Keywords

1 Introduction

1.1 MEMS Pressure Sensors

Sensors are used in modern devices for monitoring various physical parameters like temperature, pressure, humidity, etc., based on MEMS technology. Pressure sensors have wide range of applications in almost all possible areas like automobile industries, aerospace, biomedical, defense applications, etc. Pressure sensors are used to measure the applied pressure by a liquid or gas, and then the measured quantity is expressed as an electrical signal by a read circuit. Pressure sensors are based on piezoelectric, piezo-resistive, or capacitive technology. These types have their own advantages and disadvantages [1, 2]. Selection is based on application requirement like pressure range, accuracy, power consumption, and environmental condition like temperature, etc. The piezo-resistive pressure sensors are the simplest to fabricate and provide electrical output directly [3]. In MEMS piezo-resistive pressure sensors, four resistors placed in Wheatstone bridge configuration are strained in proportion to the applied pressure [4, 5]. Change in their resistance value is a measure of applied pressure.

Capacitive pressure sensors have the capability to meet up the rising demand of specialized pressure sensors because of its high sensitivity, fast response time, less temperature dependency [6, 7], and low power requirement. They have a simple design and analysis due to the simple governing equation [8].

In MEMS capacitive pressure sensors [9], two parallel and electrically isolated conducting plates by a dielectric medium like vacuum are fabricated on a silicon substrate. The bottom plate is fixed and the top membrane bends according to the applied pressure, thereby creating a capacitance change. This change is then converted into an electrical signal as a measure of applied pressure.

1.2 Comparison of Capacitive with Piezo-Resistive MEMS Pressure Sensor

Capacitive sensors have high accuracy, low power consumption [8, 9], and long term stability. The main advantage of capacitive sensor is the over pressure tolerance. Touch mode sensors have higher burst pressures [10] commonly used in biomedical applications. It can withstand up to hundred times the rated pressure without breaking and with no plastic deformation. Sensor performance is unaffected after the release of the over pressure. Capacitive pressure sensors are more suitable for medical applications [11, 12] (where high accuracy and long term stability are required) and for industry, where the occurrence of over pressure [13] is common and sensor repair is not an option. MEMS capacitive technology can be applied in demanding environments such as absolute, gauge, and relative pressure measurement [14]. But it needs more complex signal conditioning circuit and production cost is high compared with piezo-resistive sensors. The readout circuit for the capacitive sensor can be monolithically integrated [15], based on a switched mode method using reference and sensing capacitors.

2 Mathematical Modeling of Capacitive Pressure Sensors

The equation for the capacitance of a parallel plate capacitor is given by the relation indicated as

$$ c = \frac{{\varepsilon_{0} \varepsilon_{r} A}}{d} $$
(1)

Here, C denotes capacitance, expressed in terms of A = area between the two plates, εr = dielectric constant, ε0 = permittivity of free space, and d = distance between the two plates. The generalized equation for determining the maximum deflection of a square membrane whose thickness is uniform and all its edges are fixed, is given by [16] when external pressure is applied

$$ w_{\max } = \frac{{0.0138 L^{4} P}}{{Eh^{3} }} $$
(2)

In terms of poison’s ratio, central maximum deflection of the square shaped diaphragm of MEMS sensor under the applied pressure is given by [17]

$$ w_{\max } = \frac{{0.01512(1 - \mu^{2} ) L^{4} P}}{{Eh^{3} }} $$
(3)

Here, P = applied pressure, L = length of diaphragm, h = thickness of diaphragm, E = Young’s Modulus of Elasticity, and µ = Poisson’s ratio.

Now, since the spacing between the two plates of the capacitor changes due to the applied pressure, the change in capacitance is given by

$$\mathrm{\Delta C}=\iint \frac{{\varepsilon }_{0}{\varepsilon }_{r}}{d-w\left(x,y\right)} \mathrm{d}x \mathrm{d}y$$
(4)

3 Modeling MEMS Capacitive Sensor in COMSOL Multiphysics

The COMSOL Multiphysics software [18, 19] provides incorporated environment in which we can build the sensor model. The pressure sensor is part of the silicon die. One fourth of the model is considered due to the geometrical symmetry of the sensor as shown in Fig. 1. This will reduce the computational time and the memory required for computation. Using an in-build model, build-in physics structural mechanics with electro-mechanics interface and stationary study is performed. Pressure ranging from 0 to 25 kPa is applied, at T0 = 20 °C and the sensor is considered to have a die bonding temperature of 70 °C.

The boundary conditions are applied to the right side (zx plane at y = 0) and left side (zy plane at x = 0) as indicated in Fig. 1. Simulation is carried out for three different materials defined for diaphragm (silicon, graphene, and SiC). Thickness of the diaphragm is 0.01 mm and has a square shape with length and breadth as 0.5 mm. The distance between the electrodes (the cavity blue in color) is 0.003 mm, as shown in Fig. 2. The thickness of lower electrode and substrate are 0.39 mm and 0.7 mm, respectively, relatively very low thickness of diaphragm and cavity are seen as a bold horizontal line as in Fig. 3. The cavity is sealed under high vacuum, and the sides are insulated so that there is no connection between upper diaphragm and lower electrode. The boundary load is applied on upper boundary of the diaphragm. The diaphragm is allowed to move only in the z-direction to get the correct values of the capacitance change. The upper diaphragm (domain) is seized at a fixed potential of 1 V with respect to the ground at 0 V. The ground is applied to boundary of lower electrode.

Fig. 1
A geometrical representation of M E M S is a 3-D model. The top and bottom layers of the 2-D model processor can be viewed in the 3-D model. Here, the image is viewed in x, y, and z dimensions.

Geometrical model of MEMS capacitive pressure sensor

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Fig. 2
A 3-D model of the cavity has a length and width is 0.4 and 0.2 millimeters. It is viewed in the Graphics and the image is represented in three dimensions.

Cavity between the two electrodes

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Fig. 3
An illustration of M E M S in a cross-sectional view has the lower electrode which is the bottom layer. The top electrode has the diaphragm and the cavity is in between the Top and Lower electrodes.

Cross sectional view of MEMS capacitive pressure sensor

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In the physics part, solid mechanics and electrostatics are added. This automatically adds the electro-mechanical forces interface. To do the finite element analysis, over the whole capacitor MEMS structure moving mesh is applied. When pressure is applied, the center of the diaphragm is more compressed, and as a result, the applied mesh over the center of the diaphragm is also compressed. Hence, moving mesh is the appropriate to apply, instead of applying physics controlled mesh.

The material properties of different materials under consideration for simulation are as shown in Table.1.

Table1 Material property of different materials used for diaphragm
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Stationary study is computed with parametric sweep of different pressures ranging from 0 to 25 kPa and the t the corresponding displacement of diaphragm, capacitance is analyzed. An applied pressure originates deformation in diaphragm owing to that potential is not uniformly distributed in plane. Thus, plot for deformation of diaphragm versus pressure and plot of capacitance versus pressure are simulated separately. The analysis first carried out without any added stress due to sensor packaging stress and at a temperature of 20 °C. Then, it is analyzed with packaging stress and the operating temperature is varied from 290 to 300 K to find out the corresponding change in capacitance of the sensor.

4 Simulation Results

4.1 Diaphragm Displacement in Response to an Applied Pressure for Different Materials

Using switch mode simulation displacement of diaphragm as a function of pressure is simulated. Center of the diaphragm is more deflected, as shown in Fig. 4.

Fig. 4
A 3-D model of silicon has a pressure of 25000 pa with a diameter of 0.2 and 0.4 millimeters. Its color range starts from 18 to 4. A 3-D model of graphene has 25000 pa with a diameter of 0.2 and 0.4 mm. Its color range starts from 35 to 20. A 3-D model of Si C has 25000 pa with a diameter of 0.2 and 0.4mm. Its color range starts from 8 to 1.

Displacement of the diaphragm for an applied pressure of 25 kPa

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4.2 Displacement Versus Various Pressure for Square Diaphragm

The applied pressure is varied between 0 and 25 kPa. Deformation of diaphragm versus applied pressure is plotted as shown in Fig. 5.

Fig. 5
A trio of graphs of displacement versus pressure for silicon, graphene, and S i C. Each graph plots 2 rapidly decreasing curves for maximum and average displacement in micrometers.

Displacement of the diaphragm for an applied pressure range of 0–25 kPa

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4.3 Capacitance Versus Pressure for Square Diaphragm

Pressure range of 0–25 kPa is applied for which Si material shows a capacitance change from 0.75 to 1.2 pF, graphene from 0.74 to 0.77 pF, and in the case of SiC, it is 0.74–1.06 pF. Graphene material has more linear curve as compared with Si and SiC. Graphs are plotted as shown in Fig. 6.

Fig. 6
A graph of capacitance versus pressure in Silicon has a capacitance that starts from 0.75 and ends at 1.15. A linearized Analytic Capacitance starts from 0.75 and ends at 0.9. Graphene has two lines: capacitance and linearized Analytic Capacitance. Si C has 3 lines of capacitance, linearized Analytic Capacitance, and capacitance with package stress.

Capacitance in pF versus applied pressure in Pa

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4.4 Capacitance Versus Operating Temperature

To observe the effects of packaging stress on performance of device, thermal expansion is added in the model. The effect of thermal expansion is observed on entire structure as drawn in Fig. 7. The temperature is varied from 290 to 300 K.

Fig. 7
A graph of capacitance versus temperature of the silicon has a capacitance line that starts from 1.022 with a temperature of 290 and ends at 1.008. The graphene has a capacitance line that starts from 1.022 with a temperature of 290 and ends at 1.008. The Si C has a capacitance line starting from 0.24 with a temperature of 290.

Variation in capacitance with variation in operating temperature

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4.5 Displacement of Structure with Packaging Stress

The designed structure is assumed to be operating at a temperature of 20 °C. And its bonding temperature is 70 °C. The displacement of the structure at operating temperature of 20 °C with a load of 25 kPa is simulated and plotted as shown in Fig. 8. There is an interaction between the thermal stress and stress due to applied load.

Fig. 8
An illustration of the 3-D model of the silicon has a semi-circle and a layer that is the top layer. These are in the 3-dimensional view. In graphene, it has a semi-circle and a top layer. These are in the 3-dimensional view. In Si c, The semicircle and its top layer have a large diameter. It is viewed in the frontal view.

Displacement of structure with packaging stress

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5 Simulation Analysis

The simulated results are compared, shown in Table 2. Silicon has the maximum sensitivity for the pressure range 0–25 kPa, and diaphragm thickness of 0.01 mm. Sensitivity of capacitor for a change in temperature, over a range of 290–300 K is not very significant for the given pressure range proving capacitive MEMS sensor is less sensitive to temperature changes. Negligible changes are observed in the case of SiC which is widely used in high temperature application [20].

Table 2 Simulation analysis
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6 Conclusion

With the help of COMSOL Multiphysics 5.5 simulation tool, analysis and simulation of capacitive MEMS pressure sensor are carried out. The capacitor model and the physics acting upon it is studied through the simulation. Mathematical modeling is done, and the analytical value of the capacitor is compared with the simulated value by COMSOL. Geometrical shapes and diaphragm material can be changed to optimize the sensitivity and linearity of the sensor depending upon the application requirement such as pressure range operating condition, etc. Since capacitive pressure sensor are capable of measuring absolute pressure (by comparing unknown low pressure with vacuum), gauge pressure (pressure higher than atmosphering pressure), and differential pressure they find application in medical and industrial field.