*2.2.4 Case study*

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

*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*

In this section, the procedure estimating the thermal drift is discussed, as shown in **Figure 12**. A case study about the thermal drift of a MEMS capacitive acceler-

The procedure estimating the thermal drift can be divided into three distinct steps:

This step forms the base of the latter two steps. Defining the precise critical parameter that needs to be derivated depends on application requirement. For instance, the drift of frequency is critical for the application of the MEMS resonator, so the analytical formula for frequency needs to be derived. The imperfection is important in obtaining analytical formulae for critical parameters, especially for the MEMS sensors employing the differential detecting principle. If the imperfection is not considered, such as the asymmetry induced by fabrication error, the bias of MEMS sensors nulls. As such, no result on the thermal drift of bias may be obtained.

2. Calculation of the variation of dimension, stress, or material property induced

It is critical to calculate the variation of dimension, stress, or material property induced by temperature for estimating the thermal drift. For the material property, its variation with temperature is induced by the temperature coefficient. However, for the dimension and stress, the variation generally needs to be calculated by finite

second-order beam theory [60] (**Figure 11**).

*deformation, and (d) longitudinal shearing deformation.*

*Reliability and Maintenance - An Overview of Cases*

1. Deriving analytical formulae for critical parameter(s)

*2.2.3 Means of estimating thermal drift*

ometer is also presented.

*Thermal drift procedure estimation.*

**Figure 11.**

**Figure 12.**

by temperature

**100**

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 voltage was used to generate the output voltage:

$$V\_{out} = G\frac{\mathbf{C}\_A - \mathbf{C}\_B}{\mathbf{C}\_A + \mathbf{C}\_B} \tag{4}$$

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

Based on the detecting principle and the dimension shown in **Figure 13b**, the bias and scale factor are expressed as

$$B = -\frac{K\_T e}{m} \tag{5}$$

**Figure 13.**

*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, respectively.*

*Reliability and Maintenance - An Overview of Cases*

$$k\_1 = G \frac{m}{dK\_T} \tag{6}$$

where *B* and *k*<sup>1</sup> represent the bias and scale factor, respectively; *KT* is the total stiffness of the folded beams; *m* is the total mass of proof mass and moving fingers; *e* represents the asymmetry of capacitive gap induced by the fabrication error; *d* is the capacitive gap.

From the equation of bias, it can be seen that if the asymmetry of capacitive gap is not considered, then the bias nulls. In equations of bias and scale factor, the parameters varying with temperature include *KT*, *e*, and *d*. Variation of the stiffness is induced by TCEM and CTE [7]:

$$T\text{CS} = \frac{1}{K\_T} \frac{dK\_T}{dT} = a\_E + a\_s \tag{7}$$

The variations of *e* and *d* are calculated by analytical method [44]:

$$
\Delta e = \frac{K\_A - K\_B}{K\_A + K\_B} \left( a\_t - a\_{eq} \right) \Delta T L\_a \tag{8}
$$

3. In silicon structure, TDB can be reduced by middle-locating anchors for

anchors for fixed electrodes in sensitive direction.

diversity of micro-devices and develop standards.

China (Nos. 51505068 and U1530132).

**3. Conclusions**

**Figure 14.**

*Accelerometer with optimization for TDSF.*

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

**Acknowledgements**

**103**

shown in **Figure 14** [62]. As such, TDSF is suppressed significantly.

moving electrodes in sensitive direction or decreasing the stiffness asymmetry of springs, while the second part of TDSF can be reduced by middle-locating

The TDSF of the MEMS capacitive accelerometer can both be induced by the TCEM and the thermal deformation, so the structure of the accelerometer is optimized to make the TCEM and thermal deformation compensate each other, as

MEMS devices are an integrated system involving aspects of mechanics, electronics, materials, physics, and chemistry while interacting with the environment. Therefore, their reliability exhibits a great diversity of modes and mechanisms. This is a field open for further research, as it covers a vast area. However, one practical way is to conclude a few failure phenomena of a specific device for providing a guideline to study the similar behaviors appearing in other devices. This chapter only focused on the reliability problem occurring in the micro-accelerometers in storage and the thermal environment. These two factors pose the significant importance on the development of high-end microsensors for high-precise applications. The long-term stability induced by the viscoelasticity of packaging materials was first mentioned in this work to explain the performance shift after a long period of storage. The thermal effects formed by temperature change and structural layout were studied in depth and showed that the drift over temperature may be eliminated by a well-designed structure rather than perfect materials with zero CTE. It is the authors' view that this area should be further researched so as to bridge the

This work was supported in part by the National Natural Science Foundation of

$$
\Delta d = \left( da\_t + \left( l\_f + L\_f \right) \left( a\_{eq} - a\_s \right) \right) \Delta T \tag{9}
$$

where *La* expresses the distance from the anchor to the midline; *Lf* denotes the half length of an anchor for fixed fingers; *lf* defines the locations of first fixed finger, as shown in **Figure 13b**; *α<sup>s</sup>* indicates the CTE of silicon; *αeq* is called as equivalent CTE describing the thermal deformation of the top surface of the substrate and calculated by the analytical model for the MEMS die attaching proposed in literature; *KA* and *KB* stand for the spring stiffness connecting proof mass.

Deriving the differential of the bias to the temperature, the TDB is expressed as

$$\text{TDB} = \frac{\Delta B}{\Delta T} = -\frac{K\_T \Delta e}{m \Delta T} = \frac{K\_A - K\_B}{m} \left( a\_{eq} - a\_s \right) L\_a \tag{10}$$

Deriving the differential of the scale factor to the temperature, the thermal drift of scale factor (TDSF) is expressed as

$$\text{TDSF} = \frac{\Delta k\_1}{k\_1|\_{\Delta T = 0} \Delta T} = \text{TCS} - \left( a\_\circ + \frac{\left( l\_f + L\_f \right) \left( a\_{eq} - a\_\circ \right)}{d} \right) \tag{11}$$

Due to the asymmetry induced by the fabrication error, *KA* and *KB* are different from each other, and TDB is proportional to the difference between *KA* and *KB*. Therefore, the consideration on the imperfection is very important for discussing the thermal drift.

Based on the discussion on TDB and TDSF, the factors affecting thermal drift and method suppressing the thermal drift can be obtained:


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