ICSE2002 hoc.2002,PenangNalaysia

An Analytical Study on Diaphragm Behavior for Micro- machined Capacitive Pressure Sensor

Norhayati Soin andBurhanuddin Yeop Majlis, Member, IEEE

UKM-TM Microelectronic center Faculty of Engineering

Universiti Kebangsaan Malaysia, 43600, Bangi, Selangor, MALAYSIA.

... . MEMS Laboratory

Abstract Understanding the deflection behavior of micro-machined diaphragms is necessary for designing mechanical sensors such as pressure sensors. An analylkal study on the diaphragm behavior with different structures for micro-machined capacitive pressure sensor is presented in this paper. In general, analytical solutions for diaphragm behavior are desirable because of their ease and the insight they provide to the designer. Specific geometric effects can be ascertained form these solutions. However, these solutions are generally only applicable for small deflections. The behaviors of flat and corrugated diaphragms with various structural parameters and properties are analysed using the classical Timoshenko plate theory respectively,

1. INTRODUCTION

Diaphragms with a boss and corrugations are very usefnl for micro-machined capacitive pressure sensors . Such diaphragms offer longer linear travel and larger dynamic range than do planar and simple cormgated diaphragms [1,2]. In many cases, the micro- machined silicon diaphragms are in state of internal stress [3,4]. The internal stress can cause the diaphragms to be deformed and the load deflection changed.[5,6]. It was found that the effects of the internal

stress on the behavior of the diaphragms with different structures, such as planar, simple corrugated, and boss and corrugated, are different. Understanding these issues and developing appropriate processes to control, but generally takes the form of a square or circle. These shapes behave similarly to an applied the diaphragm deformation are very impottant for device design and fabrication.

In this paper, different types of diaphragm structures, including flat and cormgated diaphragm with various~cormgation depths and initial stresses. are- studied. The analytical observations have been made by 'using ,the classical Timoshenko plate theory[7]. This approach have . been .used to explore .the .performance.of different diaphragm structures. The effects on load deflection, capacitance, non-linearity, and sensitivity performance due to changes in diaphragm size, cormgation profiles, and internal shess are also explored.

It. THEORETICAL APPROACH

(i) Flat Diaphragm

In diaphragm based sensors, pressure is determined by the deflection of the diaphragm due to applied pressure. Fig.1 illustrates schematic section of a typical pressure sensor diaphragm. The reference pressure can be a sealed chamber or a pressure port so that absolute or gauge pressure are measured, respectively.

I Applied Pressure

Fig. 1 Schematic cross section of typical pressure sensor diaphragm

The shape of the diaphragm as viewed tkom the top is arhitrary, but generally takes the shape of a circle or square. These shapes behave similarly to an applied stress. For the case of a clamped circular plate with small deflections (i.e., less than half of the diaphragm thickness) the form of deflection is [71 :

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ICSE2002 Proc.2002

where w, r, a, and P are the deflection, radial distance from the center of the diaphragm, diaphragm radius and applied pressure respectively. D is the flexnral rigidity, given bY

where E, h, and Y are the Young's modulus, thickness, and Poisson's ratio, respectively of the diaphragm. From the above equations, both the shape and the amount of deflection can be determined. Moreover, it is readily apparent that the amount of deflection is directly proportional to the applied pressure. For the case of a diaphragm with large built in stress or large deflections this direct proportionality is no longer true. In general, it is desirable to use a deflection measurement scheme that is linear with pressure, since such systems are simple to calibrate and measure.

External pressure

and dielectric isoldion

Reference pressure inlet

Fig. 2. Cross section schematic of hulk micro- machined, capacitive pressure sensor.

Capacitive pressure sensors are based upon parallel plate capacitors. A typical hulk micro-machined capacitive pressure sensor is shown in Fig. 2. The capacitance, C, of a parallel plate capacitor is given by

(3)

,Penang,Malaysia

where E. A, and d are the permittivity of the gap, the area of the plates, and the separation of the plates, respectively. For a moving circular diaphragm sensor, the capacitance becomes

E rdrdb' (4) '= 5sd-u(r,b')

where w is the deflection of the diaphragm. Using the deflection of a uniform thickness, circular diaphragm &om Equation ( I ) yields

solving the integral gives

which can be expanded in a Taylor series to

(7) where CO is the undeflected capacitance given by Equation 3 and WO is the center deflection of the diaphragm(i.e. w(r=O) from Equation (4). The capacitance with respect to applied pressure, then is generally nonlinear due to the nonlinear deflected shape of the diaphragm.

The value of pressure sensitivity of the sensor for small values of measured pressure can be calculated from the simplified formulas presented in [8]. For pressure sensitivity we can write

Pressure sensitivity depends on the membrane thickness H, electrode distance d at reference pressure and edge length 2R. The deflection of flat, clamped, circular

diaphragm is given approximately by (9) :

where P is the applied pressure, R is the diaphragm radius, h is the diaphragm thickness, E is Young's Modulus, v is Poisson's ratio and WO is the center deflection of the diaphragm. For comparison pnrposes, the equivalent square diaphragm deflection is given by [IO] :

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ICSE2002 Proc.2002,Penang,Malaysia

Pa' 4.20 WO 1.58 WO ' Eh' = Z(T) +1-,.G) (I0)

-. where a is the half.sidelength . Thus a circular diaphragm with a radius equal to the half sidelength of a square diaphragm will be about 30% stiffer, for small deflections, and the non- linearity will be very similar, due to the nearly equal ratios of linear to cubic coefficients. Note that in either case the non-linearity becomes significant for deflections more than about 25 % of the thichess of the diaphragm. This is great importance in the design of very sensitive micro-machined diaphragms, particularly for capacitive sensing applications.

(ii) Corrugated Diaphragm

With the introduction of corrugations into the diaphragm structure the situation can be changed dramatically. For shallow, sinusoidal corrugations the deflection is approximately given by [IO] :

WO WO'

E'h4 h h' -_ pR4 -u,-+b,- ( 1 1 )

where

b, = 165k + 0% + 3)) (13) q2(q + 4xq + 11X2q + 1X3q + 5 )

and

and for shallows, sinusoidal profiles :

with q the corrugation quality factor and H the cormgation depth. Thus q varies from 1, for a flat diaphragm, to a value that approaches 1.22 times the ratio of corrugation depth to diaphragm thickness.

There is an additional factor which needs to be considered depending on the method of fabrication of the diaphragms. The ,m$itional approach has been to use,a heavily doped boron layer as the etch stop used to form the diaphragm. It has-been shown [Ill that'these . ' :. etch sops introduce considerable residual tension in the resulting diaphragms. For large values of initial tension the deflection of a flat diaphragm can he represented by :

PR' 4aRz WO Eh' Eh' ( h ) (I6) -=- -

where (r is the initial stress. This resistance to bending due to initial stress can be added to the terms given above using the principle of superposition to give, for a flat diaphragm : PR4

(17) For a corrugated diaohram:

i2 wo[- b i ap h'] RZ h 2.83 4 R'

P=4-- u-+-E- (18) - -

The mechanical sensitivity of a circular diauhramn is defined as

Therefore the mechanical sensitivity of the cormgated diaphragm with initial stress, for small deflections is given by

" 2

(20)

111. THEORETICAL ANALYSIS

The capacitance is a non linear function of the pressure since it varies inversely with w, diaphragm deflection. If the non-linearity is calculated for sensors of two different plate radii and identical pressure ranges, it i s apparent that the smaller device is much more linear but has low sensitivity, while the opposite is for the larger device. Fig. 3 illustrates the difference in capacitance values, sensitivity and non-linearity between a large (R= 240 pm) and small (&I30 pm) device. This discrepancy can be exploited in a linearity calibration technique, as developed in [13]. From the observation of Fig.3, it can be concluded that as the sensor radius is reduced, the pressure sensor sensitivity for a given pressure range decreases, due to the relative increase in stiffness of the membrane for a given

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pressure load. The tradeoff in non-linearity vs sensitivity can be exploited in developing schemes taking advantage of both

f 1.4 R=240um g 1.31

0.243

0.241 n

8 = 0.239 .-

0.237

0.235

IO 20 30 40 50 60

Pressure (psi)

R- 130 um

10 20 30 40 50 60

Pressure (psi)

Fig.3(a) Comparison of capacitance performance of two different sensor sizes. Both devices have h = IOW, d = 2 pm

10 1 R = 130 um

10 20 30 40 50 60 Pressure (psi)

f 0.0164 R,= 130 um 0.0162

.- 2 0.0156 1 I - 0.0154

0.0152 0.015

10 20 30 40 50 60 Pressure (psi)

Fig. 3@) Comparison of capacitance performance of hvo different sensor sizes. Both devices have h = IOpm, d = 2 pm

10 20 30 40 50 60 Pressure (psi)

R = 130 um (a) 0.2425

0.2405

j3 0.2385

n 0.2365 ' 0.2345

: 'Z CL

10 20 30 40 -50 60

Pressure (psi)

Fig. 3(c) Comparison of non-linearity performance of hvo different sensor sizes. Both devices have h = IOW, d = 2 F : (a) Linear Capaciatnce and (b) Capacitance.

aspects, by utilizing both large and small devices to contribute bigb sensitivity and low non- linearity respectively. Specifically, for a given full-scale pressure a small diameter sensor can be used for linearity while the large diameter sensor provides high resolution and sensitivity[l3]. By suitably manipulating signals from the two sensors, it may be possible to continuously linearize the output. As we can see from Fig. 4, for the range of

pressures applied, the deflection of the center of the diaphragm has been found to vary linearly with pressures. This shows that the pressures applied fall within the proportional limit of silicon. The large diaphragm is found to deflect greater compared to the smaller one.

0 2 4 6 8 1 0

Pressure (kPa)

Fig. 4 Pressure-load deflection of a diaphragm with various size : (a) 6 5 0 p @) 600 p, (c ) 550 p, and (d) 500 p .

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Fig. 5 shows the calculated central deflections versus the corrugation numbers (N) of a rectangular cormgated diaphragm. As can be seen that smaller cormgation numbers normally result in larger central deflections under definite initial stress, while for large corrugation number m=lO means the whole diaphragm area is occupied by the corrugations), very small static deflection can be expected. Thus for the purpose of reducing the static deflection (as is preferred in capacitive sensors), cormgations occupying the whole diaphragm area are necessary. Fig. 6 shows the calculated results of

mechanical sensitivities for different cormgation depths of square and circular cormeated dianhraems. The initial stress and

I . I

Young's Modulus are assumed to be Mpa and 2OOGpa respectively.

3.5 1

0 10 20 30 40 50 60 70 80 Pressure (kPa))

F i g 3 Calculated pressure-cenml deflection cwes vs cormgation numbers (N) for a rectangular cormgated diaphragms (Young's Modulus E=200 Gpa, initial stress IT = 70 Mpa, H = 1.2 pm, H= IO pm). : (a) N=2 @) N = 5 and (c) N= IO..

0.25

$ 0.2 1 7

It can be seen that both diaphragms exhibit a peak at the corrugation depth, H = 4 - 5 p m at which the mechanical sensitivity ' reaches the maximum.. There are' no obvious peaks for-H larger that 5-7 ' -k m. For 'the cnmigated diaphragms with initial stress the mechanical sensitivity is higher for small cormgation depths compared to relatively large cormgation depth. It was found from the.calculated results that f0r.a square cormgated diaphragm, the curve remain relatively flat i.e ., the diaphragm mechanical sensitivity is not sensitive to the large cormgation depths of at least IO p m.

Fig.7 shows that the mechanical sensitivity of a corrugated diaphragm with two values of initial stress. It can he seen that the mechanical sensitivity is increased when the initial stress is increased for small corrugation depth. In this case it may be corrugated diaphragm with a cormgation depth of at least 8p.

tal 0.3

0.2 - m $ 0.1 3 g o 'a .- m-0.1

-0.2 m

-0.3 corrugation Depth (urn)

Fig.7 The calculated mechanical sensitivity of a circular cormgated diaphragm versus corrugation depths (H) under different values of initial stress: (a) Initial stress = 10' Pa and @) IO' Pa

0.3

3 ._ e.-..* 5 0.05 *....

3 5 7 10 12 15 (a) 2 4 6 8 1 0 1 2 q wlues

rn

Corrugation Depth (urn) Fig. 8 Calculated relationship of sensitivity versus q values

Corrugated diaphragms with different q (2.65, 5, 7.42, 9.89, 12.2, and 14.7) values have been investigated. The results are shown in figure 8. As we can see, the larger value of q would produce the lower mechanical sensitivity.

Fig. 6. Calculated mechanical sensitivities versus cormgation depth of different diaphragm : (a)

'Omgated diaphragm and @) flat cormgated diaphragm.

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iV. CONCLUSION

The development of high-performance diaphragm structures is of critical importance in the successful realization of micro- machined pressure sensors. Pressure sensors for different pressure ranges can be designed hy adjusting the parameters such as diaphragm thickness and cormgation depth. The sensitivity of the diaphragm can be optimized by choosing appropriate structure parameters. The q value (cormgation quality factor) is an

important parameter that determines the mechanical sensitivity and the rigidity of the hack-plate. Optimal structures can be obtained hy choosing an appropriate q value.

V. ACKNOWLEDGEMENTS

The authors would like to thank the Electrical Department of University Malaya for their support and contributions.

VI. REFERENCES

[ I ] M. Giovanni,”Flat and Corrugated Diaphragm Design Handbook,” Marcel Dekker, Inc., New York, 1982. [2] I. Jerman,’The Fabrication and Use of Micro-machined Cormgated Silicon Diaphragms,” Sensors and Actuators, A23 (1990) 988. [3] K. E. Peterson,“Silicon as a mechanical material“, Proceedings of the IEEE pp.420- 457 (Mq 1982) [4] X . Ding, W. KO and J. Mansour,”Residual stress and mechanical properties of Boron Doped P’ - Silicon Films,’’ Sensor and Actuators, A23 (1990) 866. [ 5 ] X. Ding, W. Ko,”Buckling Behavior of Boron Doped PI - Silicon Diaphragms,” Transducers ’91, Digest of Technical Papers, pp. 201. [6] F. Maseeh and S. Senturia,”Plastic Deformation of highly Doped Silicon,” Sensors and Actuators, A23 (1990) 861. [7] S. P. Timoshenko, Theory of Plates And Shells New York: Mc Craw Hill, 1959. [8] Husak,M, On-Chip Integrated Resonance Circuit with the Capacitive Pressure SensorJonma1 of Micromechanics and Micro- engineering 7(1997),pp. 173.178. [9] Bert, C.W. and Martindale, J.L., An Accurate, Simplified Method for Analyzing Thin Plates Undergoing Large Deflections, J.AIAA, vol. 26, n. 2,Feb. 1998, p.235.

[IO] Haringx, J.A., Design of Corrugated Diaphragms, ASME Transactions, vol. 79, 1957,~. 55. [ l I ] Hin-Leung Chau and Wise, loc. Cit. [I21 J. A. Llyod, Phd theses& Integrated Circuit Pressure Sensing System with Adaptive Linearity Calibration., Massachussets institute ofTechnology, Jun 1997

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