Efficient in-depth analysis and optimum design parameter estimation of MEMS capacitive pressure sensor utilizing analytical approach for square diaphragm

Capacitive pressure sensors have become more popular as compared to piezoresistive pressure sensors as they yield superior sensitivity and lesser nonlinearity. Efficient analysis for modeling capacitive pressure sensors is thus increasingly becoming more important due to their innumerable use cases. The higher sensitivity of square diaphragm for the same side length in comparison to circular diaphragm makes it ideal for sensor design. In this work, a complete formulation for analysis of capacitive pressure sensor with the square diaphragm in normal and touch mode operation has been presented as these two modes are established operating modes for these sensors. A comprehensive study of sensor parameters like capacitance, diaphragm deflection, capacitive and mechanical sensitivity has been formulated to aid the choice of sensor characteristics. This work also focuses on the method to determine core design parameters for optimal operation. Computationally complex methods have been used in the past for analysis of square diaphragms. The analytical approach presented in this research is less complex and computationally efficient, in comparison to the finite element method. MATLAB has been used to compute and simulate results.


Introduction
Micro-ElectroMechanical Systems (MEMS)-based capacitive pressure sensors have a pivotal role in MEMS devices for real-world applications [1]. These sensors have been extensively used in the field of medical science, automobile industry, avionics, industrial and commercial applications [2,3]. Thus, due to their mission critical use case these pressure sensors must be optimally designed and should be reliable if used in extreme conditions. These devices should also offer high sensitivity, low temperature drift and low noise. Initially piezoresistive sensors were used, but they were replaced by capacitive pressure sensors (CPS). The reason for this advance from piezoresistive to capacitive pressure sensors was driven by the higher sensitivity and lesser nonlinearity delivered by the CPS. Capacitive pressure sensors also have a much higher linear performance range. Reliable performance is witnessed even in high temperature conditions and harsh environments [4]. These sensors use ceramic or silicon diaphragms and can operate in two modes, the normal mode, and the touch mode.
In comparison to normal mode operation, touch mode delivers near-linear output characteristics and large over range pressures and this has led to the design of Single Touch Mode Capacitive Pressure Sensor (STMCPS) with circular diaphragm [1]. To further enhance performance, Double Touch Mode Capacitive Pressure Sensor (DTMCPS) were proposed. In this design, an additional notch was etched at the bottom of the substrate providing higher linearity and enhanced sensitivity compared to STMCPS [2,3]. A novel upgrade to both STMCPS and DTMCPS was suggested by etching two back-to-back Touch Mode Capacitive Pressure Sensor (TMCPS)s in one substrate. This structure was called the Double-sided Touch Mode Capacitive Pressure Sensor (DSTMCPS) [4,5]. This enriched design provided exceptional results for differential pressure applications but entailed increased manufacturing and design costs [6][7][8]. Silicon Carbide (SiC) has emerged as an appealing material for high-temperature applications owing to its excellent electrical, mechanical and chemical properties of high-thermal stability, conductivity, inertness, high critical electric field, hardness, resistance to wear and tear and mechanical robustness [4,9]. Thus further innovations for circular diaphragms based CPS were brought about by using a sandwich structure composed of SiC and aluminum nitride (AlN) for sensor construction [10,11].
For a given side-length, square diaphragm based CPS are more sensitive to changes in applied pressure as compared to their circular counterparts [6,9]. In addition, square diaphragms are preferred due to ease in fabrication and better yield [12]. Previous work shows the mathematical modeling for square diaphragm CPS employs computationally expensive and complex methods [8,13,14]. An analytical solution for normal mode square diaphragms CPS has been illustrated in [6]. As touch mode capacitive pressure sensors offer better performance in many applications, there is a need for efficient analysis of touch mode operation. This work therefore elucidates an in-depth, step-by-step derivation of the capacitance, capacitive and mechanical sensitivity of a silicon-based CPS with square diaphragm in both normal and touch mode operation. An approach for choice of sensor dimensions for optimum design has been provided using MATLAB simulations. The analytical solution provided is based on small deflection theory and assumes linear elastic deformation of the square diaphragm. The model developed is efficient and eliminates the need for computationally exhaustive finite element method (FEM) modeling [15,16]. FEM necessitates the use of complex software such as COMSOL, which requires a machine with significant compute and memory resources. In contrast, the proposed mathematical model can generate quick MATLAB simulations by tweaking the necessary design variables, enabling a faster analysis without using any complex simulation software. This model can be used as a solid foundation to decide on the operating range, experimental design parameters and application. It will not only save manufacturing resources but also reduce time in tuning the parameters to check results.
The paper is organized as follows: The second section deals with the general theory for analysis of square diaphragm CPS and demonstrates the different operating modes of the sensor. The third section describes the complete stepby-step approach for evaluating the capacitance, capacitive sensitivity, and mechanical sensitivity for normal and touch mode operation. The results, analysis, and comparison of various parameters, for the exact design of a MEMS pressure sensor have been discussed in Sects. 4 and 5.

Theory
A capacitive pressure curve ( Fig. 1) is used to examine the behavior of square diaphragm-based CPS. Figure 1 illustrates four operation zones normal, transition, touch, and saturation mode.
During normal mode, the diaphragm does not touch the bottom electrode or the substrate which can be observed in Fig. 2. The pressure range for normal mode spans from zero to the touch point pressure. The touch point pressure is the minimum pressure that is applied so that the diaphragm just touches the electrode. When the plate touches the bottom of the cavity, a highly nonlinear relation between capacitance and pressure exists and the sensor is said to be in transition mode. The transition mode creates noise and hence is not utilized for sensor operation.
In touch mode operation of the device, the diaphragm is in contact with the bottom electrode. This is depicted in Fig. 3. A layer of dielectric material is placed over the bottom electrode to prevent short circuit. The touch mode was introduced to enhance sensor performance. In saturation mode, the diaphragm completely touches the bottom electrode as a result, very less variation in capacitance takes place and ultimately capacitance saturates to a peak value.

Analytical solution
The mathematical modeling presented in this work is based on the theory of small deflection for plate bending. The theory assumes that diaphragm deflections are small in comparison with the plate thickness. Figure 2 illustrates a square diaphragm of thickness (h), Poisson's ratio ( ) , Young's Modulus (E) and side length (2a). The diaphragm deflection, w(x, y) is a function of x and y co-ordinates given by [6,16].
where D is the flexural rigidity and can be defined as resistance offered by a non-rigid structure when it is undergoing bending, or when a force couple is applied to the structure [9,15,16] and P is the applied pressure. Flexural rigidity can be expressed as The central deflection for square diaphragm can be evaluated by substituting x = 0 and y = 0 in (1) The touch-point pressure is defined as the minimum pressure at which the square diaphragm just touches the surface of the bottom electrode. At touch-point pressure ( P = p t and w 0 = d 0 ), where d 0 is the separation between the plates at zero deflection. Touch-point pressure is evaluated as

Normal mode capacitance analysis
For a parallel plate capacitor, capacitance C can be expressed as where 0 , A and d are the gap permittivity, plate area and plate separation, respectively. This equation can be used  (7) Assuming

is simplified as
On evaluating the integral in (10), C can be expressed as Substituting p t from (4) in (11), capacitance for normal mode operation is evaluated as

Touch mode capacitance analysis
Touch mode contact for a square diaphragm CPS is illustrated in Fig. 3. Touch mode capacitance constitutes (1) touchdown capacitance of diaphragm C t and (2) cavity capacitance C ut [1]. C t can be evaluated using the concept of parallel plate capacitance. To solve for C ut axial symmetry of boundary condition approximation is used [1]. Electrical field flux lines can be approximated as directional arcs due to axis symmetry of the diaphragm and underlying substrate. This is shown in Fig. 4 According to Gauss law, where is total charge density, D is electric displacement, is permittivity of medium. Considering electric flux as directional arcs, electric field intensity is where V is the applied voltage between electrodes and is the angle between the tangent through diaphragm. L is expressed as Here, d is the separation between the capacitor plates and is equal to (d 0 − w(x, y)) . Elemental touch mode capacitance can be expressed as where, dC ut , the elemental cavity capacitance is the function of L and thus The touch region between the diaphragm and the bottom electrode is assumed to be square in shape. Elemental touch down capacitance, dC t is expressed as The light-colored region in Fig. 5 indicates the touch region. This part contributes to C t , the touch region capacitance.
C ut emerges out of the dark colored area in Fig. 5. Hence, solving for C ut , where b is half length for the touch region. From Fig. 5 it can be observed that at a point (b, b) deflection of the diaphragm is d 0 . Substituting x = b , y = b and w(b, b) = d 0 in (1), b can be evaluated as (20) and using binomial approximation, the equation is modified as The integral in (23) can be further simplified as As depicted in Fig. 5 for the integral C 1 , the limits for x will be from b to a and similarly, the limits for y will be from −a to a.

Fig. 5 Top view in touch mode
Similarly, solving for C 2 with the following limits, x = −b to b and y = b to a (shown in Fig. 5) Assuming, Adding C 1 and C 2 and substituting (28) and (29) in the sum, C ut is obtained as From (19) and (30), C touch can be mathematically written as, (25)

Capacitive sensitivity for touch mode operation
Capacitive Sensitivity is defined as change in capacitance for a given change in applied pressure. It can be mathematically represented as Substituting the values of C t and C ut in (32) from (19) and (30), respectively where,

Mechanical sensitivity
Mechanical sensitivity is defined as the change in deflection for a given change in pressure Substitute x = 0 and y = 0 in (1), S mec is obtained as

Design specifications and considerations
The MATLAB simulation produced in Figs. 6, 7, 8, 9, 10, 11 is based on parameters specified in Table 1. It is evident from Table 1, that the sensor works in Normal Mode for the pressure range (0-0.023 MPa) and operates in Touch Mode for the pressure range (0.5-3 MPa). The highly nonlinear transition region between these two modes is observed in the pressure range (0.023-0.5 MPa). In order to validate and compare the results of this analytical approach, values from the research conducted in [1] have been used and the comparison is presented in Figs. 12, 13.

Results and discussion
Normal and Touch mode operation are elucidated in Sect. 5.1 and 5.2, respectively. The motivation behind the choice of magnitude of the parameters involved in sensor design has been illustrated in Sects. 5.3, 5.4, 5.5 and 5.6. Comparison between the sensor performance for circular diaphragm CPS [1] and square diaphragm CPS has been detailed in Sects. 5.7 and 5.8.

Capacitance variation with pressure in normal mode
Variation in capacitance with applied pressure for normal mode operation is depicted in Fig. 6. It can be observed that the capacitance varies linearly within the pressure range (0-0.023 MPa). The maximum capacitance variation is 2.1 pF for this mode. Figure 6 also depicts touch-point pressure ( p t ). The transition mode exists during the switch from normal mode to touch mode which can be observed in Fig. 7. Transition mode is highly nonlinear as the sensing diaphragm just starts to touch the bottom electrode. The pressure range for the transition region is (0.023-0.5 MPa). Figure 7 indicates capacitance variation with applied pressure for touch mode operation within the pressure range (0.5-3 MPa). Touch mode can be further divided into two sub-regions (1) linear and (2) saturation. The linear region forms the basis of touch mode sensor operation, as it can be accurately calibrated in terms of pressure. In the saturation region, the diaphragm completely touches the bottom electrode, hence no further change in capacitance is observed.

Variation in capacitive sensitivity with half-length of diaphragm for touch mode operation
The capacitive sensitivity plays a crucial role in determining the performance of a sensor. For P = 1.0 MPa, Fig. 8 examines the capacitive sensitivity variation with half-length of diaphragm for touch mode operation. It indicates a negligible variation in capacitive sensitivity for (0-250 m) with a sudden increase after (300 m). Even though capacitive sensitivity is further enhanced with increase in half-length of diaphragm beyond 300 m), there is a reduction in the linear operating range of the sensor. Considering the trade-off between sensitivity and linear range, a = 300 m has been selected for the sensor design as it offers higher sensitivity without compromising the linear operating range.

Variation in capacitive sensitivity with gap depth for touch mode operation
For P = 1.0 MPa, Fig. 9 illustrates the capacitive sensitivity variation with gap-depth for touch mode operation. A significant decrease in capacitive sensitivity is observed after a gap depth of 0.5 μ m. The design presented in this paper has a gap depth of 2 μ m which acts as the limiting value for optimum sensor operation.

Variation in capacitive sensitivity with applied pressure for touch mode operation
Variation of capacitive sensitivity with applied pressure for touch mode operation is presented in Fig. 10. Capacitive sensitivity decreases with increase in applied pressure and the variation with pressure is almost constant after 3 MPa.
In the pressure range between (0.023 MPa and 0.5 MPa), the value of captative sensitivity is considerably large, and this demonstrates the high nonlinearity of the transition region. As depicted in Fig. 10, linear operating range for this design is (0.5-3 MPa) and the sensor enters the saturation region after 3 MPa where the capacitive sensitivity is significantly reduced. Figure 7 also supports these observations. Figure 11 demonstrates the variation of mechanical sensitivity with diaphragm thickness. Increase in diaphragm thickness reduces mechanical sensitivity due to reduced diaphragm deflection. It can be concluded that for the given design parameters (Table 1), if the thickness is increased beyond 5 μm , the mechanical sensitivity drops by a significant amount. Therefore, the diaphragm thickness is constrained to be 5 μm [1]. Figure 12 presents the capacitance variation with applied pressure in normal mode operation for both square and circular diaphragm CPS. It can be inferred that the capacitance, capacitive sensitivity, and full pressure range of normal mode operation is higher for square diaphragms. Figure 13 illustrates the capacitance variation with applied pressure in touch mode operation for both square and circular diaphragm CPS. A square diaphragm occupies a larger area in comparison to a circular diaphragm with the same half-length ( r = a , where r is the radius of the circle and 2a, is the square's side length). As an increase in the area leads to an increase in capacitance ( C = A∕d ), hence higher capacitance and sensitivity are achievable with a square diaphragm CPS which is substantiated by Fig. 13. The results obtained and the interpretations have been summarized in Table 2 (Table 3; Fig. 14).

Conclusions
This study presents an analytical model for the determination of normal and touch mode operation of a capacitive pressure sensor with a clamped square diaphragm. Accurate analysis of these modes of operation is central to evaluating the operation range and stability of capacitive pressure transducers. Derived parameters like capacitive and mechanical sensitivity provide constructive insights for efficient sensor design. A justification for the sensor dimensions for optimum operation has been provided to further validate the presented analysis. Finite element method enables accurate prediction and modeling of normal and touch mode capacitance. However, the resource and time complexity of these tools make their usage less pragmatic. This analytical method offers significant ease of computation and clearly examines the higher sensitivity resulting out of the square geometry of the diaphragm. It also aims to provide a fast analysis model for prototyping the sensor to circumvent the need for complex simulation software. The examined theory and simulated results fully validate the superiority of square diaphragms in terms of ease of manufacturing, higher sensitivity, increased capacitance, and robustness.