Modelling and simulation of a silicon micro ...

Modelling and simulation of a silicon micro-diaphragm piezoresistive pressure sensor using Finite Element Analysis (FEA) tools Ah Ju Pang and Marc P.Y. Desmulliez MIcroSystems Engineering Centre (MISEC) Department of Computing and Electrical Engineering, Heriot-Watt University, Edinburgh, U.K. EH14 4AS Tel: +44 (0) 131 449 5111 (ext. 4167) Fax: +44 (0) 131 451 3327 Email: [email protected] Abstract In this paper are presented the diaphragm deflection and output voltage obtained by applying a series of pressure loads to a silicon piezoresistive micro-diaphragm with built-in edges. Sensors of different length and thickness of diaphragms as well as various resistors lengths have been modelled and simulated using Finite Element Analysis methods. The resulting values of the membrane deflection and output voltage are compared with experimental and numerical data. The absolute value of the output voltage and its variation with respect to physical parameters are shown to depend on i.) the resistor length, ii) its placement within the diaphragm and iii.) the thickness of the diaphragm. Keywords: Piezoresistive, Pressure sensor, Finite Element Analysis (FEA), Micro-diaphragm, Modelling, Silicon micromachining

Introduction In pressure sensors, sensing is implemented by a micro-diaphragm that undergoes deflection under strain; this deflection is converted into output electrical signals. This article aims to study piezoresistive sensors for which the electrical resistance of a resistor placed on the deflected diaphragm changes in response to mechanical stress exerted on the membrane. The first application of the piezoresistive effect was for metal strain gauges to measure parameters such as force, weight and pressure. In 1954 C.S. Smith discovered [1] that, depending on orientation of silicon and germanium crystal, the piezoresistive effect in these crystals is 100 times bigger than with metals. The first pressure sensors based on diffused (impurity doped) resistors on to a silicon diaphragm were demonstrated in 1969 by Gieles [2]. Since then vast research effort and development in this area have been carried out. A typical silicon diaphragm pressure sensor would consist of a thin silicon diaphragm used as the elastic material and piezoresistive resistors made by diffusing impurities into the diaphragm. It is customary to connect the resistors as a Wheatstone bridge configuration as shown in figure 1. This paper aims to model and simulate such a pressure sensor using FEA methods and compares the results with analytical expressions found in the literature. In particular, this work demonstrates that there exists an optimum length and placement of the resistors for best pressure sensor performance. L

V x

y

L

R11

Diaphragm

R +

∆ R

∆ R

Diaphragm

54.7 R11

R11

R12

∆V

R12

R12

R +

R +

∆ R

R +

0

∆ R

Bulk area

(a)

(b)

(c)

Figure 1 (a) resistors layout and dimensions of the studied diaphragm pressure sensor . (b) schematic representation of the four piezoresistors arranged as a Wheatstone bridge. (c) typical bulk silicon micromachined pressure sensor

1

Analytical expression of the diaphragm deflection This section describes the mechanical response of a thin diaphragm subject to an applied stress. The classical thin plates theory is used in order to obtain a closed form expression of the deflection [3]. L

z (inward) L

( 0, 0 )

x ( x/L , 0 )

y ( 0 , y/L ) Figure 2. Diagram of a square diaphragm oriented in the x-y plane

Considering a thin square diaphragm, of length L, as shown in figure 2, the deflection, W, in response to pressure, Pa, on the x-y plane (in Cartesian co-ordinates), is given by two equilibrium equations namely [4,5, 6]:   ∂ 2 W ∂ 4F ∂ 4F ∂ 4F +2 + = E  ∂x 4 ∂x 2∂y 2 ∂y 2  ∂x ∂y 

2   2 2  − ∂ W ∂ W   ∂x 2 ∂y 2   

Equation 1

and ∂4W ∂4W ∂ 4 W 12 (1 − v 2 ) +2 2 2 + = 4 ∂x ∂x ∂y ∂y 2 Eh 3

  ∂2F ∂2W X Pa + h  2 2   ∂x ∂y −2

Equation 2

∂ 2 F ∂ 2 W ∂ 2 W ∂ 2 F   + ∂x ∂y ∂x ∂y ∂x 2 ∂y 2 

where E is the Young’s modulus, v is the Poisson’s ratio, h is the diaphragm thickness and F is the force. The details of deriving the partial differential equations 1 and 2 can be found in [5,6,7]. The two-coupled partial differential equations are difficult to solve. The problem is to determine W and F that satisfy the boundary conditions of square silicon diaphragm with built-in-edges of length L (as in figure 2). These boundary conditions are defined as [3]: W=0

and

∂W =0 ∂x

at x = ±

L 2

Equation 3

2

W=0

and

u = 0 and

d = 0 and

∂W =0 ∂y ∂ 2F ∂x ∂y ∂ 2F ∂x ∂y

at y = ±

L 2

= 0 at x = ±

L 2

Equation 5

= 0 at y = ±

L 2

Equation 6

Equation 4

where u and d are the mid-plane displacements of a diaphragm in the x and y directions; these are defined as [7] : x

u= 0

∫

2  1  ∂2F ∂ 2 F  1  ∂W     2 −v 2 −   dx  ∂x  2  ∂x    E  ∂y

∫

2  1  ∂ 2F ∂ 2 F  1  ∂W        dy v − −  E  ∂x 2 ∂y 2  2  ∂y   

y

d= 0

Equation 7

Equation 8

The deflection W(x,y), which satisfies the boundary conditions given by equations 3 and 4 can be approximated by: W(x,y) = h f(E,v,L,h,Pa) cos 2 (πx / L ) cos 2 (πy / L )

Equation 9

where f(E,v,L,h,Pa) is a function that depends on E,v,L,h and Pa. For the sake of clarity, this function is written in the rest of the article as f(Pa). It is worth noting that the general expression of W(x,y) is in the form of an infinite series of squared cosine terms as discussed in [3]. However, the second or successive terms in the series provide a negligible contribution to W [6, 8]. At first order approximation, and using equations 1 and 9:

∂ 4F ∂ 4F ∂ 4F π4h 2E +2 + =− [f (Pa )]2 4 2 4 2 2 2L ∂x ∂x ∂y ∂y

  2πx   2πy   4πy  X cos  + cos  + cos  L L     L   

Equation 10

 4πy   2πx   2πy  + cos  + 2 cos  cos   L   L   L   4πx   2πy   2πx   4πy  + cos  cos  + cos cos   L   L   L   L 

3

The solution of equation 10 that satisfies the boundary conditions as presented in equations 5 and 6 is: F(x,y) = −

 3π 2 Eh 2 [f (Pa )] 2  (x 2 + y 2 ) 2 32 12L (1 − v)  2πy   2πx  − cos   − cos  L   L  1  2πx   2πy  1  4πx  − cos  − cos  cos  2 L L 16  L     

Equation 11

1  4πy  1  2πx   4πy  − cos cos −  cos 16  L   L   L  25 Substituting equations 9 and 11 into equation 2 yields an error function e(x,y) that can be written as [7] : e(x,y) =

∂4W ∂x 4

+2

∂4W ∂x 2 ∂y 2

+

∂ 4W ∂y 2

−

12 (1 − v 2 ) Eh 3

  ∂2F ∂2W X Pa + h 2  ∂x ∂y 2  

Equation 12

∂ 2 W ∂ 2 F   ∂x ∂y ∂x ∂y ∂x 2 ∂y 2  In order to minimize the error function (equation 12), as discuss in [7], the Ritz-Galerkin method is employed [9]. It is required that e(x,y) is to be orthogonal to the deflection W over the domain of the diaphragm so that: −2

∫ e( x, y)W ( x, y)

∂2F

∂2W

+

∂x ∂y = 0

Equation 13

Using equation 9, 12 and 13 yields a cubic equation to be satisfied by f:

 3π 2 h 2 (1 − v 2 )  9 3 + 0.1666[f (Pa )]  2 2 L  64(1 − v )  +

Equation 14

3(1 − v 2 ) L2 2π 4 h 2 [ ( ) ] f Pa − Pa = 0 L2 Eh 2

Numerical and simulated analysis of diaphragm displacement The solution of the cubic equation 14 provides f(Pa) as a function of Pa, E, v, L and h. This solution can itself be fed back into equations 9 and 11 in order to calculate W and F, respectively. The calculation of the output voltage, given W, F and f(Pa) is carried out in later sections. Considering a square silicon diaphragm of the following parameters: h = 9.8 µm, L = 0.87 mm, v = 0.066 [7, 8] and E = 1.68 x1011 Nm-1, equation 14 becomes: 1.254[f(Pa)]3 + 2.472f(Pa) – 1.664x10-5 Pa = 0,

Equation 15

4

where Pa is in Pascal. Equation 15 has only one real root. Using equation 9, the diaphragm deflection W(x,y) can be found at any position on the diaphragm as illustrated in figures 3 and 4.

6.0

Quas i -anal yti cal [7] 5.0

µ

E xper i mental [5] ANSY S-s i mul ati on

4.0

3.0

2.0

1.0

0.0 100

150

200

250

300

350

400

450

500

550

600

A ppl i e d pr e s s u r e ( mm H g)

Figure 3 Diaphragm deflection as a function of applied pressure Pa

Diaphragm deflection W(x,y=0)

1 0.9

Quasi - anal y t i cal [ 7]

0.8

ANS YS - si mul at i on

0.7

E x per i ment al [ 5]

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Po s itio n o n d iap h r ag m (x/L )

Figure 4 Diaphragm deflection W(x,y=0) as a function of position on the diaphragm (x/L) at 100mmHg.

Three curves are represented in figures 3 and 4: the quasi-analytical results obtained in [7], the experimental results from [5] and our ANSYS simulated results. Figure 3 shows the deflection of the diaphragm W at x=0,y=0, with respect to different input pressures Pa. Figure 4 shows the deflection distribution W at y=0 along the x-axis of the diaphragm at a pressure Pa = 100mmHg. The FEA-simulated data are quite close to the quasi-analytical values and experimental results from [5]. The deflection coefficient in the quasi-analytical technique is calculated to be around 79 µm(mmHg)-1 at the center of the diaphragm, compared 75 µm (mmHg)-1 in the FEA case. This slight variation may be due to the effect of the meshing density and the type of nodal element used in the FEA method.

Stress calculation on the diaphragm In this section a full bridge with 4 p-type piezoresistors laid out on the pressure sensor diaphragm is used as a reference for stress analysis as shown in figure 1. Comparisons of both analytical values and the FEA simulation results are carried out. It is reported in [10, 11] that maximum pressure sensitivity is achieved using two parallel and perpendicular resistors oriented in the

5

[1,1,0] crystal direction ,when placed near to the edge of the diaphragm as shown in figure 1. This placement is optimum because of the maximum effective mechanical stress found at the edge of a diaphragm where x/L = ± 0.5 and y/L = ± 0.5 (figure 2). Once the deflection W(x,y) is known, the total stress, parallel to the surface of diaphragm is defined as [3,5, 8]:

σ xx = σ yy =

∂ 2F ∂y 2 ∂2F ∂x 2

σ xy = −

−

Eh  ∂ 2 W ∂ 2 W  v + 1 − v 2  ∂x 2 ∂y 2 

Equation 16

−

Eh  ∂ 2 W ∂ 2 W  v + 1 − v 2  ∂x 2 ∂y 2 

Equation 17

∂2F ∂x ∂y

−

Eh ∂ 2 W 1 − v ∂x ∂y

Equation 18

The first and second terms in equation 16 and 17 represent the membrane and bending stress, respectively [5 ,8]. The functions  ∂ 2F  W and F obey equations 9 and 11, as before. For our pressure sensor, the contribution of the membrane stress  2  to the  ∂x    overall x- or y-stress component is relatively small (less than 0.1% of the overall value), compared to the bending stress Eh  ∂ 2 W ∂ 2 W  v . The membrane stress can therefore be neglected in the calculations mentioned below. + 1 − v 2  ∂x 2 ∂y 2 

Resistance calculations The change of resistance due to an applied stress is related to the piezoresistive coefficient. For a diffused resistor arranged in parallel and perpendicular orientation on the diaphragm, the resistance change is [10]:

∆R R = π || σ yy + π ⊥ σ xx

Equation 19

where π|| and π⊥ are the piezoresistive coefficients in longitudinal (y-direction) and transverse (x-direction). For silicon, there are only 3 non-zero coefficients π11, π12 and π44. These coefficients depend on the level of doping and the temperature. The magnitude of π44 varies from 90 x 10-12 cm2 dyn-1 (or 90 x 10-11 m2 N-1) to around 138 x 10-12 cm2 dyn-1 (or 138 x 10-11 m2 N-1) due to doping and temperature variations [10]. The common values used for π11 and π12 are 3 x 10-12 cm2 dyn-1 and -1 x 10-12 cm2 dyn-1, respectively. When the resistors are arranged in direction, the longitudinal and transverse piezoresistive coefficients can be approximated by [10]:

π|| = π11 − 2 (π11 − π12 − π 44 ) 1 / 4 = 40.5 x 10-11 m2 N-1

π ⊥ = π12 − 2 (π11 − π12 − π 44 ) 1 / 4 = -38.5 x 10-11 m2 N-1

Equation 20 Equation 21

From [7-10] the piezoresistive coefficients can be approximated to:

π || = −π ⊥ = 1 / 2 π 44

Equation 22

The resistance changes of R11 and R12 as shown in figure 1 are given by:

 ∆R    = π || σ yy + π ⊥ σ xx = α  R 11

Equation 23

6

 ∆R    = π|| σ yy + π ⊥ σ xx = β  R 12

Equation 24

Substituting Equation 22 to both equation 23 and 24 yields:

π 44 π  ∆R   ∆R  s   = −  = 44 (σ1y − σ1x ) = 2 2  R 11  R 12

Equation 25

where ‘s’ is defined as the effective mechanical stress. For example, assuming that one of the resistors R11 is placed at the edge of the diaphragm, where the coordinate is x/L=0 and y/L = 0.5, the solution for effective stress ‘s’ is at first approximation: s = (σ1y − σ1x ) =

Eh ∂ 2 W (1 − v ) 1 − v 2 ∂y 2

Equation 26

Output voltage calculation

Assuming a reference supply voltage Vref and a differential output voltage of ∆V , the dimensionless output voltage of the bridge becomes [7, 8, 10]:

(∆R R )11 − (∆R R )12 ∆V  β − α  ∆V  or , =  = Vref  2 + α + β  Vref 2 + (∆R R )11 + (∆R R )12

Equation 27

with α, β defined in equations 23, 24. Substituting equation 25 into 27 yields a simpler version of equation 27:

∆V 1 = π 44 ( σ 1 y − σ 1 x ) V ref 2

Equation 28

When the calculated effective stress ‘s’ is substituted into equation 28, the dimensionless variation of output voltage as a function of applied pressure Pa is plotted as shown in figure 5.

0.04 Dimensionless output voltage

Numerical [5]

0.03

Quas i-analytical [7] F E A-s imulation

0.03 0.02 0.02 0.01 0.01 0.00 0

50

100

150

200

250

300

350

Applied pressure (mmHg)

Figure 5 Variation of the dimensionless bridge output voltage as a function of pressure Pa (L=1mm, h=10 µm

7

The FEA-simulated results are in good agreement with the calculated values obtained from [7]. Looking at the slopes of the different curves in figure 5, the sensitivity of the output voltage is calculated to be around 82 µV/V.mmHg in the FEA case, 89 µV/V.mmHg for the analytical model and 94 µV/V.mmHg for the numerical method of [10]. It is observed that in the low pressure range (0 to 150 mmHg) the dimensionless output voltages are similar.

Effect of different diaphragm and resistor’s dimension The above discussion on the voltage sensitivity assumes that the resistors are point transducers placed at the edge of the diaphragm where maximum levels of stress are found. In any real device, resistor size has to be traded off against manufacturability and pressure sensitivity. The sensor sensitivity decreases with increasing resistor size due to the stress averaging over the finite resistor dimensions, whilst manufacturability of the device is eased for large resistors. In this section, a series of simulations are carried out using a sensor with larger diaphragm dimension (L=3.4 mm). Thicknesses of 10 and 20 µm are considered and analysed with various resistor sizes. Three different resistor lengths, namely 160 µm, 200 µm, and 260 µm are used for the simulation. Simulated output voltages for diaphragm thicknesses of 10 and 20 µm are plotted in figures 6 and 7 with different resistor lengths used as parameters.

000.0E+0

000.0E+0 1000

2000

3000

4000

5000

6000

7000

8000

9000 10000

1000

-10.0E-3

4000

5000

6000

-30.0E-3

RL260;dt20

-40.0E-3

8k-10k Pascal

-50.0E-3 -60.0E-3 -70.0E-3

7000

8000

9000 10000 RL160;dt10

RL200;dt20

Output Voltage (Volt)

Output Voltage (Volt)

3000

-20.0E-3

RL160;dt20

-20.0E-3

2000

RL200;dt10

-40.0E-3 RL260;dt10

-60.0E-3

8k-10k Pascal

-80.0E-3

-100.0E-3

1k-4k Pascal -80.0E-3

1k-4k Pascal Pressure (Pascal)

Figure 6 Graph showing output voltage vs pressure, for different resistor lengths with diaphragm thickness of 20 µm.

-120.0E-3 Pressure (Pascal)

Figure 7 Graph of output voltage vs pressure, showing different resistor lengths with diaphragm thickness of 10 µm.

As the resistor length increases, the output voltage decreases relatively. The region in which maximum stress occurs (the edge of the diaphragm) is independent of the values of the pressure that is being applied on the diaphragm. As the resistor length increases, the total amount of effective stress that acts on the resistor area is reduced, which in turn leads to a decrease in the output voltage. This phenomenon is more pronounced with diaphragm thickness of 10 µm (figure 7) in the pressure range 8-10k Pascal. The output voltage produced by different resistor lengths at low pressure range (1-4k Pascal) are also very close to each other. At higher pressure (range of 8-10k Pascal) and thin diaphragm (10 µm), the voltage deviation is strongly dependent on the resistor lengths.

Voltage sensitivity for diaphragm thickness of 20 µm with different resistor lengths The sensitivity of a sensor depends upon the geometry of its diaphragm, its cross-sectional area, thickness and shape. It also depends on the shape and size of the resistor and its position on the surface of the diaphragm. The sensor sensitivity is defined as [10, 12]: Ap=

1 ∂ Vout Vref ∂ Pa

Equation 29

8

where Vref , the supply voltage is assumed to be 5V for all simulations. From figure 6, for a diaphragm thickness of 20 µm, different resistor lengths have little effect on the output voltage for the whole pressure range of 1-10k Pascal. Therefore the profile of the curve can then be calculated: Vout =

1 7 x 10-6 Pa + C Vref

Equation 30

where ‘C’ is a constant for the case of non-zero Vout at zero pressure. By taking the derivative of equation 30, the sensitivity for diaphragm thickness of 20µm and length of 3.4mm is found to be 1.4 µV/V.Pascal or 187 µV/V.mmHg. Comparing to the work in [7, 10] for diaphragm length = 1 mm and thickness = 10 µm, with sensitivity of 95 µV/V.mmHg, the sensor here is about 50% more sensitive. This value is reasonable, as the diaphragm length is larger in our case.

Voltage sensitivity for diaphragm thickness of 10 µm with different resistor lengths Taking the derivative of all the graphs in figure 7, the sensitivity of the sensor is plotted as a function of pressure as shown in figure 8. The sensitivity graphs are not linear in nature. There are two distinct categories of sensitivity, one in the low- and the other on the higher-pressure range. By taking the average of the first and last three values on the graphs, the low- and highpressure range sensitivity are obtained as shown in table 1. Of all the resistor lengths shown in table 1, the 160 µm length case produces the best sensitivity for all range of pressure from 1-10k Pascal in all simulations that have been performed. Sensitivity with resistor length 160 µm is about 2.5 times higher for a diaphragm thickness of 10 µm compared to the 20 µm case. This observation is important when it comes to design and manufacture of pressure sensors for low-pressure applications.

Gr a p h o f s e n s itivity V s P r e s s ur e for d iffe r e nt r e s is to r le ng ths a t dia p hr a g m th ic k n e s s o f 1 0 µ m 5 .0 4 .5

dv /dp R L 1 6 0 ;dt1 0 dv /dp R L 2 0 0 ;dt1 0

4 .0

dv /dp R L 2 6 0 ;dt1 0

3 .5

µ

3 .0 2 .5 2 .0 1 .5 1 .0 0 .5 0 .0 1

2

3

4

5

6

7

8

9

10

P r e s s u r e ( k P a s c a l)

Figure 8 Graph of sensitivity as a function of pressure for different resistor lengths at diaphragm thickness

Resistor Length 160 µm

Sensitivity Low- pressure range (1-4k) High- pressure range (8-10k) (µV/V.Pascal) (µV/V.Pascal) 3.5 1.25

200 µm

3.2

1.15

260 µm

3.0

1.00

Table 1 Sensitivity for different resistor lengths at diaphragm thickness of 10 µm

9

Conclusions This paper has reviewed some properties and characteristics of thin-diaphragm silicon pressure sensors. The equations that govern the diaphragm deflection, stress levels, and piezoresistance have been presented. Using these equations, the pressure sensitivities of the sensors can be calculated. A sensor with diaphragm length of 1mm and thickness of 10 µm is used as a reference in order to compare and discuss the FEA-simulated results, numerical and experimental results of the deflection and output voltage. A series of simulations have been carried out using a sensor of diaphragm width of 3.4mm, thicknesses of 10 and 20 µm and different resistor lengths. The pressure sensitivities of the sensor with different resistor lengths have been presented and explained. By reducing the resistor length and placing the resistor optimally on the diaphragm, maximum sensitivity can be achieved.

Acknowledgement The author would like to thank R.Buchhold from Dresden University of Technology (Germany), for his help and invaluable advice, Dr Susan M. Flockhart and Dr Julian A.B. Dines for their help in using the FEA-software tools. The work was sponsored under the aegis of Scottish Entreprise and the Institute for System Level Integration (ISLI).

Reference: [1]

C.S Smith “Piezoresistance effect in germanium and silicon.” Phys. Rev. 94 pp42-49 1954

[2]

Gieles, C.M, “Subminiature silicon pressure sensor transducer” Digest IEEE international solid-state circuits conference, Philadelphia, PA, Feb. 19-21, 1969 pp 108-109

[3]

S. Timoshenko and S. Woinowsky-Krieger, “Theory of plates and shells,” McGraw-Hill, New York, 1959, pp.415-428

[4]

William Pomeroy, “Surface Micromachined Pressure sensor”, Dissertation, PhD. University of New Mexico, (May1997) pp82-104

[5]

K.W. Lee and K.D. Wise, SENSIM: a simulation program for solid-state pressure sensore, IEEE Trans. Electron Devices, ED-29 (1982) pp. 34-41

[6]

H.F. Banuer, “Nonlinear response of elastic plates to pulse excitations, J.Appl. Mech., 35 (1968 47-52)

[7]

H.E. Elgamel “Closed-form expressions for the relationships between stress, diaphragm deflection, and resistance change with pressure in silicon piezoresistive” Sensor and Actuators A 50 1995 pp.17-22

[8]

H.E. Elgamel, “Aninvestigation of the piezoresistance of silicon with reference to pressure sensors” M.Sc. Thesis, Cairo university 1998 pp. 101-113

[9]

B. Banerjee and S. Datta, “A new approach to analysis of large deflections of thin elastic plates”, J.Non-Linear Mech, 16 (1981) pp.47-52

[10]

Samuel K. Clark and Kensall D. Wise “Pressure sensitivity in anisotropicall etched thin-diaphragm pressure sensors” IEEE trans. Electron devices vol. ED-26 December 1979 pp.1887-1896

[11]

Samaun, K.D Wise and J.B. Angell, “ An IC piezoresistive pressure sensor for biomedical instrumentation” IEEE Trans. Biomed. Eng., vol. BME-20 pp101-109, Mar 1973

[12]

Julian W. Gardner, “Microsensors Principles and Application” John Wiley & Sons, Inc 1994 pp184

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