from temperaturefrequency measurements in a MEMS gyroscope. We set forth a parametrization of the gyroscope’s stiffness that distinguishes stress-induced components from other terms. This formulation enables an inverse problem approach to identify the substrate CTE through stiffness perturbation caused by the anchors’ thermal expansion. As the average estimate from two temperature tests, we obtain the glass’s CTE as 3.0510 + 0.0018 (T-20) - 1.4393×10-5 (T-20)2 (ppm/°C) for T ∈[20,100]°C. Compared to silicon, glass has a larger CTE with significantly less temperature variation (1% versus Si’s 20% over 20-100°C). Keywords—Silicon-on-glass, Coefficient of thermal expansion, MEMS gyroscope, Thermal stresses I. INTRODUCTION The CTE mismatch between different constituent material layers is a critical factor that can adversely affect the performance of MEMS devices by inducing thermal stresses [1]. In gyroscopes, such stresses perturb the ideal stiffness distribution, giving rise to frequency drifts, phase errors, and undesired mode couplings [2]. Although the temperature dependence of the silicon’s CTE has been studied in detail [3], the thermal behavior of CTE of the common glass substrates remains largely unexamined. The datasheets often report an average value (e.g., [4] gives the value 3.25ppm/°C for BOROFLOAT over 20-300°C). However, differentiating the temperature sensitivities of CTEs of silicon and glass is essential for accurate modeling of small quantities, such as stiffness asymmetry or temperature-frequency hysteresis [5]. A notable exception in the literature is Ref. [6], which presented experimental data for the Borosilicate glass CTE over 130800K. We work with a ring gyroscope (3.2mm-diameter, 58kHz, operating at 𝑛=2 degenerate modes) whose SEM is shown in Fig. 1-(a). The device is fabricated by a silicon-on-glass (SOG) process using (111) wafers followed by wafer-level vacuum packaging. Sixteen pairs of electrodes are placed around the outer ring for the drive, sense and control purposes. Moreover, a symmetric array of 16 capacitive stress sensors is distributed inside and outside the ring gyroscope (Fig. 1-(b)). These sensors pick up the mechanical stresses (i.e., the total stress minus the free thermal expansion) at the substrate level. This task is achieved by using a bridge-type mechanical structure that converts and amplifies 𝑥-direction displacements in the 𝑦direction set by the imbalance angle (Δ𝑦/Δ𝑥 =1/tan (𝛼), see Fig. 1-(c)) [7]. Capitalizing on the recently developed analytical model [5], we present an optimal estimation framework where the obtained fitting coefficients represent physical quantities such as stiffness and CTE. Extracting these key parameters is instrumental in understanding the underlying physics of temperature effects. The method presented here attempts to bridge the gap between pure data-driven [7] and analytical modeling [5] approaches to performance improvement of MEMS gyroscopes since, in the former, the repeatability of fitting coefficients across different tests is not guaranteed. II. MODELING TEMPERATURE EFFECTS In this section, we give a brief overview of the mathematical model of temperature effects on stiffness. Our modeling is based on the key observation that, under temperature variation Δ𝑇= 𝑇−𝑇 ( 𝑇 is the initial stress-free temperature), the total displacement field of the gyroscope is the superposition of three states [2]: 1. (State I) Nominal 2θ vibration of the wineglass modes, which occurs in a fast time scale; 2. (State II) Thermal expansion/contraction of the moving structure (two rings and 16 beams) while it is fixed at the innermost boundary; 3. (State III) The moving structure is pushed/pulled due to thermal deformation of the anchored internal structure. States II and III have a much slower time scale than State I. This superposition, visualized in Fig. 2, helps us differentiate various temperature-induced stiffness types. Accordingly, we decompose the gyroscope’s total stiffness as the sum of four components: This work was supported by the European Union’s European Research Council (ERC) Starting Grant under the grant agreement 101116162 – 0drift – ERC-2023STG. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of European Union. Neither the European Union nor the granting authority can be held responsible for them. 2025 IEEE International Symposium on Inertial Sensors and Systems (INERTIAL) | 979-8-3503-8932-6/25/$31.00 ©2025 IEEE | DOI: 10.1109/INERTIAL63280.2025.11037086 Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on November 05,2025 at 13:16:43 UTC from IEEE Xplore. Restrictions apply. 1. Nominal mechanical stiffness (𝑘) corresponding to State I; 2. Thermal stiffness (𝑘) corresponding to State II; this term is due to thermal stresses in the moving structure itself; 3. Stress stiffness (𝑘) corresponding to State III; this term accounts for the effect of anchored internal region on the moving structure; 4. Electrostatic ( 𝑘) and Duffing ( 𝑘 ) softening; temperature affects this type of stiffness via gap variations. The drive mode’s stiffness is therefore given by 𝑘  = 𝐸 ( 𝑇 ) 𝐸 ( 𝑇  ) 𝑘  + 𝐸 ( 𝑇 ) 𝐸 ( 𝑇  ) 𝜀  ( 𝑇 ) 𝜀  ( 𝑇  ) 𝑘  + 𝐸(𝑇) 𝐸 ( 𝑇  ) 𝑘   Δ Δ      − 𝑘  − 𝑘  , (1) where 𝐸(𝑇) is the temperature-dependent Young’s modulus, 𝜀(𝑇)=∫𝛼(𝑠)d𝑠   is the thermal strain at the silicon layer. We calculate 𝑘 for the unit displacement Δ=10nm across the [0,45°] portion of the ring and then scale it with the beam displacements, Δ. Substrate’s CTE, 𝛼 enters the above equation through the displacements Δ, 𝑖 =1:8, of the inner (supporting) beams. As explained in Fig. 2, Δ’s given by Δ  = 𝑅   𝛼  ( 𝑠 ) d 𝑠   󰆊 󰆎 󰆎 󰆎 󰆎 󰆎 󰆋 󰆎 󰆎 󰆎 󰆎 󰆎 󰆌  (  ) + Δ 𝑅  𝛼  ( 𝑠 ) d 𝑠   󰆊 󰆎 󰆎 󰆎 󰆎 󰆋 󰆎 󰆎 󰆎 󰆎 󰆌  (  ) + Δ     (  ) (2) are made up of three parts: 1. Displacements due to anchors’ thermal expansion evaluated by the glass’ CTE; 2. Displacement due to anchor-to-beams expansion evaluated by the silicon’s CTE; 3. Displacements due to mechanical strains that mainly stem from die-attach epoxy; we obtain these terms by performing an interpolation process on the stress sensors’ measured data [8]. We account for the temperature dependency of the silicon’s material properties by adopting the following models for the temperature range [20,100]°C [3], [9]: Fig. 1. Ring gyroscope’s SEM (a); arrangement (b) and working principle (c) of stress sensors. Fig. 2. Superposition basis for parametrizing the drive mode’s stiffness and forming a regression model for glass CTE estimation. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on November 05,2025 at 13:16:43 UTC from IEEE Xplore. Restrictions apply. 𝐸 ( 𝑇 ) = 170  1 − 60 × 10   ( 𝑇 − 20 )  [ GPa ] , 𝛼  ( 𝑇 ) = 2 . 3332 + 9 . 2618 × 10   𝑇 − 1 . 8901 × 10   𝑇  [ ppm / ∘ C ] . (3) A. Parameter estimation Based on Equation (1) and assuming the polynomial form 𝛼  ( 𝑇 ) ≈ 𝑎  + 𝑎  ( 𝑇 − 20 ) + 𝑎  ( 𝑇 − 20 )  (4) for the glass CTE, we form the regression model 𝑦 [ 𝑛 ] = 𝜙  [ 𝑛 ] 𝜃 + 𝜖 [ 𝑛 ] , 𝑛 = 0 , 1 , 2 , … (5) to estimate the unknown parameters 𝜃=[𝑘,𝑘,𝑘,𝑘𝑎,𝑘𝑎,𝑘𝑎] using recursive leastsquares (RLS) [10]. A suitable initialization is crucial for the estimates to converge to physically meaningful values as we use a single equation for six unknowns. Hence, we set the analytical modeling-based stiffness values [𝑘,𝑘,𝑘]=[7×10,−0.6,0.2] (N/m) along with the condition 𝛼(20∘𝐶)=3.25 (ppm/°C) (that is, the datasheet value [4]) as the initial conditions of the RLS algorithm. In the regression model (5), we calculate the electrostatic/Duffing stiffness using the fixed glass CTE 3.25 (ppm/°C). As a result, the error signal 𝜖[.] in (5) has nonstationary characteristics. To alleviate the consequent errors, we resort to an adaptive forgetting factor technique in the RLS algorithm that emphasizes the most recent data whenever the estimation error increases. We note that RLS offers several advantages over usual batch least-squares [10]: 1) adaptation and faster response to any changes or perturbations affecting the estimation model, 2) enhanced robustness against nonstationary noises, 3) memory and computational efficiency that makes it a better choice when real-time implementation is considered. III. RESULTS We expose our gyroscope to controlled temperature cycles using the setup in Fig. 3. The daughter board contains the MEMS die and an integrated PCB heater. The temperature sensor (PTAT) and gyro front-ends are positioned underneath the daughter board. This configuration enables heating the MEMS die while minimizing heat exposure of the electronics. Fig. 3. Test setup: the on-PCB heater targets beneath the MEMS die for temperature cycling; a PTAT under the daughter board measures temperature. Fig. 4. Temperature-frequency measurements for two reported tests. Fig. 5. Convergence of RLS parameter estimation for Test#1. Fig. 6. Estimated substrate CTE from Tests#1 and 2 for 20-100°C. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on November 05,2025 at 13:16:43 UTC from IEEE Xplore. Restrictions apply. The outputs of the stress sensors are AC modulated and multiplexed. The control loops are all implemented in a Zurich instruments digital lock-in amplifier. During testing, we record the temperature, stress sensor outputs, and drive mode frequency at a sampling rate of 1 Hz. We here report the data for two tests whose temperature and frequency profiles are shown in Fig. 4. We implemented the estimation model and RLS algorithm in MATLAB. The measured temperature and stress data were used to construct the regressor vector 𝜙[.] in Equation (5). Strain/displacement interpolation was used to calculate the inner beams’ displacements as well as gap variations. The measured frequency along with electrostatic/Duffing stiffness was used to obtain 𝑦[.] in Equation (5). The time trajectories and convergence of the RLS parameter estimation results for Test#1 are depicted in Fig. 5. The largest settling time for the convergence is about 50% of the samples. The estimated glass CTEs for the two tests are plotted in Fig. 6 for the temperature range [20,100]°C. The difference between the two results is attributable to the non-repeatable die-attach properties and our limited capability to capture substrate strain. As the average of these two estimates, we have 𝛼  ( 𝑇 ) = 3 . 0510 + 0 . 0018 ( 𝑇 − 20 ) − 1 . 4393 × 10   ( 𝑇 − 20 )  ( ppm / ∘ C ) . (6) Borosilicate CTE reported by [6] is also provided as a comparison reference. Our results have a DC error of less than 3% with respect to this reference. The slopes and curvatures, however, are very similar and noticeably different from those of the silicon’s CTE. For example, the CTE of silicon varies by more than 20% over 20-100°C, while for the glass, this variation is about 1%. The estimated drive mode frequencies (reconstructed using the estimated parameters) are compared with the actual experimental observations in Fig. 7. The estimated curves follow the experimental one, successfully capturing the hysteresis loops. We noticed that distinguishing the temperature variations of the CTEs of silicon and glass is the key to correctly modeling the shapes of these hysteresis loops. IV. CONCLUSION By leveraging an analytical temperature effects model, we presented a method to estimate the temperature-dependent glass substrate CTE in a MEMS gyroscope. As a function of temperature, the glass CTE has a distinctly different slope and curvature compared to silicon. The larger variation of silicon’s CTE is pertinent not only to undesired mismatch-induced stresses, but also contributes to higher temperature sensitivity of thermoelastic damping. As a future direction, we will examine the potential of our approach for calibration and performance improvement of MEMS gyroscopes. Unlike conventional inputoutput data fitting, the parameters estimated by this method have well-understood physical meaning (that is, thermally induced stiffness and CTE-temperature coefficients). Theoretical aspects, such as the persistency of excitation and guarantees for convergence to the true parameter values, require further investigation. REFERENCES [1] D. Erkan and E. Tatar, “Improving the Temperature Stability of MEMS Gyroscope Bias with on-chip Stress Sensors,” in 2024 IEEE International Symposium on Inertial Sensors and Systems (INERTIAL), Mar. 2024, pp. 1–4. [2] M. Hosseini-Pishrobat, D. Erkan, and E. Tatar, “On Temperature Effects in a MEMS Ring Gyroscope,” in 2024 IEEE International Symposium on Inertial Sensors and Systems (INERTIAL), Mar. 2024, pp. 1–4. [3] Y. Okada and Y. Tokumaru, “Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500 K,” J. Appl. Phys., vol. 56, no. 2, pp. 314–320, Jul. 1984. [4] “SCHOTT BOROFLOAT® technical details.” Accessed: Jan. 23, 2024. Available: https://www.schott.com/en-gb/products/borofloatP1000314/technical-details [5] M. Hosseini-Pishrobat and E. Tatar, “Modeling Temperature Effects in a MEMS Ring Gyroscope: Toward Physics-Aware Drift Compensation,” J. Microelectromechanical Syst., pp. 1–0, 2025. [6] L. S. Sinev and I. D. Petrov, “Linear Thermal Expansion Coefficient (at Temperatures from 130 to 800 K) of Borosilicate Glasses Suitable for Silicon Compounds in Microelectronics,” Glass Ceram., vol. 73, no. 1, pp. 32–35, May 2016. [7] B. E. Uzunoglu, D. Erkan, and E. Tatar, “A Ring Gyroscope With OnChip Capacitive Stress Compensation,” J. Microelectromechanical Syst., pp. 1–12, 2022. [8] M. Hosseini-Pishrobat, D. Erkan, and E. Tatar, “Analytical and experimental study of stress effects in a MEMS ring gyroscope,” Sens. Actuators Phys., vol. 362, p. 114639, Nov. 2023. [9] M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s Modulus of Silicon?,” J. Microelectromechanical Syst., vol. 19, no. 2, pp. 229–238, Apr. 2010. [10] P. A. Ioannou and B. Fidan, Adaptive control tutorial. in Advances in design and control, no. 11. Philadelphia, Pa: Society for Industrial and Applied Mathematics, 2006. Fig. 7. Hysteresis loops in frequency variations against temperature; results obtained using estimated parameters (glass CTE and stiffness) are compared with the experimental values. RLS algorithm attempts to minimize the error between the estimated and the experimental. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on November 05,2025 at 13:16:43 UTC from IEEE Xplore. Restrictions apply.