# Analysis of Frequency Stability and Thermoelastic Effects for Slotted Tuning Fork MEMS Resonators

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{20}cm

^{−3}(see e.g., [3,11]). Alternative approaches have been put forward in the literature such as temperature compensation methods that utilize either a tri-mode operation scheme (see [12]) or a nonlinear amplitude-frequency coupling (see [13]). Other solutions consist in the design of lateral micromechanical resonators supported by proper mechanical structures that introduce stresses to counteract temperature induced frequency shifts (see [14]), or of etch holes in Lamè resonators to modify their thermal drift (see [15]). Finally, active electronic compensations techniques are an alternative viable solution (see e.g., [16]).

## 2. Mechanical and Thermal Properties of Single-Crystal Silicon

^{19}cm

^{−3}. The elastic constants and their temperature dependences for such level of doping concentration are obtained by fitting the experimental results reported in Table 1. They read ${c}_{11}=161.41$ GPa, ${c}_{12}=66.13$ GPa, ${c}_{44}=78.56$ GPa, $T{c}_{{11}_{1}}$ = −30.37 ppm/°C, $T{c}_{{11}_{2}}$ = −81.30 ppb/°C

^{2}, $T{c}_{{12}_{1}}$ = −133.86 ppm/°C, $T{c}_{{12}_{2}}$ = −8.70 ppb/°C

^{2}, $T{c}_{{44}_{1}}$ = −71.69 ppm/°C and $T{c}_{{44}_{2}}$ = −30.39 ppb/°C

^{2}. Please note that, if not otherwise specified, only the data from [8] for the P-doping are used in the following for the sake of simplicity.

## 3. Analytical Model

#### 3.1. Temperature Variation of Frequency

^{−20}cm

^{−3}.

#### 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 $\tilde{\Delta}f$ 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

^{−6}means that the algorithm stops if changes of the objective function are smaller than 1 × 10

^{−6}). Lower and upper bounds are introduced in the optimization procedure in order to mimic feasibility criteria of the resonator (e.g., no negative dimensions and no slots radius smaller than 1 µm are allowed). Moreover, an upper bound for the in-plane thickness of the cantilever (i.e., W< 35 µm) is chosen in order to obtain a relatively small footprint of the MEMS resonator.

^{19}cm

^{−3}and an out-of-plane thickness of the device equal to 20 µm are fixed. Please note that it is in principle possible to add such parameters in the optimization variables reported in Equation (16) without any further modification of the optimization procedure. A Matlab routine has been implemented in order to combine the CMA-ES algorithm with the FEM Fortran code already presented for the computation of the natural frequencies and the quality factor of the resonator. At each iteration of the optimization procedure, a new mesh is generated and the objective function is computed on the basis of the results of the FEM code.

#### 5.1.1. Q Maximization

#### 5.1.2. Multi-Objective Function

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Lam, C.S. A review of the recent development of MEMS and crystal oscillators and their impacts on the frequency control products industry. In Proceedings of the IEEE Ultrasonics Symposium, Beijing, China, 2–5 November 2008; pp. 694–704. [Google Scholar]
- SiTime. SiT1532 Ultra-Small 32 kHz Oscillator. Available online: www.sitime.com (accessed on 9 June 2018).
- Jaakkola, A.; Prunnila, M.; Pensala, T.; Dekker, J.; Pekko, P. Design rules for temperature compensated degenerately n-type doped silicon MEMS resonators. J. Microelectromech. Syst.
**2015**, 24, 1832–1839. [Google Scholar] [CrossRef] - Mussi, G.; Bestetti, M.; Zega, V.; Frangi, A.; Gattere, G.; Langfelder, G. Resonators for real-time clocks based on epitaxial polysilicon process: A feasibility study on system-level compensation of temperature drifts. In Proceedings of the IEEE Micro Electro Mechanical Systems (MEMS), Belfast, UK, 21–25 January 2018. [Google Scholar]
- Ng, E.J.; Wang, S.; Buchman, D.; Chiang, C.-F.; Kenny, T.W.; Muenzel, H.; Fuertsch, M.; Marek, J.; Gomez, U.M.; Yama, G.; et al. Ultra-stable epitaxial polysilicon resonators. In Proceedings of the Solid-State Sensors, Actuators and Microsystems Workshop, Hilton Head Island, SC, USA, 3–7 June 2012. [Google Scholar]
- Hsu, W. Resonator miniaturization for oscillators. In Proceedings of the IEEE International Frequency Control Symposium, Honolulu, HI, USA, 19–21 May 2008; pp. 392–395. [Google Scholar]
- Varshni, Y.P. Temperature dependence of the elastic constants. Phys. Rev. B
**1970**, 2, 3952–3958. [Google Scholar] [CrossRef] - Jaakkola, A.; Prunnila, M.; Pensala, T.; Dekker, J.; Pekko, P. Determination of doping and temperature dependent elastic constants of degenerately doped silicon from MEMS resonators. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2014**, 61, 1063–1074. [Google Scholar] [CrossRef] [PubMed] - Ng, E.J.; Hong, V.A.; Yang, Y.; Ahn, C.H.; Everhart, C.L.M.; Kenny, T.W. Temperature Dependence of the Elastic Constants of Doped Silicon. J. Microelectromech. Syst.
**2015**, 24, 730–741. [Google Scholar] [CrossRef] - Mirzazadeh, R.; Saeed Eftekhar, A.; Mariani, S. Micromechanical characterization of polysilicon films through on-chip tests. Sensors
**2016**, 16, 1191. [Google Scholar] [CrossRef] [PubMed] - Shin, D.D.; Heinz, D.B.; Kwon, H.-K.; Chen, Y.; Kenny, W. Lateral diffusion doping of silicon for temperature compensation of MEMS resonators. In Proceedings of the IEEE International Symposium on Inertial Sensors and Systems (INERTIAL), Lake Como, Italy, 26–29 March 2018. [Google Scholar]
- Chen, Y.; Shin, D.D.; Flader, I.B.; Kenny, T.W. Tri-mode operation of higly doped silicon resonators for temperature compensated timing references. In Proceedings of the IEEE 30th International Conference on Micro Electro Mechanical Systems (MEMS), Las Vegas, NV, USA, 22–26 January 2017. [Google Scholar]
- Defoort, M.; Taheri-Tehrani, P.; Horsley, D.A. Exploiting nonlinear amplitude-frequency dependence for temperature compensation in silicon micromechanical resonators. Appl. Phys. Lett.
**2016**, 109, 153502. [Google Scholar] [CrossRef] - Hsu, W.-T.; Clark, J.R.; Nguyen, T.-C. Mechanically temperature compensated flexural-mode micromechanical resonators. In Proceedings of the International Electron Devices Meeting, San Francisco, CA, USA, 11–13 December 2000. [Google Scholar]
- Luschi, L.; Iannaccone, G.; Pieri, F. Temperature compensation of silicon Lamé resonators using etch holes: theory and design methodology. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2017**, 64, 879–887. [Google Scholar] [CrossRef] [PubMed] - Zadeh, S.A.G.; Saha, T.; Allidina, K.; Nabki, K.; El-Gamal, M.N. Electronic temperature compensation of clamped-clamped beam MEMS resonators. In Proceedings of the 53rd IEEE International Midwest Symposium on Circuits and Systems, Seattle, WA, USA, 1–4 August 2010. [Google Scholar]
- Frangi, A.; Cremonesi, M. Semi-analytical and numerical estimates of anchor losses in bistable MEMS. Int. J. Solids Struct.
**2016**, 92–93, 141–148. [Google Scholar] [CrossRef] - Fedeli, P.; Frangi, A.; Laghi, G.; Langfelder, G.; Gattere, G. Near vacuum gas damping in MEMS: Simplified modeling. J. Microelectromech. Syst.
**2017**, 26, 632–642. [Google Scholar] [CrossRef] - Frangi, A.; Fedeli, P.; Laghi, G.; Langfelder, G.; Gattere, G. Near vacuum gas damping in MEMS: numerical modeling and experimental validation. J. Microelectromech. Syst.
**2016**, 25, 890–899. [Google Scholar] [CrossRef] - Prabhakar, S.; Vengallatore, S. Thermoelastic Damping in Hollow and Slotted Microresonators. J. Microelectromech. Syst.
**2009**, 18, 725–735. [Google Scholar] [CrossRef] - Asadi, S.; Sheikholeslami, T.F. Effects of slots on thermoelastic quality factor of a vertical beam MEMS resonator. J. Microsyst. Technol.
**2016**, 22, 2723–2730. [Google Scholar] [CrossRef] - Candler, R.N.; Duwel, A.; Varghese, M.; Chandorkar, S.A.; Hopcroft, M.A.; Park, W.-T.; Kim, B.; Yama, G.; Partridge, A.; Lutz, M.; et al. Impact of geometry on thermoelastic dissipation in micromechanical resonant beams. J. Microsyst. Technol.
**2006**, 15, 927–934. [Google Scholar] [CrossRef] - Abdolvand, R.; Johari, H.; Ho, G.K.; Erbil, A.; Ayazi, F. Quality factor in trench-refilled polysilicon beam resonators. J. Microsyst. Technol.
**2006**, 15, 471–478. [Google Scholar] [CrossRef] - Hopcroft, M.A.; Nix, W.D.; Kenny, T.W. What is the Young’s modulus of silicon? J. Microelectromech. Syst.
**2010**, 19, 229–238. [Google Scholar] [CrossRef] - Jaakkola, A. Piezoelectrically Transduced Temperature Compensated Silicon Resonators for Timing and Frequency Reference Applications. Ph.D. Thesis, Aalto University, Helsinki, Finnland, 2016. [Google Scholar]
- Hall, J.J. Electronic effects in the elastic constants of n-type silicon. Phys. Rev.
**1967**, 161, 756–761. [Google Scholar] [CrossRef] - Okada, Y. Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500 K. J. Appl. Phys.
**1984**, 56, 314. [Google Scholar] [CrossRef] - Maycock, P.D. Thermal conductivity of silicon, germanium, III-V compounds and III-V alloys. Solid-State Electron.
**1967**, 10, 161–168. [Google Scholar] [CrossRef] - Kim, B.; Hopcroft, M.A.; Candler, R.N.; Jha, C.M.; Agarwal, M.; Melamud, R.; Chandorkar, S.A.; Yama, G.; Kenny, T.W. Temperature dependence of quality factor in MEMS resonators. J. Microelectromech. Syst.
**2008**, 17, 755–766. [Google Scholar] [CrossRef] - Corigliano, A.; Ardito, R.; Comi, C.; Frangi, A.; Ghisi, A.; Mariani, S. Mechanics of Microsystems; Wiley: Hoboken, NJ, USA, 2018; ISBN 978-1-119-05383-5. [Google Scholar]
- Hansen, N.; Müller, S.D.; Koumoutsakos, P. Reducing the Time Complexity of the Derandomized Evolution Strategy with Covariance Matrix Adaptation (CMA-ES). Evol. Comput.
**2003**, 11, 1–18. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hansen, N.; Ostermeier, A. Completely Derandomized Self-Adaptation in Evolution Strategies. Evol. Comput.
**2001**, 9, 159–195. [Google Scholar] [CrossRef] [PubMed][Green Version] - Marini, F.; Walczak, B. Particle swarm optimization (PSO). A tutorial. Chemom. Intell. Lab. Syst.
**2015**, 149, 153–165. [Google Scholar] [CrossRef] - McCall, J. Genetic algorithms for modelling and optimisation. J. Comput. Appl. Math.
**2005**, 184, 205–222. [Google Scholar] [CrossRef] - Auger, A.; Hansen, N. Performance Evaluation of an Advanced Local Search Evolutionary Algorithm. In Proceedings of the IEEE Congress on Evolutionary Computation, Scotland, UK, 2–5 September 2005; pp. 1777–1784. [Google Scholar]
- Hansen, N.; Niederberger, S.P.N.; Guzzella, L.; Koumoutsakos, P. A method for handling uncertainty in evolutionary optimization with an application to feedback control of combustion. IEEE Trans. Evol. Comput.
**2009**, 13, 180–197. [Google Scholar] [CrossRef] - Fukagata, K.; Kern, S.; Chatelain, P.; Koumoutsakos, P.; Kasagi, N. Evolutionary optimization of an anisotropic compliant surface for turbulent friction drag reduction. J. Turbul.
**2008**, 9, 1–17. [Google Scholar] [CrossRef] - Gagne, C.; Beaulieu, J.; Parizeau, M.; Thibault, S. Human-competitive lens system design with evolution strategies. Appl. Soft Comput.
**2008**, 8, 1439–1452. [Google Scholar] [CrossRef][Green Version] - Capellari, G.; Chatzi, E.; Mariani, S. Structural health monitoring sensor network optimization through Bayesian experimental design. ASCE-ASME J. Risk Uncertain. Eng. Syst. Part A Civ. Eng.
**2018**, 4, 04018016. [Google Scholar] [CrossRef] - UQLAB. Available online: www.uqlab.com (accessed on 9 June 2018).
- CMA-ES. Available online: http://cma.gforge.inria.fr/cmaes_sourcecode_page.html (accessed on 26 June 2018).

**Figure 1.**Material orientation of the local ${x}_{1},{x}_{2}$ axes with respect to the wafer [100] direction.

**Figure 2.**Tuning fork resonator. (

**a**) Schematic view of the tuning fork resonator with out of plane thickness t. (

**b**) First bending mode of the resonator. The contour of the displacement field is shown in color.

**Figure 3.**Frequency variation f

_{0}(T) − f

_{0}(25 °C) relative to f

_{0}(25 °C) for the tuning fork shown in Figure 2 for different orientations $\vartheta $.

**Figure 4.**Maximum temperature variation of the natural frequency of the tuning fork in the range [−35 °C–85 °C] for different orientations of the device with respect to the silicon wafer and for different dopings of the silicon.

**Figure 5.**(

**a**) $\tilde{\Delta}f$ for different orientations of the device with respect to the silicon wafer and for different n-dopings of the silicon. The white dotted line represents the minima of the contour plot. (

**b**) Minimum temperature variation of the natural frequency of the resonator for different n-dopings of the silicon.

**Figure 6.**Temperature variation of the quality factor of the tuning fork in the range [−35 °C–85 °C] for different orientations $\vartheta $ of the device with respect to the silicon wafer.

**Figure 7.**Temperature variation of the frequency of the tuning fork in the range [−35 °C–85 °C] for different orientations of the device with respect to the silicon wafer. Dotted lines denote numerical results, while continuous lines represent the analytical solution shown in Figure 3.

**Figure 8.**Temperature variation of the quality factor of the tuning fork in the range [−35 °C–85 °C] for different orientations of the device with respect to the silicon wafer. Dotted lines denote numerical results, while continuous lines represent the analytical solution shown in Figure 6.

**Figure 10.**Influence of the hole position on the (

**a**) quality factor and (

**b**) on the variation of the frequency in the range [−35 °C–85 °C]: only the results for the orientation that minimize $\tilde{\Delta}f$ in the case of the SETF of Figure 2 is reported for the sake of clarity. In this analysis LH = 73 µm, R = 3 µm and the other geometric dimensions of Table 2 are employed.

**Table 1.**Doping concentration dependence of the elastic constants of silicon and their temperature dependences. Elastic constants are expressed in GPa, while $T{c}_{{ij}_{1}}$ in ppm/°C and $T{c}_{{ij}_{2}}$ in ppb/°C

^{2}.

Doping Type | Concentration [cm^{−3}] | c_{11} | c_{12} | c_{44} | ${\mathit{Tc}}_{{11}_{1}}$ | ${\mathit{Tc}}_{{12}_{1}}$ | ${\mathit{Tc}}_{{44}_{1}}$ | ${\mathit{Tc}}_{{11}_{2}}$ | ${\mathit{Tc}}_{{12}_{2}}$ | ${\mathit{Tc}}_{{44}_{2}}$ |
---|---|---|---|---|---|---|---|---|---|---|

dop-n | 3.00 × 10^{13} [26] | 165.64 | 63.94 | 79.51 | −63.4 | −78.7 | −55.4 | −35 | −56 | −7 |

dop-n | 1.98 × 10^{19} [26] | 163.94 | 64.77 | 79.19 | −39.2 | −116.2 | −58.7 | −118 | NaN | −28 |

P | 4.10 × 10^{19} [8] | 163 | 65.4 | 79.2 | −34.5 | −133.7 | −67.8 | −115 | 22 | −51 |

P | 4.66 × 10^{19} [8] | 162.5 | 65.7 | 79.1 | −32.5 | −131.8 | −68.7 | −110 | 18 | −43 |

P | 6.60 × 10^{19} [9] | 164 | 66.7 | 78.2 | −34.2 | −135.17 | −67.8 | −103.04 | −1.1 | −40.26 |

P | 7.47 × 10^{19} [8] | 161.4 | 66.1 | 78.5 | −30.7 | −134.9 | −71.9 | −78 | −12 | −31 |

As | 1.20 × 10^{19} [9] | 164.2 | 65.6 | 78.6 | −46.58 | −124.61 | −63.12 | −105.41 | 31.73 | −45.21 |

As | 1.66 × 10^{19} [8] | 164 | 64.3 | 79.5 | −48.5 | −114.7 | −63.7 | −111 | 25 | −58 |

As | 2.46 × 10^{19} [8] | 163.8 | 64.9 | 79.4 | −44.2 | −124.6 | −65.1 | −111 | 34 | −55 |

Sb | 1.30 × 10^{18} [9] | 165.6 | 64.4 | 79.3 | −65.5 | −85.08 | −60.92 | −67.85 | −28.1 | −52.81 |

**Table 2.**Geometric dimensions of the tuning fork shown in Figure 2.

L | 195 µm |

HB | 45 µm |

W | 20 µm |

LB | 34 µm |

t | 20 µm |

**Table 3.**Optimal geometries computed through the CMA-ES optimization algorithm starting from the geometry shown in Figure 9. The employed objective function reads: f

_{obj}= −Q(@25 °C). All the geometric dimensions are reported in µm and the angles in degrees.

Geometry | Optimization Options | Results |
---|---|---|

${x}_{0}$ = [110 3 73 195 20 34 0 45] 0.3 MHz < ${f}_{0}$ < 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] f _{obj} = −Q(@25 °C) = −237831.19 ${f}_{0}$ = 0.30 MHz $\tilde{\Delta}f$ = 1115.21 ppm | |

${x}_{0}$ = [110 3 73 195 20 34 0 45] 0.4 MHz < ${f}_{0}$ < 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] f _{obj} = −Q(@25 °C) = −82910.63 ${f}_{0}$ = 0.40 MHz $\tilde{\Delta}f$ = 936.86 ppm |

**Table 4.**Optimal geometries computed through the CMA-ES optimization algorithm starting from the geometry shown in Figure 9. The objective function reads: ${f}_{obj}=100\tilde{\Delta}$f − Q(@25 °C). All the geometric dimensions are reported in µm and the angles in degrees.

Geometry | Optimization Options | Results |
---|---|---|

${x}_{0}$ = [110 3 73 195 20 34 0 45] 0.3 MHz < ${f}_{0}$ < 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] ${f}_{obj}(\mathrm{x})=-45416.74$ Q(@25 °C) = 62534.74 ${f}_{0}$ = 0.31 MHz $\tilde{\Delta}f$ = 171.18 ppm | |

${x}_{0}$ = [10 3 73 195 20 34 0 45] 0.3 MHz < ${f}_{0}$ < 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] ${f}_{obj}(\mathrm{x})=-12126.73$ Q(@25 °C) = 28164.73 ${f}_{0}$ = 0.45 MHz $\tilde{\Delta}f$ = 160.38 ppm | |

${x}_{0}$ = [110 3 73 195 20 34 0 45] 0.4 MHz < ${f}_{0}$ < 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] ${f}_{obj}(\mathrm{x})=-14215.97$ Q(@25 °C) = 30955.97 ${f}_{0}$ = 0.44 MHz $\tilde{\Delta}f$ = 167.4 ppm |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Zega, 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