1 Introduction

Micro-Electro-Mechanical System (MEMS) is developed and popular technology. Integrated circuit (IC) processing adopted this technology for fabricating the micro-sensors for the measurement of the various process variables like pressure, distance, temperature, fluid flow, acceleration, angular velocity, and rotation, etc. (Mishra and Kumar 2019). MEMS based pressure sensor can detect the very low range of pressure in micrometers and small feature size, good performance, low power requirements, and feasible mass-production in the micromachining process (Tran et al. 2017). The measurements of intracranial pressure (ICP), intrauterine pressure (IUP), the inlet and outlet pressures of blood and intraocular pressure (IOP) have great significance for biomedical applications. The ICP and IOP are kept usually in range 0.78–1.76 kPa and 1.47–2.79 kPa, respectively. The blood pressure measurements 25.7 mm Hg, 3.43 kPa and 0.1 mm Hg in a 20 mm Hg is required like IUP. Therefore, a pressure sensor with a high sensitivity and linearity (smaller than 0.5%) in 0–5 kPa is required for monitoring biomedical applications (Hossain and Mian 2017; Huang and Zhang 2014b; Guan et al. 2016). Different principles of sensing for senor design have been considered to adapt to various environments. These principles are optical fibre grating sensor, strain gauge, piezo-capacitance, piezoelectric, resonant sensing mechanisms, and piezoresistive sensor, etc. few measuring restrictions are found in most of the principles (Niu et al. 2014). Piezoresistive pressure sensors have high sensitivity and low cost in comparison with other sensors (Sosa et al. 2015). In this paper, A Centre Squared Bossed Diaphragms (CSBD) is proposed to improve Non-Linearity and sensitivity of the pressure sensor. CNTs use as piezoresistor elements because of strong sensing capabilities, and as centred bossed to concentrate the more stress at the edges of the diaphragms thus obtain the good linearity at the same thickness.

2 Piezoresistive pressure sensor

The piezoresistive pressure sensor consists of four piezoresistors, which are arranged in a configuration of the Wheatstone bridge to measure the output voltage. When no pressure is applied, the bridge is in a balanced state, i.e., Vout = 0 V because each piezo resistor has equal resistance. This principle has the advantages of good accuracy, higher piezoresistance coefficient, simple signal transduction and easy fabrication (Jakati et al. 2016; Lu et al. 2007; Xu et al. 2017). According to the effect with the applied pressure the diaphragm of the pressure sensor would deflect and cause the change in piezoresistor’s values. There are few fundamental factors to be considered for improvement of the sensitivity and linearity of the pressure sensor, first the placement of piezoresistors need to be done at the maximum stress region. In the case of the square diaphragm pressure sensor, the maximum stresses are produced at the centre of each edge of the diaphragm. Hence the four piezo resistors are pledges of the diaphragms as shown in Fig. 1. The width of centre boss need to be optimized to address the design limitations of nonlinearity errors. The centre boss is usually moulded by selectively design from the backside of a wafer with the sidewalls of 54.7.

Fig. 1
figure 1

a Schematic of a piezoresistive pressure sensor (top view), b cross sectional schematic of the pressure sensor

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Second the shape, thickness, material, dimensions of the diaphragms, thirdly the doping concentration of both piezo resistor and diaphragm play very effective roles in the sensitivity of the low-pressure sensor. The performance of the sensor can also improve by changing the position and shape of the piezo resistors.

2.1 Nano material as sensing elements

Silicon Carbide, Silicon, Germanium, polysilicon, CNTs, bismuth sulfide (Bi2S3) and Graphene materials have a piezoresistive effect (Zhu et al. 2013; Song et al. 2020b; Bala and Khosla 2018; Ali et al. 2017). Compared to bulk silicon, silicon nanowires have seven times the piezoresistive effect (Shaby et al. 2015; Kim et al. 2009). The materials with the piezoresistive effect can be used as a transducer and produced an electrical signal as a function of applied mechanical stress (Tran et al. 2017). Sensitive materials such CNTs affecting the performance of pressure sensors due to its remarkable electronic properties (Wang et al. 2020).

The foremost properties of CNTs are that bandgap of the CNT material is inversely proportional to the diameter of the tube, Single-walled carbon nanotube (SWCNT) and Multi-walled nanotube (MWCNT) are two basic structure of the CNT showed in Fig. 2. SWCNTs are an imperative because they exhibit electric properties that are not shared by MWCNT (Aqel et al. 2012). The basic principle of CNT nano sensor has arrived from the fact that radial deformation and electrical transition induced in armchair SWNT with hydrostatic pressure (Li et al. 2017). MWCNTs have several layers in concentric cylinders with diameter from 3 to 30 nm rolled one inside others and they were closed at both ends as in Fig. 2b. CNTs behave like a metallic as well as semiconductor depending on chiral vector as c = na1+ ma2 here (m, n) is chiral number and a1 and a2 is unit vector.

Fig. 2
figure 2

a Internal view of single wall carbon nanotube (SWCNT) and b Multiwall nanotube (MWCNT) (Gupta et al. 2019)

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In fig 3a, the circumference of CNT is \(=\left|\mathrm{C}\right|=\sqrt[a]{({n}^{2}+mn+{m}^{2})}\). Here a is unit vector. The diameter (\(d=\frac{c}{\pi }\)) of CNTs depends on the size of the catalyst particles (Gupta et al. 2019). Variation in strain effect the band gap as given in (1), here \(v\) is passion ratio, \(\varepsilon\) is axial strain, \(\gamma\) torsional strain, or the hopping integer (\(\sim\) 2.6 Ev). The resistance of the CNTs can be occurred due to electron transport due to thermal activation as given below in Eq. (2)

$$\Delta {E}_{gap}=\mathrm{sin}(2p+1)3{t}_{o}[\left(1+v\right)\varepsilon \mathrm{cos}3\theta +\gamma \mathrm{sin}3\theta )]$$
(1)
$$R={R}_{C}+\frac{1}{{|t}^{2}| }\frac{h}{8{e}^{2}}\left [1+\mathrm{exp}(\frac{{E}_{g}}{KT})\right]$$
(2)
$${E}_{g}={E}_{gap}^{o}+\frac{d{E}_{gap}}{d\varepsilon }\epsilon$$
(3)

\({|t}^{2}|\) is the electron transmission probability, Rc presents the contact resistance in series, h denotes the plank’s constant, T is temperature, electron charge is represented by e, k provides Boltzmann’s constant, \({E}_{g}\) is bandgap energy that depends upon the strain Eq. (3) for a semiconducting CNTs is \({E}_{gap}^{o}=\frac{{t}_{o}{a}^{2}}{4{d}^{2}}\). It is concluded from the second equation that when strain is applied changed CNTs resistance which building it as best material for piezoresistive (Katageri and Sheeparamatti 2015).

Fig. 3
figure 3

a Chiral vector of CNTs and b the first Brillouin zone (BZ) of graphene

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2.2 Different structure

The designing of a new efficient structure and optimize the locations of the piezo resistor on the diaphragm, the dimensions of the piezo resistors and diaphragm, structure of the diaphragm, and the shapes of components. This approach has been established for the designing of the several diaphragms structures (Tran et al. 2017). A pressure sensor of combined peninsula island with a bossed diaphragm structure with dimensions 3500 × 3500 um2 was designed by Xu et al. the nonlinearity of 0.33% FSS and a sensitivity of 0.066 mV/V/Pa was reported (Tran et al. 2018). Four Peninsulas with a centre Bossed diaphragm structure was demonstrated at low pressure to obtain high sensitivity and linearity (Huang and Zhang 2014a) peninsula-island combined with a bossed diaphragm structure was proposed to diminish the trade-off between nonlinearity and sensitivity for ultra-low pressure (Xu et al. 2016). A peninsula–island combined with the bossed diaphragm was tested with a non-linearity of 0.42% FS and sensitivity of 66 μV/V/Pa (Zhao et al. 2016). Crossed beam on the top of silicon membrane was etched for stress concentration to the detection pressure (0.5–40 kPa) (Yu and Huang 2015). The pressure sensor was designed using four grooves with a working range of 0–500 Pa and had a high sensitivity of 0.06 (Xu et al. 2017). A shuriken-structured diaphragm (SSD) was design to improve both linearity and sensitivity (Guan et al. 2016).

3 Small deflection diaphragm theory

To design a pressure sensor, mostly the conventional square diaphragm flat membrane is used for high linearity and accuracy. The square diaphragm has other meaning full advantages are there like low nonlinearity and simultaneous high sensitivity as well as decrease the influence of process fluctuations on the output characteristics of the pressure sensor (Li et al. 2020). The performance of sensor is measured maximum linearity and produced stress. Smaller diaphragm with same thickness offered high linearity and poor sensitivity and vice versa and true for larger diaphragm vice versa. Based on theory of electricity, the maximum stress induced in diaphragm is calculated by the given Eq. (4) (Li et al. 2017).

$$stress (\sigma )=0.308P{\left(\frac{L}{h}\right)}^{2}\left(1-{\mu }^{2}\right)$$
(4)

where P is the applied pressure, µ is the passion ratio, L/h is the length to thickness ratio of square diaphragm. To obtain more stress this ratio is to improve. When the deflection of the squared diaphragms exceeds the certain value compared to its thickness, the large deflection theories work, and then there will be nonlinear relation of defection with applied pressure. A combination method was developed to resolve the large deflection. Applied pressure P is divide into bending stress and shearing stress (P = Pa + Pb) and small deflection theory was used to derive Pa by using Eq. (5) Large deflection theory was used to derive Pb by using Eq. (6). Both the equations are separately deduced and at end combined together Eq. (7) (Suja et al. 2015).

$${P}_{a}=71.3\frac{\omega E{h}^{3}}{{L}^{4}}$$
(5)
$${P}_{b}=31.1\frac{{\omega }^{3}Eh}{{L}^{4}}$$
(6)
$$\frac{P{L}^{4}}{E{h}^{4}}=71.3\left(\frac{\omega }{h}\right)+31.1{\left(\frac{\omega }{h}\right)}^{3}$$

or

$$\frac{P{L}^{4}}{E{h}^{4}}=\frac{4.2}{\left(1-{\mu }^{2}\right)}(\frac{\omega }{h})+\frac{1.58}{\left(1-{\mu }^{2}\right)}{\left(\frac{\omega }{h}\right)}^{3}$$
(7)

From Eq. (7) relative load is \(\frac{P{L}^{4}}{E{h}^{4}}\) and the reative deflection is the ratio of deflection to thicknes of diaphragm as slowen in Fig. 4.

Fig. 4
figure 4

Normalized deflection for small and large theory

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It can be seen from the large deflection curve, that when the relative deflection is goes beyond 0.2, the relationship of deflection and load is no longer remain linear. The deflection of diaphragm must be less than 1/5th of the thickness of diaphragm according to small deflection theory for obtaining low nonlinearity.

4 Design of the centre squared bossed diaphragms (CSBD)

CSBD with piezo resistors was proposed to measure pressure ranging from 0 to 5 kPa, shown in Fig. 5a. Four piezo resistors are connected in such a way to form wheat stone bridge at centre of the edges of the diaphragms. The rigidity of the diaphragm needs to be increased for avoid deformation of the diaphragm (Huang and Zhang 2014a; Zou et al. 2017). The centre bosses on the backside are added to set the diaphragm, as shown in Fig. 5b. This structure relieved the discrepancy between the linearity error and sensitivity. In the proposed structure the side length (L = 800 µm) and the thickness (h = 10 µm) of the diaphragm has been taken for high sensor outputs. The distribution of stress intensity for the CSBD diaphragm has been investigated under 5000 Pa pressure.

Fig. 5
figure 5

a Cross-section of bossed diaphragm sensor structure. b Arrangement of piezoresistor as a wheat stone bridge

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5 Structure analysis and simulation results

5.1 Diaphragm deformation

The maximum deformation of the diaphragm is a function of diaphragms geometry and shape as given by Eq. (8) to achieve good sensitivity a thinner diaphragm is preferred. However, as the thickness reduces, the stiffness/stability is also diminishing.in order to maintain the linearity canter bossed is added (Meena et al. 2017).

The novel structures have been designed based on a local stiffening of the membrane to reduction in nonlinearity. The flexural stiffness is given in Eq. (8) (Song et al. 2020a; Kumar and Tanwar 2020; Zhang et al. 2014).

$$\omega =\frac{0.0159P\left(1-{\mu }^{2}\right){L}^{4}}{E{h}^{4}}$$
(8)
$$D=\frac{E{h}^{2}}{12(1-{\mu }^{2})}$$
(9)

D is flexural stiffness, E defines the Young modulus; μ represented Poisson ratio (Zhao et al. 2016). It is analyzed from the pressure deformation response that the squared diaphragm provides linear deformation performance over the specified pressure range. Figure 6 shows simulation results of stresses and displacement for CSBD structure. In addition, at the center of the diaphragm displacement is maximum as shown in Fig. 6a. Maximum stress is located at the center (red area) of the edge on the diaphragms given in Fig. 6b. Therefore, two piezoresistive gauges required to be placed at the center of the line with (x = 0; y =  ± 400 μm) and the other two gauges must be situated in the middle of the edges respectively. For example, their center was at the points (x =  ± 400 μm, y = 0). According to small deflection theory, the maximum deformation should be less than 1/5 of the diaphragm thickness \(\omega\)max ≤ 0.2 h. accordingly, if the thickness is h = 10 μm, it allowed a maximum deformation 2 μm. Figure 7a shows the maximum equivalent stress and (b) total deformation at the centre of the diaphragm for variation in diaphragm size. It has been determining that stress and deflection increases by large diaphragm and the sensor have good sensitivity due to more deformation, but linearity is almost vanished.so to obtain better results for sensitivity and the linearity, the centre of the diaphragm should be stiffening. Squared bossed is added at the centre to increase the stiffness of the diaphragm. Comparison of total deformation of different bossed size and bossed with CNTs Material in response of applied pressure ranging from 0 to 5000 Pa showed in Fig. 7c and analyzed from that with increased bossed size, maximum deformation at center is going to decreased and bossed with cnt material provide less deformation at the center and stiffen the diaphragm at the center.

Fig. 6
figure 6

Simulation results for a stress and b, deformation at applied pressure

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Fig. 7
figure 7

a Maximum equivalent stress and b maximum deflection in the centre of the diaphragm for different diaphragms dimensions, c maximum deflection in the centre of the diaphragm for different bossed size

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5.2 Final output of the sensor

Four piezoresistors in Wheatstone bridge, (R1 and R3) are placed in longitudinal and R2 and R4 are in transverse direction. N-type silicon material is used for designing of the diaphragms and all the piezoresistor are of CNT material. The longitudinal and transverse stress is induced on all the piezoresistors with the applied pressure on the diaphragms. σx1 is longitudinal stress and σy1 is transverse stress on R1 and R3 similarly σy2 is longitudinal stress and σx2 is transverse stress on R2 and R4. The cross-sectional area, the length and resistivity of the resistor are related to the resistance. The internal resistance of the piezoresistor will changed with these stresses. The change in length and cross section area are the dimensional effect and can be neglected. The change in resistivity is the physical effect and related to the transversal and the longitudinal (Yu and Huang 2015). Some positive change ∆R1 with the resistance R1–R3 and same negative change ∆R2 with the resistance R2–R4 (Gupta et al. 2019). The overall change in the resistance is represented by the Eq. (10) where π44 is the piezoresistive coefficient. Out the output voltage of the Wheatstone bridge circuit can be calculated in terms of input voltage Vin is 5 V.

$$\begin{aligned} \frac{{\Delta R}_{3}}{{R}_{3}} &=\frac{{\Delta R}_{1}}{{R}_{1}}=\left(\frac{{\pi }_{44}}{2}\right) \left({\sigma x_{1}}-{\sigma y_{1}}\right)= \left(\frac{{\pi }_{44}}{2}\right){\sigma_{LT1}}\\ \frac{{\Delta R}_{4}}{{R}_{4}} & =\frac{{\Delta R}_{2}}{{R}_{2}}=\left(\frac{{\pi }_{44}}{2}\right)\left({\sigma x_{2}}-{\sigma y_{2}}\right)=\left(\frac{{\pi }_{44}}{2}\right){\sigma_{LT2}}\end{aligned}$$
(10)
$${V}_{out}={V}_{in}\frac{(\frac{{\Delta R}_{1}}{{R}_{1}}-\frac{{\Delta R}_{2}}{{R}_{2}})}{2+\frac{{\Delta R}_{1}}{{R}_{1}}+\frac{{\Delta R}_{2}}{{R}_{2}}}={V}_{in}\frac{\mathrm{\sigma LT}1-\mathrm{\sigma LT}1}{\frac{{\pi }_{44}}{2}+\mathrm{\sigma LT}1+\mathrm{\sigma LT}1}$$
(11a)
$${V}_{out}=P\frac{{a}_{2}}{{h}_{2}} \left(1-v\right){\pi }_{44}{V}_{in}$$
(11b)

Overall output voltage of the bridge is Vout represented by Eqs. (11a, 11b) and that is prepositional to the applied Pressure P, ratio of diaphragm’s length (L) to weight (h) and given input.

5.3 Sensor sensitivity and linearity

The trade-off between Linearity and sensitivity conflicts at low-pressure. Since sensitivity of a sensor is proportional to the ratio of length and thickness of diaphragms (L/h). However, the nonlinearity found to be increased with L/h ratio. To achieve an enhanced sensitivity and attenuate the contradiction between the Linearity and sensitivity simultaneously, many structurally sensitive membranes were developed. (Li et al. 2018).

The sensitivity, which is the most important factor reflecting the capability of the proposed were developed pressure sensor, can be defined as

$$S=\frac{\left[{V}_{out}{(P}_{\mathrm{max}})-{V}_{out}({P}_{min})\right]}{\left({P}_{max}-{P}_{min}\right)}=\frac{{V}_{FS}}{\left({P}_{max}-{P}_{min}\right)}$$
(12)

where \({P}_{max}\) and \({P}_{min}\) are the maximum and minimum applied pressure,\({V}_{out}{(P}_{\mathrm{max}})\) and \({V}_{out}{(P}_{\mathrm{min}})\) are calculated output voltage at \({P}_{max}\) and \({P}_{min}\), respectively and \({V}_{FS}={V}_{out}{(P}_{\mathrm{max}})-{V}_{out}({P}_{min})\) is the Full -Scale Span (FSS).

$${NL}_{t}=100\%*\left[{V}_{out}\left({P}_{t}\right)-\left({P}_{t}\right)\frac{{V}_{out}\left({P}_{max}\right)}{{P}_{max}}\right]/\frac{{V}_{out}}{\left({P}_{max}\right)}$$
(13)

where \({P}_{t}\) is pressure measured at tested points.\({V}_{out}{(P}_{\mathrm{max}})\) is the FSS output voltage at maximum input pressure \({(P}_{\mathrm{max}}\)). Hence the nonlinearity can be either negative or positive depending on the calibration point.

The sensitivity(S) of the sensor under 5000 Pa were simulated and are shown in Fig. 8a and b. Compared with different bossed size and it was found that with increased diaphragms thickness the sensitivity is decreased and increased with improved bossed size.

Fig. 8
figure 8

a Sensor sensitivity as a function of a diaphragm thickness, b different bossed size

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6 Different meander configurations

Three different types of piezo resistors were introduced in the design as shown in Fig. 9. The length of the all piezoresistive bars should be equal. To calculate the response of pressure sensor used by various designs of piezo resistor, the average transverse and longitudinal stresses are derived by Finite element analysis (FEA). The output of the pressure sensor with various designs can be computed from the stresses and piezoresistive coefficients of CNT by using Eq. (11). However, in Eqs. (12) and (13), defines the non-linearity error and sensitivity.

Fig. 9
figure 9

Four different piezoresistor configurations

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The input voltage of the sensor Vin is fixed at 5 V. Figures 10 and 11 demonstrated the relationship of the applied pressure (0–5000 Pa) versus output voltages and non-linearity errors for dissimilar designs of piezoresistors likes design I with 0-turn, design II with 1-turn, design III with 2-turn and design IV with 3-turns. It was observed that the changes in sensitivity and nonlinearity causes due to shifting of resistors position. Linearity and sensitivity are mainly function of piezoresistor locations. In design III piezoresistors are breaks into two turns to increase the average stress over the four piezoresitors.

Fig. 10
figure 10

Simulated sensor output of four different piezoresistor configurations

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Fig. 11
figure 11

Nonlinearity error for different of four different piezoresistor configurations

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The design III found to have a maximum sensitivity of 27.82 mV/kPa. Design II gives the tremendous non-linearity error of − 0.27% FSS. Table 1 represented the comparison of sensitivity and linearity of the pressure sensor. It has been seen from the results that the design III can be the considered for the proposed sensor.

Table 1 Sensor characteristics of different meander configurations
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7 Conclusion

In this work, a squared bossed diaphragm piezoresistive pressure sensor has been designed and analysed for low pressure range. The simulation results show that the proposed structure stiffened the diaphragm to improve the linearity and developed more strain energy in the SCR to assure the higher sensitivity. Further, CNTs based piezoresistive pressure sensor provides tremendous performance. The sensitivity and linearity of the piezoresistive pressure sensors have also been studied for square-shape diaphragms. It has been found that the different configuration of piezoresistors and the thickness (h) of the diaphragm affects the characteristics of sensitivity and nonlinearity error. In addition, four different meander configurations are applied to analyse the linearity and sensitivity. The simulation results shows that the 2-turn meander configuration provides ultimate sensitivity of 27.82 mV/kPa and the 1 turn meander configuration is found to give the least non-linearity of 0.5051%. Therefore, either of these meander configurations may be considered for the improvement of overall performance of the sensor.