*2.1.2 Viscoelasticity-induced stability problem of MEMS*

The viscoelasticity-related issue has become one of the most critical steps for assessing the packaging quality and output performance of highly precise MEMS sensors [50]. Applying the viscoelastic property to model the MEMS devices could yield a better agreement with the results observed in experiments than the previous elastic model [51]. The packaging stress in the MEMS was influenced not only by the temperature change but also by its change rate due to the time-dependent property of polymer adhesives [52]. Besides, the viscoelastic behavior influenced by moisture was recognized as the cause of the long-term stability of microsensors in storage [53].

The temperature profile in the simulation started from 60 (curing temperature) to 25°C (room temperature), and then the sensor was kept at 25°C for 12 months. The variation of the bias of the sensor was shown in **Figure 9**. The bias decreased over time due to stress relaxation and declined by about 21 mg in 12 months. After the sensor was kept at 25°C for about 10 months, the bias reached a steady state (<1 mg per month). The variation trend of the bias is generally consistent with the master curve of the relaxation modulus (**Figure 5**), which indicates that the package-induced stress will gradually be released because of adhesive viscoelas-

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

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 approx-

The performance of MEMS devices is influenced by TCEM through the stiffness. In

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

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].

tic characteristic in the long-term storage period.

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

imately 64 ppm/°C at room temperature [54].

*2.2.2 Thermal stress/deformation*

**97**

thermal stress/deformation are the factors studied mostly.

**2.2 Thermal drift of MEMS devices**

*2.2.1 TCEM*

**Figure 7.**

*Loading history for the analysis.*

**Figure 6.** *Prony series fitted to master curve.*


#### **Table 1.**

*Prony pairs of the die attach adhesive.*

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

It is evident that the output of the sensor after each thermal cycle will not change if the adhesive is assumed to be linear elastic.

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

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

**Figure 7.** *Loading history for the analysis.*

The temperature profile in the simulation started from 60 (curing temperature) to 25°C (room temperature), and then the sensor was kept at 25°C for 12 months. The variation of the bias of the sensor was shown in **Figure 9**. The bias decreased over time due to stress relaxation and declined by about 21 mg in 12 months. After the sensor was kept at 25°C for about 10 months, the bias reached a steady state (<1 mg per month). The variation trend of the bias is generally consistent with the master curve of the relaxation modulus (**Figure 5**), which indicates that the package-induced stress will gradually be released because of adhesive viscoelastic characteristic in the long-term storage period.
