3.1. Commercial of the Shelf MEMS Sensor Modeled as a Single Degree of Freedom Model
As a proof concept, we first used a model for a cantilever beam with a proof mass pinned from the middle, as shown in
Figure 2a. This device, produced by Sensata Technologies, is a commercially available capacitive sensor that has been extensively investigated in the literature, and a good match between its simulation results and experimental data was reported [
12]. A schematic of the MEMS device is shown in
Figure 2b.
The MEMS sensor-proof mass, forming one side of a capacitor electrode, has a length
l (m), width
b (m), and thickness
h (m). An initial gap
d (m) separates the proof mass from the substrate beneath. The following single-degree freedom model can describe the dynamic response of the beam:
where
is the deflection, negative upwards and positive downwards (
), the dot operators represent temporal derivatives,
is the time in seconds (
), the effective mass
is given by
,
is the natural frequency of the system
, and
Fe is the electrostatic force given by:
where
represents the AC excitation frequency (
) and
the area of overlap between the microbeam-proof mass and the substrate beneath it (
).
As a demonstration of the THS operation,
Figure 2c shows the normalized MEMS response at different helium concentrations with a bias voltage slightly above the pull-in voltage. The figure shows a minimum detection level threshold of 50,000 PPM can be achieved. While the sensor minimum detection level can be improved by optimizing the MEMS design, electrical resonance activation results in multiple order of magnitude sensitivity improvement (
Figure 2d) [
19,
20]. In this operation, a small AC voltage signal with a frequency that matches the electrical resonance of the MEMS circuit will significantly increase the MEMS capacitive sensitivity to changes in air permittivity. Thus, the commercial capacitive sensor can achieve a minimum sensitivity of up to 300 PPM.
3.2. Micro-Continuous Beam Finite Element Model
In the second case study, we present a complete detailed analysis of a continuous micro-beam response to the presented helium sensing concept. The finite element software package COMSOL Multiphysics was used for this study. The electromechanics physics interface was used, which combines electrostatics and solid mechanics and solves the electromechanical forces (by solving the Minkowski electromagnetic stress tensor). The mesh used in this study is physics-controlled tetrahedral mesh with an extremely fine setting as defined by COMSOL Multiphysics.
COMSOL simulations were conducted using the UNL supercomputer cluster CRANE. They use Intel Xeon CPUs, including Xeon E5-2697 V4 and Xeon Gold 6248. Simulations used a variable number of nodes 1–16, each containing 32 cores, with waiting time from several hours up to two weeks and computing times taking up to several days even when using coarse mesh. Most of our simulations were conducted using the time-dependent solver of COMSOL. Output sizes reached more than 100 GB files, especially with long simulation times that passed 100 µs. The default time stepping was used because fixing the time step to the smallest size taken by the solver will make it highly unlikely to finish the simulations in the allowable run time rules.
A polysilicon beam base design (
Figure 3) with a width of 300 µm (x-dimension), a depth of 8 µm (y-dimension), and a height of 20 µm (z-dimension) was used for in-plane actuation (negative y-direction). The gap thickness is 10 µm. The dielectric constant of air was approximated to 1.0005, and for full helium was considered 1.0001, the vacuum permittivity was 8.854
10
−12 F·m
−1, and the nominal capacitance was calculated to be 5.3151
−15 F. To account for the parasitic capacitance, 10 times the nominal capacitance was used, leading to a total system capacitance of 5.8466 × 10
−15 F. In this work, the change in the dielectric constant due to helium presence (from 1.0005–1.0001) was assumed to follow a linear rule of mixtures. Thus, the helium detection level (HDL) is calculated by the following equation:
The detected dielectric constant is the maximum dielectric constant less than 1.0005 (full air), at which the beam will not suffer a pull-in event. This study used the dielectric constants of 1.0004, 1.00043, 1.00045, 1.00048, and 1.0005 corresponding to helium detection levels of 25, 17.5, 12.5, 5, and 0% (full air), respectively. While a helium detection level of less than 5% can theoretically be achieved, we limit our study to 5% due to the demanding computation effort. When a DC voltage is ramped from 0 to pull-in voltage, with a course variable voltage step size, the beam displacement response difference between full helium (ε
r of 1.0001) and full air (ε
r of 1.0005) is not noticeable, as shown in
Figure 4. In this simulation, we used the stationary solver of COMSOL Multiphysics. While a noticeable detection level can be achieved by lowering the voltage step size, similar to
Figure 2, this will not be significant. Moreover, lowering the step voltage requires significant computing power and may last weeks, even using a supercomputer.
Next, we added a series resistance of 40 Ω and an inductance of 30 mH to form an RLC circuit with a resonance frequency of 3.8 MHz. A small amplitude AC signal that matches the MEMS electrical resonance frequency was applied. The amplification Equation (3) was used to calculate the amplitude of the terminal voltage (beam voltage) at the actuation AC frequency. By triggering the MEMS electrical resonance, the MEMS electrostatic sensitivity to the input voltage and hence to the dielectric constant of the medium (gas mixture of air and helium in our case) increases.
Figure 5 validates this point.
Figure 5a shows the beam response due to different helium concentrations, represented by the different dielectric constants.
Figure 5a shows that with a dielectric constant of 1.0005 (full air), the electrostatic force was enough to cause the beam to pull in (switch on). However, by reducing the dielectric constant to 1.0004 (helium concentration of 25%) or less, the beam gained stability and escaped the pull-in (switch off). Moreover,
Figure 5b confirms the sensitivity of the terminal voltage (effective voltage across the MEMS) to the dielectric constant values. In conclusion, compared with
Figure 4,
Figure 5 shows the effective use of MEMS electrical resonance to increase its sensitivity to helium detection. Another advantage of utilizing electrical resonance, as reported in [
19,
20], is the massive reduction of the actuation voltage to achieve pull-in.
Figure 6 pushes the minimum detection level of helium lower by simulating a dielectric constant of 1.00048, which deviates only by 0.00002 from the dielectric constant of air. Surprisingly, the figure shows that the beam still escapes the pull-in even with this minimal dielectric constant reduction.
Next, we will discuss the effect of the AC excitation frequency on the sensor sensitivity.
Table 2 summarizes the different excitation frequency values and their effect on the minimum helium detection level. The table shows that with the reduction of the excitation frequency from the electrical resonance value, two behaviors are observed: (1) the increase in the required input voltage to achieve pull-in for full air and (2) and the reduction in helium detection performance (i.e., the increased in the minimum detection level). For example, at 3.78 MHz frequency, an input voltage of 7.479 Volt is required to achieve pull-in at full air. Moreover, the helium detection level increased to 25% (
Figure 7). This behavior can be attributed to the reduced amplification of the input voltage; hence, less sensitivity to the ε
r value, as seen in
Figure 8.
Table 2 also shows that the minimum helium detection level increased to 50% (ε
r = 1.0003) at 3.77 MHz. This further emphasizes the importance of electrical resonance physics.
In summary,
Table 2 shows that the minimum helium detection level increases when the excitation frequency departs from the electrical resonance frequency, degrading the sensor’s performance. Using electrical resonance, the magnitude of voltage amplification depends on the dielectric constant value and the excitation frequency, achieving the max amplification at the electrical resonance frequency. Thus, the sensor sensitivity to changes in helium is reduced as the excitation frequency is moved away from the electrical resonance frequency.
One of the critical parameters in the RLC circuit is the resistor, which acts as the electrical damping element. Increasing the resistance value while keeping the excitation frequency at 3.8 MHz increases the input voltage and reduces the helium detection performance, as shown in
Table 3. The table shows that increasing resistance up to 100 Ω, the detection level remains at 5%. However, the minimum detection increases to 25% for 1 kΩ. This increase can be explained by plotting the terminal voltage variation in
Figure 9 versus the dielectric constant for 40 Ω and 1 kΩ resistances.
Figure 9 shows that terminal voltage variation at higher dielectric constant values is almost saturated for the high resistance value (1 kΩ). Thus, a much lower value of ε
r = 1.0004 is needed to produce a noticeable variation in the terminal voltage compared with the 40 Ω resistance (ε
r =1.00048).
Any MEMS electrical circuit may suffer parasitic capacitance due to wiring and the different circuit parts. Next, in
Figure 10, we investigate the parasitic capacitance effect on MEMS detection sensitivity.
Figure 10 shows that as the parasitic capacitance increases, the total system capacitance increases, and hence the electrical resonance frequency decreases (
Figure 10a). This increases the input voltage while matching the excitation frequency to the new electrical resonance frequency (
Figure 10b). Despite these changes, our simulations reveal that the minimum possible helium detection level remains at 5%. However, accepting a large parasitic capacitance in the MEMS circuit design may come at the cost of decreasing the sensor stability against noise and environmental conditions.
To elaborate more on this, in
Figure 11, we compare the maximum beam displacement and terminal voltage at 5% helium (corresponding to ε
r = 1.00048) to the last stable displacement and terminal voltage before pull-in at full air, as we vary the parasitic capacitance. The plots in
Figure 11 show that while there is still a clear pull-in (full air) versus no pull-in (with 5% helium) at all parasitic capacitance levels, the difference in displacements and terminals voltages decreases with increased parasitic capacitance. Theoretically, the big difference at lower patristics capacitance means the sensor may be able to detect a lower level of Helium than 5%. However, in this study, as we limited it to 5%, this big difference translates to more stability in the system response to the noise.
Finally, in
Figure 12, we show the terminal voltage amplitude at different dielectric constants for the parasitic capacitance of 10, 100, and 1000. The figure confirms the voltage reduction across the full range of dielectric constant as the parasitic capacitance increases. The relationship becomes almost linear at 1000 PCF, explaining the behavior observed in
Figure 11a,b.
The effect of beam thickness on the sensing performance is discussed at the end. The beam thickness varies from 6 to 20 µm while keeping other parameters unchanged, as shown in
Figure 13. The figure shows that the input voltage amplitude increases as the beam thickness increases, increasing the beam’s mechanical stiffness (
Figure 13a).
Figure 13b shows that, despite the voltage reduction, there is almost negligible impact on the beam displacement difference between full air and 5% helium concentration.
It is worth mentioning this work has some limitations. For example, the model assumes an ideal inductance, which ignores any internal resistance that may impact the electrical resistance amplification and reduces the sensor sensitivity. Currently, there are efforts to create an experimental set-up to validate some of the simulations presented in this paper.
Finally, to check the effect of temperature on the sensor’s performance, we performed a simulation for a cantilever beam at full air condition, as shown in
Figure 14. The fixed end of the cantilever beam had a temperature of 293.15 K, and the other boundaries varied from 239.15 to 493.15 K at 40 K steps. The simulation showed no appreciable difference in displacement as a function of temperature at a constant excitation voltage. This implies that temperature has less impact on these capacitive systems; hence, the presented gas-sensing method is immune to temperature fluctuations. Similar findings on the minimum effect of temperature variation on the MEMS dielectric constant measurement systems in air and with the presence of helium and other gases were presented in [
21].