**2.2 Thermal drift of MEMS devices**

The thermal drift of MEMS devices is related to its material, structure, interface circuit, and so on. The temperature coefficient of elastic modulus (TCEM) and the thermal stress/deformation are the factors studied mostly.

#### *2.2.1 TCEM*

In the following, the output stability of a capacitive micro-accelerometer was investigated using both simulation and experimental methods. The simulation introduced the Prony series modulus into the whole finite element model (FEM) in Abaqus software to acquire the output of the micro-accelerometers over time and temperature. The thermal experiment was carried out in an incubator with an accurate temperature controller. The full loading history used in both simulation and the experiment is shown in **Figure 7**. The red-marked points represent the starting or ending points of a loading step. The bias and sensitivity of the accelerometers subjected to the thermal cycles are shown in **Figure 8**. The observed output drift in the simulation and the experiments indicates that the viscoelasticity of adhesive was the main cause of the deviation of zero offset and sensitivity. The underlying mechanism can be attributed to the time- and temperature-dependent stress and defor-

*i Ei***/***E***<sup>0</sup>** *τ<sup>i</sup>* 0.08510 3041.87694 0.14589 981765.85865 0.22654 243.32699 0.11248 2083.25650 0.15906 48993.21024 0.21617 31.16988 0.03923 5052.23180 0.00025 6992.77554 0.00676 3321.33054

mation states of the sensitive components of the micro-accelerometers.

time due to the internal strain changing with the relaxation of stress.

if the adhesive is assumed to be linear elastic.

**Figure 6.**

*Prony series fitted to master curve.*

*Reliability and Maintenance - An Overview of Cases*

*E0 = 2744.76252 MPa.*

*Prony pairs of the die attach adhesive.*

**Table 1.**

**96**

It is evident that the output of the sensor after each thermal cycle will not change

The storage long-term drift of the accelerometer was also assessed by simulation and experimental methods based on the viscoelasticity of polymer adhesive. The residual stress formed in the curing process of the packaging would develop over

Due to the excellent mechanical property, single crystal silicon is suitable for high-performance sensors, oscillators, actuators, etc. However, the elastic modulus of single crystal silicon is temperature dependent. Because the single crystal silicon is anisotropic, the temperature behavior of elasticity is more properly described by the temperature coefficients of the individual components of the elasticity tensor, Tc11, Tc12, etc., as shown in **Table 2**. In order to simplify the designing, the value of TCEM for typical axial loading situations is usually employed and equal to approximately 64 ppm/°C at room temperature [54].

The performance of MEMS devices is influenced by TCEM through the stiffness. In fact, the temperature coefficient of stiffness (TCS) is the sum of TCEM and CTE (coefficient of thermal expansion). CTE is 2.6 ppm/°C at room temperature and much smaller than TCEM. Therefore, TCS is mainly determined by TCEM. The effect of TCEM on performance is dependent on the principle of the MEMS device. If the MEMS device is oscillating at a fixed frequency for time reference, sensing or generating Coriolis force, its frequency has a thermal drift of TCS/2, because the frequency is related to the square root of stiffness. On the other hand, the thermal drift of the MEMS device, which the mechanical deformation is employed for sensing, such as the capacitive sensor, is equal to TCS, because the performance is related to the stiffness [55].

#### *2.2.2 Thermal stress/deformation*

Single crystal silicon expands with temperature and has a CTE of 2.6 ppm/°C at room temperature. The expansion induced by CTE can cause the variation of

**Figure 8.** *Results comparison: (a) bias and (b) sensitivity.*

geometric dimension, such as gap, width, and length, and consequently induce the performance drift of MEMS device. However, the performance drift induced by CTE is generally very small compared to that induced by CTE mismatch.

> drift of MEMS accelerometer. Besides the CTE mismatch in MEMS die, another source of thermal stress/deformation is the CTE disagreement between the MEMS die and the package. The package material is in most cases ceramic, metal, and polymer, whose CTE values differ from the single crystal silicon. For instance, the CTE of a ceramic package is over twice as much as that of single crystal silicon [58]. In order to calculate the thermal stress/deformation, finite element method is

**TCE p-type (4 Ω cm, B) n-type (0.05 Ω cm, P) p-type (4 Ω cm, B) n-type (0.05 Ω cm, P)**

TCEC11 73.25 0.49 74.87 0.99 49.26 4.8 45.14 1.4 TCEC12 91.59 1.5 99.46 3.5 32.70 10.1 20.59 11.0 TCEC44 60.14 0.20 57.96 0.17 51.28 1.9 53.95 1.8

**/K) Second-order (<sup>10</sup><sup>6</sup>**

**/K<sup>2</sup> )**

**Figure 9.**

**Table 2.**

**Figure 10.**

**99**

*CTE of silicon and borosilicate glass [61].*

*The variation of the bias of the sensor in 12 months.*

*Reliability of Microelectromechanical Systems Devices DOI: http://dx.doi.org/10.5772/intechopen.86754*

**First-order (<sup>10</sup><sup>6</sup>**

*Temperature coefficients of the elastic constants given by Bourgeois et al. [56].*

Besides the single crystal silicon, there often exist the layers made of other material in a MEMS die, such as glass, SiO2, metal, and so on. The CTE of these materials is generally different from the single crystal silicon. Even for the borosilicate glass that has a CTE very close to the single crystal silicon, there still exists a CTE mismatch, as shown in **Figure 10**. In literature [57], research shows that the CTE difference between single crystal silicon and borosilicate glass induces bias

*Reliability of Microelectromechanical Systems Devices DOI: http://dx.doi.org/10.5772/intechopen.86754*

#### **Figure 9.**

*The variation of the bias of the sensor in 12 months.*


#### **Table 2.**

*Temperature coefficients of the elastic constants given by Bourgeois et al. [56].*

**Figure 10.** *CTE of silicon and borosilicate glass [61].*

drift of MEMS accelerometer. Besides the CTE mismatch in MEMS die, another source of thermal stress/deformation is the CTE disagreement between the MEMS die and the package. The package material is in most cases ceramic, metal, and polymer, whose CTE values differ from the single crystal silicon. For instance, the CTE of a ceramic package is over twice as much as that of single crystal silicon [58]. In order to calculate the thermal stress/deformation, finite element method is

geometric dimension, such as gap, width, and length, and consequently induce the performance drift of MEMS device. However, the performance drift induced by CTE is generally very small compared to that induced by CTE mismatch.

**Figure 8.**

**98**

*Results comparison: (a) bias and (b) sensitivity.*

*Reliability and Maintenance - An Overview of Cases*

Besides the single crystal silicon, there often exist the layers made of other material in a MEMS die, such as glass, SiO2, metal, and so on. The CTE of these materials is generally different from the single crystal silicon. Even for the borosilicate glass that has a CTE very close to the single crystal silicon, there still exists a CTE mismatch, as shown in **Figure 10**. In literature [57], research shows that the CTE difference between single crystal silicon and borosilicate glass induces bias

element analysis or other analytical methods. The imperfection is also important for

imperfection is not considered, the impact of the variation of dimension and stress is operating in common mode. As such, these variations cannot induce the variation of the bias for the MEMS sensors employing the differential detecting principle.

3. Discussion on the factors affecting the thermal drift and how to suppress the

Based on the variation of dimension, stress, or material property induced by temperature, the thermal drift can be acquired by deriving the differential of the temperature. Then, the factors affecting thermal drift and how to suppress this will

In the following, a MEMS capacitive accelerometer is employed to showcase the procedure estimating the thermal drift. The detection of the accelerometer is based on the open-loop differential capacitive principles, as shown in **Figure 13**. The acceleration force makes the proof mass move and is balanced by elastic force generated by folded beams. The moving proof mass changes the capacitances of the accelerometer. The detected variation amplitude of the capacitance difference between capacitors *CA* and *CB* via modulation and demodulation with preload AC

*Vout* <sup>¼</sup> *<sup>G</sup> CA* � *CB*

Based on the detecting principle and the dimension shown in **Figure 13b**, the

*<sup>B</sup>* ¼ � *KTe m*

*MEMS capacitive accelerometer employed as case study. (a) SEM picture and (b) open-loop differential capacitive principle.* �*Va is preload AC voltage. CA and CB represent capacitors on the bottom and top sides,*

where *G* is the gain that depends on the circuit parameters.

*CA* þ *CB*

(4)

(5)

the MEMS sensors employing the differential detecting principle. If the

*Reliability of Microelectromechanical Systems Devices DOI: http://dx.doi.org/10.5772/intechopen.86754*

thermal drift

be discussed.

*2.2.4 Case study*

**Figure 13.**

*respectively.*

**101**

voltage was used to generate the output voltage:

bias and scale factor are expressed as

**Figure 11.**

*Four components of the deformation in the die or substrate. (a) Longitudinal normal deformation induced by the thermal expansion, (b) longitudinal normal deformation induced by the shear stress, (c) transverse bending deformation, and (d) longitudinal shearing deformation.*

#### **Figure 12.** *Thermal drift procedure estimation.*

widely employed. However, the finite element method generally generates a model with high degrees and is time-consuming. For the thermal stress/deformation induced by the package, the analytical model based on strength of material is also largely employed, while taking up less time. In the analytical model, the elastic foundation for the adhesive layer is generally employed [59], and the deformation inside the die or substrate is by and large divided into four components, which are shown in **Figure 10**. The four components can be described by the first-order or second-order beam theory [60] (**Figure 11**).
