Analysis of Frequency Stability and Thermoelastic Effects for Slotted Tuning Fork MEMS Resonators
Abstract
:1. Introduction
2. Mechanical and Thermal Properties of Single-Crystal Silicon
3. Analytical Model
3.1. Temperature Variation of Frequency
3.2. Temperature Coefficient of Quality Factor
4. Validation on the Real 3D Structure
- For a given level of doping and resonant mode type (e.g. bending-mode) the material orientation has a strong impact on and a clear minimum can be achieved. This value is essentially independent of the mode-order and geometric dimensions. The same minima are obtained analytically and numerically, although they might correspond to slightly different rotations of the material axes.
- The impact of material orientation on the Q value is minimal, and the rather low Q is an intrinsic limitation.
5. Optimization of the Tuning Fork Resonator
5.1. Covariance Matrix Adaptation Evolution Strategy Optimization
5.1.1. Q Maximization
5.1.2. Multi-Objective Function
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Doping Type | Concentration [cm−3] | c11 | c12 | c44 | ||||||
---|---|---|---|---|---|---|---|---|---|---|
dop-n | 3.00 × 1013 [26] | 165.64 | 63.94 | 79.51 | −63.4 | −78.7 | −55.4 | −35 | −56 | −7 |
dop-n | 1.98 × 1019 [26] | 163.94 | 64.77 | 79.19 | −39.2 | −116.2 | −58.7 | −118 | NaN | −28 |
P | 4.10 × 1019 [8] | 163 | 65.4 | 79.2 | −34.5 | −133.7 | −67.8 | −115 | 22 | −51 |
P | 4.66 × 1019 [8] | 162.5 | 65.7 | 79.1 | −32.5 | −131.8 | −68.7 | −110 | 18 | −43 |
P | 6.60 × 1019 [9] | 164 | 66.7 | 78.2 | −34.2 | −135.17 | −67.8 | −103.04 | −1.1 | −40.26 |
P | 7.47 × 1019 [8] | 161.4 | 66.1 | 78.5 | −30.7 | −134.9 | −71.9 | −78 | −12 | −31 |
As | 1.20 × 1019 [9] | 164.2 | 65.6 | 78.6 | −46.58 | −124.61 | −63.12 | −105.41 | 31.73 | −45.21 |
As | 1.66 × 1019 [8] | 164 | 64.3 | 79.5 | −48.5 | −114.7 | −63.7 | −111 | 25 | −58 |
As | 2.46 × 1019 [8] | 163.8 | 64.9 | 79.4 | −44.2 | −124.6 | −65.1 | −111 | 34 | −55 |
Sb | 1.30 × 1018 [9] | 165.6 | 64.4 | 79.3 | −65.5 | −85.08 | −60.92 | −67.85 | −28.1 | −52.81 |
L | 195 µm |
HB | 45 µm |
W | 20 µm |
LB | 34 µm |
t | 20 µm |
Geometry | Optimization Options | Results |
---|---|---|
= [110 3 73 195 20 34 0 45] 0.3 MHz < < 0.7 MHz R < W/2–2.5 µm Y −R > −HB + 2.5 µm Y + LH + R < L −2.5 µm | x = [81.86 14.94 92.05 191.36 34.88 69.95 2.034 68.71] fobj = −Q(@25 °C) = −237831.19 = 0.30 MHz = 1115.21 ppm | |
= [110 3 73 195 20 34 0 45] 0.4 MHz < < 0.6 MHz R < W/2–4 µm Y − R > −HB + 4 µm Y + LH + R < L −4 µm | x = [−7.27 10.37 64.18 155.44 28.75 66.82 0.09 69.11] fobj = −Q(@25 °C) = −82910.63 = 0.40 MHz = 936.86 ppm |
Geometry | Optimization Options | Results |
---|---|---|
= [110 3 73 195 20 34 0 45] 0.3 MHz < < 0.7 MHz R < W/2–4.5 µm Y −R > −HB + 4 µm Y + LH + R < L −4 µm | x = [73.69 11.23 122.39 229.05 32.44 37.35 13.32 51.97] Q(@25 °C) = 62534.74 = 0.31 MHz = 171.18 ppm | |
= [10 3 73 195 20 34 0 45] 0.3 MHz < < 0.7 MHz R < W/2–2.5 µm Y − R>−HB + 2.5 µm Y + LH + R < L −2.5 µm | x = [47.19 7.11 83.50 241.25 31.07 11.95 −12.996 93.35] Q(@25 °C) = 28164.73 = 0.45 MHz = 160.38 ppm | |
= [110 3 73 195 20 34 0 45] 0.4 MHz < < 0.6 MHz R < W/2–4.5 µm Y −R > −HB + 4 µm Y + LH + R < L −4 µm | x = [90.10 9.68 87.17 239.46 33.20 74.95 −12.834 60.95] Q(@25 °C) = 30955.97 = 0.44 MHz = 167.4 ppm |
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Zega, V.; Frangi, A.; Guercilena, A.; Gattere, G. Analysis of Frequency Stability and Thermoelastic Effects for Slotted Tuning Fork MEMS Resonators. Sensors 2018, 18, 2157. https://doi.org/10.3390/s18072157
Zega V, Frangi A, Guercilena A, Gattere G. Analysis of Frequency Stability and Thermoelastic Effects for Slotted Tuning Fork MEMS Resonators. Sensors. 2018; 18(7):2157. https://doi.org/10.3390/s18072157
Chicago/Turabian StyleZega, Valentina, Attilio Frangi, Andrea Guercilena, and Gabriele Gattere. 2018. "Analysis of Frequency Stability and Thermoelastic Effects for Slotted Tuning Fork MEMS Resonators" Sensors 18, no. 7: 2157. https://doi.org/10.3390/s18072157