# 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

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**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 |

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**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