Optimum design considerations for silicon piezoresistive pressure sensors

SEgRS

ACTWORS A E LS E V I E R

Sensors and Actuators A 62 (1997) 539-542

PHYSICAL

Optimum design considerations for silicon piezoresistive pressure sensors Yozo Kanda a,,, Akio Yasukawa b "Department of Electrical and Electronic Engineering, Faculty of Engineering, Toyo University, Kawagoe,350, Japan h Mechanical Engineering Research Laboratoo', Hitachi Ltd., Tsuchiura, 300, Japan

Abstract Optimum design considerations for silicon piezoresistive pressure sensors are carried out on the basis of the design restriction release produced by recent progress in microwave plasma etching technology. The optimum shape and plane of a diaphragm and the optimum direction of gauges are analysed by taking into account the effects of anisotropy of the piezoresistivity and elasticity and large deflection of diaphragms. A new index 7/, which expresses the relative performance of sensors, is introduced for the optimal gauge direction on the optimal diaphragm plane. It is found that the optimum design is as follows: four gauges are aligned on a (111 ) direction on a {110} plane square diaphragm with a centre boss. © 1997 Elsevier Science S.A. Keywords: Pressure sensors; Piezoreslstive; Si( 110); Optimum design

1. Introduction The diaphragms of conventional silicon pressure sensors are formed by an anisotropic wet etching process. Therefore the wafer orientation and line direction of a square diaphragm are limited to (001) and along (110), respectively. Longitudinal gauges R~ and transverse gauges R~ are parallel and perpendicular to (110), respectively, as shown in Fig. 1. Recent progress in microwave plasma etching (MPE) [ 1,2] of crystalline silicon releases the limits of usual anisotropic wet etching and extends the freedom of design and fabrication of MEMS. Namely, MPE can etch an arbitrary shape on an arbitrary wafer orientation. The maximum longitudinal piezoresistive coefficient H~ is along a (11 i) orientation, which is on a {011 } wafer orientation as shown in Fig. 2 [3]. In order to get high sensitivity, we hope to use this orientation. Even if the sensitivity is high, the signal-to-noise ratio is the most important characteristic. Here we use noise in the broad sense. For highly precise sensors non-linearity is the most important and is regarded as noise. We shall consider again the optimum design for piezoresistive sensors in this situation. 2. Square or circular A square is superior to a circular shape for a sensor diaphragm. This is because the preferred sensor chip shape is *Corresponding author. Fax: +81 492 33 I855. E-mail: yozo@ krc.eng,toyo.ac.jp 0924-4247/97/$17.00 © 1997 Elsevier Science S.A, All rights reserved PIIS0924-4247(97) 0 1545-8

<110>

m

I

l

\ diaphragm

V//////////////////////A V/d ,OOll W/A Fig. 1. Conventional silicon sensor.

square from productivity considerations. We consider two situations: one is a square diaphragm made in a square chip and the other is a circular diaphragm made in a square chip. It is assumed that the chip Size and the minimum sealing width are the same for both cases. Then the size of one side of the square and the diameter of the circular diaphragm are the same. In this case the theory of plates [4] teaches us that the maximum stress of the square diaphragm is 1.64 times as large as that of the circular diaphragm when their diaphragm thickness is the same. This means that better characteristics are obtained in a square than in a circular diaphragm.

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K Kanda, A, Yasukawa / Sensors and Actuators A 62 (1997) 539-542

4. New index for optimizing crystal plane and direction

,' -

\

'

0 i~i o iooqo " 80 9D 2 "04-~0" L

)

t

-

°

~

0a-O .~1 60 70 ~ I~I IO0llO0/.l

~

09t

"

f\\,X",.

"

In this section a new index r/is proposed for obtaining the optimal crystal planes of a diaphragm and the optimal crystal directions of gauges. This index enables a relative comparison of the sensor performance between various crystal planes and directions to be made. In this index the effect of diaphragm dimensions is eliminated to obtain general results because optimal dimensions are different for the required pressure measurement range and precision in various applications. In the case where gauges are aligned to the x direction as shown in Fig, 3, the output voltage Vr is expressed by Vra %~'~r.,+ % 2'cr.,.-I- %~,"r,:,.

(1)

where "rh ~', %2' and ~r~,' are the longitudinal, transverse and shear piezoresistive coefficients, respectively, and are obtained from the three fundamental piezoresistive coeNcients ,rq ~, %2 and 7r,),~by coordinate transformation ~3/. o-,, ¢r~. and %. are the longitudinal, transverse and shear stress components, respectively, and am expressed by

Fig, 2, Piezoresistance coefficients in the (011 ) plane for p-type Si ( 10-~t

pa-I).

3. With or without centre boss

or.,.= E,:,.' er

(2)

or,, = E,.,' er

(3)

and %,= E~,.'er

A diaphragm with a centre boss as shown in Fig. 3 is superior to a diaphragm without a centre boss. A diaphragm which is connected with a centre boss and thick outer support portion is formed. Two pairs of p-type gauges (the outer and inner gauge pairs) are located above and close to the outer and inner edges of the diaphragm. A Wheatstone bridge is formed by connecting the four gauges. The outer and the inner gauges experience tensile and compressive stress, respectively, when front pressure is applied. This situation minimizes the non-linearity for both front and back pressures. Therefore this structure is also suitable for differential pressure sensors. The precise non-linear analysis of sensors with and without a centre boss was carried out in Ref. [5].

........ lY .

(4)

where E,..,.', E~,,' and E.,,/are the longitudinal, transverse and shear components of elastic rnodulus relating to longitudinal strain, respectively, and are obtained from the three fundamental compliance coefficients sz ~, sj~_and s44 by coordinate transformation [6]. The e r is the full-scale strain and is expressed by [5] ero~ pe/ ( Eh 2)

(5)

where Pr is the full-scale pressure, h is the thickness of the diaphragm and E is the equivalent elastic modulus of a diaphragm and is obtained from the three fundamental compliance coefficients st ~, s~2 and s44 for any crystal plane by the method in Ref. [6]. From Eqs. ( 1 ) - ( 5 ) we obtain Vr~ ( "n'tt 'E~. + 7rtz'E~.~+ "trt<,'E~)pr/(Eh 2)

(6)

The non-linearity NL, which is one of the most important properties of sensor accuracy, is resolved into components as follows [5]:

I I 1 ( I 1

I

~°°,

X

[

I J I

gauge

L~--

...... . . . . .

a i

Fig. 3, Sensor structure with a centre boss.

NL = N L v _ , + N L .... + NL, _ I,

(7)

where NLv_, is the non-linearity between output voltage and strain E, NL~_,. is the non-linearity between strain and deflection v and NL,,_p is the non-linearity between deflection v and pressure p. NLv_, is caused by the non-linear characteristics ofpiezoresistive gauges and is small when the strain value is small [7]. N L ..... is caused by the effect that the membrane strain component is non-linear to the deflection v. However, most of this component can be cancelled by forming gauges

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E Kanda, A. Yasukawa / Sensors and Actuators A 62 (1997) 539-542

at tensile strain positions and compressive strain positions, as shown in Fig. 3. On the other hand, NL,_~, cannot be cancelled and becomes the most important factor. As a result, the total non-linearity NL is approximately expressed as

NL=NL,._t,

(8)

The linear component of deflection v~ is expressed by [4] vl

oc pt'/ Eh 3

(9)

Table I Fundamental piezoresistive coefficients and compliance coefficients used in calculation 7rll ( 1 0 - " Pa - I ) 7rl2 ( 1 0 - " Pa - t ) "n'.~4(10 - l l Pa -~) sit (10 -Ij Pa -I) Slz (10 - t l Pa - I ) s4a ( 1 0 - " Pa -I)

6.6 - 1.1 138.1 0.765 -0.214 1.256

The non-linear component of deflection A v is expressed by [5] A U ~/.:13/,//2

'

]'Y

..........

( 1 O)

From Eqs. ( 8 ) - ( 1 0 ) we obtain

NL,._p -- A v/th ct vi2/h 2 =pr2/(E"h 8)

( 11 ) <111> ,

From Eqs. (6) and (11) the following equation is obtained:

Vr/(NLJ/4pl./2) c~ ~7

(12)

where r/is defined by r/= (~'l I'E,~ + "rfi2'E~.,+ rrIr'E.~.,.)/E 1/2

7/becomes a maximum in the optimum gauge direction of the optimum diaphragm plane because Vr is maximized when NL and Pr are held constant, r I represents a signal-to-noise ratio in a wider sense, in which the non-linearity is also regarded as noise. The index r/does not include the size of the diaphragm.

5. Results Fig. 4 shows calculated results using Eq. (13), where 0 is the angle from (110) and (100) on (110) and (100) diaphragm planes, respectively. The data used in the calculations are tabulated in Table 1, assuming p-type gauges. From Fig. 4 it is found that the optimal directions are ( 111 ) in the case o f a { 110} diaphragm and (1 I0) in the case of a 500 400

300 -

-

-

'

200 100

°o

-

-

~

~

.-"

110 direction

........ i .......... ..'~,.

-

1'0

f .....

(13)

i

,,

: .....

L ......

a

!

...........

Fig. 5. An acceleration sensor; the batohed portion is etched off.

{ 100 } diaphragm. In the former case, the value for'O is larger than in the latter case. Therefore a design with ( I 11 ) gauges on a {110} diaphragm is superior to one with (110) gauges on a {100} diaphragm. Specifically, "0(<111) on {110})/ r/( ( 110} on { 100}) = 1.4. This means that the output voltage in the (111 ) direction on {I l0 } is 1.4 times as ]argo as that of (110) on {100} when the non-linearity and full-scale pressure are the same. If the hatched portion of Fig. 5 is etched off this sensor can be used as an acceleration sensor.

6. Conclusions A new index, '0, which expresses the relative performance of sensors, is proposed for obtaining the optimal gauge direction on the optimal diaphragm plane. ",?represents a modified signal-to-noise ratio in which non-linearity is regarded as noise. Based on calculations of this index, it is found that a design with ( 111 ) gauges on a {110} diaphragm is superior to one with (110) gauges on a {100} diaphragm.

Acknowledgements

20'

angleO (degree) Fig. 4. Influence of the gauge direction on the performance index r/, The angle 0 of the solid and the dashed lines starts fi'om the (110) and the (100) directions on the (110) and the (100) planes, respectively.

]'his work is partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture (07455013) and by the Inoue Enryou Memorial Foundation for Promoting Science from Toyo University.

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E Kanda, A. Yasukawa / Sensors and Actuators A 62 (1997)539-542

References [ 1] K. Tsujimeto, S. Okudaira and S. Tachi, Low-temperature microwave plasma etching of crystalline silicon, Jpn. J. Appl. Phys., 30 (1991) 3319-3326, [2] W.H. Juan and S.W. Pang, High-aspect-ratio Si etching for microsensor fabrication, Z Vac. Sci. Technol. A, 13 (1995) 834--838. [3] Y. Kanda, Graphical representation of piezoresistance coefficients in silicon, IEEE Trans. Electron Devices, ED-29 (1982) 64-70. [4] S, Timoschenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, New York, 1959, [5] A. Yasukawa, M. Shimazoe and Y. Matsuoka, Simulation of circular silicon pressure sensors with a center boss for very low pressure measurement, IEEE Trans. Electron Devices, ED-36 (1989) 12951301. [6] A. Yasukawa, S. Shimada, Y. Matsuoka and Y. Kanda, Design considerations for silicon circular diaphragm pressure sensors, Jpn. Z AppL Phys., 21 (1982) 1049-1052. [7] K. Matsuda, K. Suzuki, K. Yamamura and Y. Kanda, Nonlinear piezoresistance effects in silicon, J. Appl. Phys., 73 (1993) 1838-1847.

Biographies Yozo Kanda received his B.Eng. degree in applied physics and the Dr.Sc. degree in physics from Waseda University in 1957 and 1968, respectively. For 12 years he worked on solidstate sensors and actuators at the Central Research Laboratory of Hitachi Ltd., Kokubunji, Tokyo. Between 1970 and 1972 he worked in the Sensor Physics Section, Measurement Phys-

ics Branch, Instrument Research Division of NASA Langley Research Center, Hampton, VA, USA, as a National Academy of Sciences Visiting Scientist. From June 1974 to March 1992 he was professor of physics at Hamamatsu University School of Medicine, Hamamatsu, Japan. Since April 1992 he has been professor, Department of Electrical and Electronic Engineering, Faculty of Engineering, Toyo University, Kujirai, Kawagoe, Saitama, Japan. His research interests are solidstate sensors and actuators and solid-state physics. He has attended all the International Conferences on Solid-State Sensors and Actuators (Transducers). Professor Kanda is a member of the Physics Society of Japan, the Japanese Society of Applied Physics, the Japanese Society of Medical Electronics and Biological Engineering and the Institute of Electrical Engineers of Japan.

Akio Yasukawa received the B.S. and Dr.Eng. degrees in mechanical engineering from Waseda University, Tokyo, in 1973 and 1994, respectively. Since 1973 he has been with the Mechanical Engineering Research Laboratory, Hitachi Ltd., where he has been engaged in research on stress analysis and in the development of software for designing semiconductor package structures. He is a senior researcher. Dr Yasukawa is a member of the Mechanical Engineers of Japan and the Society of Instrument and Control Engineers of Japan,