Piezoresistive pressure transducers

w 1. Design withfiat diaphragms

241

Chapter 6

Piezoresistive pressure transducers

w

Designs with flat diaphragms

At the early stages of their development, the diaphragms for silicon pressure transducers were circular ones processed by mechanical drilling (sometimes combined with isotropic chemical etching). Therefore, the crystal planes of the diaphragms were not restricted by the process. The most popular designs used (100) and (110) planes. With the replacement of mechanical drilling by anisotropic etching technology, the diaphragms of present day pressure transducers are almost exclusively based on the (100) plane with edges in the <110> directions. This means that the diaphragms are either square or rectangular. Therefore, the designs discussed in this section will be restricted to square or rectangular diaphragms of the (100) plane only.

w

Designs with square diaphragms

Square diaphragms can be formed by anisotropic etching of a (100) silicon wafer with the edges of the etching window in the <110> directions. As the sidewalls of the etching cavity are { 111 } planes, the size of the diaphragm (the bottom of the cavity) is smaller than that of the etching window by x/2d, where d is the etching depth of the cavity. In other words, the etching windows should be larger than the required diaphragm sizes by x/2d. Here let us suppose that the size of the diaphragm formed is 2a. For a square diaphragm, the maximum stresses are at the edge centers of the diaphragm, where the sensing elements are usually located. Now we discuss the designs for two types of sensing elements. The Wheatstone bridge and the four-terminal single sensing element (i.e., the "Hall-like" sensing element, or, Motorola's "X-ducer" design).

(1) The Wheatstone bridge design

242

Chapter 6. Piezoresistive pressure transducers

A typical design of a pressure transducer with a Wheatstone bridge is shown in Fig. 6.1. The four piezoresistors are located near an edge center of the diaphragm. According to w the approximate values of the stress components at the edge centers of the square diaphragm are: a2 a2 Txx = 1.02p ~-T' Trr = 1.02vp ~-T' Txr = 0

(6.1)

where h is the thickness of the diaphragm and p is the pressure applied. According to w for piezoresistors in the <110> directions, the components of the piezoresistive coefficient are: 1 1 ~l = -2~44' /i;t = - - 2 ~ 4 4 ' ~s = O

(6.2)

Therefore, for piezoresistors perpendicular to the edge (Rz and R3), the piezoresistance is: (___~_)! = 2~44 1 X 1.02P~22(1 v)

(6.3)

For piezoresistors in parallel to the edge (R~ and R4), the piezoresistance is" (___~_) t = - 2 1 ~44 x 1.02p-;-T(1 ta2 / - - v)

(6.4)

Therefore, the output of the Whetstone bridge is: 2 V o u t "-

0.51(1- v ) p ~a - rc44Vs

(6.5)

where V s is the supply voltage of the bridge. Note that r~44 is a function of the doping level of the piezoresistors. y [

edge of the diaphragm

I

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii ...... ;" o

v ..... Ji,i

~'"" ..... iiiiiii

x "

El

Fig. 6.1. Pressure transducer with a Wheatstone bridge

w 1. Design with fiat diaphragms As

an

example,

let

us

suppose

that

a-lmm,

243

h-30ktm,

~44-

80 x 10 -ll / Pa, p - lOOkPa,V s - 5V and v - 0.3. Using these data values, the output of the pressure transducer is found to be Vout - 158 m V . The calculated result has overestimated the sensitivity of the pressure transducer, as the resistors have to be away from the edge center by some distance for process considerations. Therefore, the stresses are reduced accordingly. According to Eq. (2.177), for x - 0.ga and y - 0, we find: a2

Txx - Tr,r, = 0.72p-~-T(1- v) Therefore, the sensitivity of the pressure transducer is reduced by about 28%, i.e., the output of the pressure transducer is reduced to about 110 mV.

(2) The four-terminal sensing element design A typical pressure transducer with a four-terminal sensing element design is shown in Fig. 6.2. The sensing element is also located near an edge center of the square diaphragm, with an inclined angle of 45 ~ to the edges. Y edge of the diaphragm ................................ Z : i:!::.::.:iVo2

o-

~i,','~;~=~i~..~Vo~

.......................................

".

~

2a

Fig. 6.2. Pressure transducer of a single sensing element design

According to w the voltage output of a four-terminal sensing element piezoresistive sensor exactly at edge center of a diaphragm is: AV=I

a2

-~ ~ 44 0.51p ---h-f( Txx - Tyy )

~(b) b V ~ S

(6.6)

Chapter6. Piezoresistivepressure transducers

244

where b and L are the width and the length of the sensing element, respectively, f3(b/L)is a correction factor caused by the short circuit effect of the electrodes. According to w

the larger the b/L ratio the smaller the correction factor f3(b/L), but for a very large b/L ratio, ~(b/L)b/L approaches unity. However, the larger the b/L ratio the smaller the input resistance and the larger the power consumption. As a compromise, a b/L ratio of 0.5 to 1.0 is often used in practical applications. If b/L=0.6--0.7 is used, ~(b/L)b/L is found to be about 0.5. Therefore, for this condition, the output of a single element pressure transducer is smaller than that of a Whetstone Bridge type pressure transducer by a half. In addition, the input impedance of the sensor is usually in the range of 0.5-1.0 kf2 (whereas the input impedance of a Wheatstone bridge can be 5 kf2 or higher). For the same power consumption or current limitation, low input impedance restricts the supply voltage, which, in turn, reduces the sensitivity. Although a single-element pressure transducer has the advantages of low noise, high yield, etc., it is not as widely used as the Wheatstone bridge design. Motorola Inc. may be the only major company that produces the single element pressure transducer in large quantities.

w

Designs with rectangular diaphragms

For a rectangular diaphragm, the piezoresistive sensing elements can either be located near a longer edge center of the diaphragm or at the center of the diaphragm. The designs can either be a piezoresistive Wheatstone bridge type or a four-terminal single element type. Fig. 6.3 shows the designs of Wheatstone bridge type pressure transducer for the two locations. The design for the four-terminal sensing element is similar; the only difference is that the four resistors which formed the Wheatstone bridge are replaced by a fourterminal sensing element and, as a result the sensitivity is reduced almost by a half. Therefore, only the Wheatstone bridge designs will be considered in the following discussions.

(1) Designs near the longer edge centers For a Wheatstone bridge design at the edge, the arrangement of the piezoresistors is shown in Fig. 6.3(a).

w 1. Design with fiat diaphragms

245

Y edge of the

edge of the

............................. I:

diaphragm ...................................

t,~

.......... i....ill,fillVsl ~! -- ([-.....-.~: ]]...:....:':.':':" V :! 2 ~ : ~:i:-::i:

2bl

4

X

,,

I

.............. i__}i~iiiii VsI ff "1 . . . . . . . . . . i"" ".'.'.'. ii 2tL~..................!....!iiii~i~Vo2J ii

ok'

X

..................i i i i iv4 "

Fi ....i!i!ilil """....!iiiiiii

o

I

....

i

...................... i .... ~i~i~i~i G

!

,:...............~;;-..............;,

. . . . . . . . . . . . . . .

(a)

. . . . . . . . . . . . . .

.J

(b)

Fig. 6.3. Wheatstone bridge designs on rectangular diaphragm (a) sensing element at the edge, (b) sensing element at the center According to w the stresses at the center of a longer edge of a rectangular d i a p h r a g m are" b4

a 2

Txx = 2 a4 + b4 p-h-f ,

Trr

vTxx

(6.7)

where a is half the width and b is half the length of the diaphragm. For b> >a, a2 T x x - 2P-h-f , i.e., the stresses can be twice as large as those in a square diaphragm with an edge length of 2a. Therefore, for the same width of 2a, the sensitivity of a pressure transducer with a rectangular d i a p h r a g m can be doubled if 2b is much larger than 2a. As the area of the rectangular diaphragm is proportional to its length, we examine the ratio of sensitivity to the area of the diaphragm (A=4ab) with a constant width " 2 a " when the length is variable. If b=~a, we have:

Tx x _ Tx x A 4ab

~3 p 2(1+~4)h e

The value of t~ for a m a x i m u m Txx is ~ A the stress: a2

4 ~ _ 1.32. In this case, we have

a2

Txx - 1 . 5 p ~ - , Try - 1 . 5 v p ~ -

(6.8)

Chapter 6. Piezoresistive pressure transducers

246

(2) Designs at the diaphragm center The stress components of Txx and Trr are equal and Txr is always zero at the center of a square diaphragm due to the symmetric property in the x- and y- directions. Therefore, neither the Wheatstone bridge or the four-terminal sensing element at the center of a square diaphragm is sensitive to the pressure applied on the diaphragm. However, the stress components at the center of a rectangular diaphragm are not symmetrical. Therefore, the sensing elements at the center of a rectangular diaphragm are sensitive to the pressure applied on the diaphragm. For a Wheatstone bridge design, the arrangement of the piezoresistors is shown in Fig. 6.3(b). According to Eq. (2.184) in w the stress components at the diaphragm center are: a2

Pa 2 1 + V b--T Txx = h2 a4

a2

,

Pa 2 - ~ + v Trr = h2 an , Txr = 0

1+~

(6.9)

1+~

b4

b4

Therefore, the output voltage of the Wheatstone bridge is" a2 2 71;44

Vout = V S T

( zXX - Tyy ) -

/I;44 V

2

a

s P -~

(1-v)~

b2 a4

(6.10)

l+m b4

It is obvious that the sensitivity of a center design is about half as large as that of an edge design. Similar to the discussions on the edge design, we consider the situation that a is constant but b=c~a where c~ is a variable. If the ratio of stress to diaphragm area,

( Txx - Trr ) o

, is considered, the tx value for a maximum ratio 4txa is found to be tx= 1.93. (Two other invalid solutions are ct = -1 and tx = 0.27 ). In this case, we have:

(Txx - Trv ) = O.68~ 2 (1- v)

(6.11)

Another condition is also interesting: the length 2b of the diaphragm is fixed but the width a is variable. Let a=c~b, as the maximum sensitivity ~2 _~4

corresponding to the maximum of (1+c~4 / \

the ~ value for a maximum

/

sensitivity is ~/~f2-1 = 0.645, i.e. a=0.645b. In this case, we have:

w

Pressure transducers with sculptured diaphragm structure

pb2 " v) (Txx - Trr )max : 0.207--~- {1 -

247

(6.12)

Though the sensitivity for the design with sensing elements at center is always smaller than that for the design with sensing element at edge by about half, the central design is still useful sometimes. The reason is that the variation of stress with position at center region is very moderate when compared with that at edge. Therefore, the alignment for resistors (on the front side) and diaphragm (on the back side) for the central design is not as critical as that for an edge design.

w w

Pressure transducers with a sculptured diaphragm structure T w i n - i s l a n d structure

For a flat diaphragm as discussed in w

with a lateral dimension of 2a and a2 a thickness of h, the stress in the diaphragm is proportional to p-~-. It seems that a very sensitive pressure transducer can be developed if the ratio of a to h is made extremely large. However, as the nonlinearity of the pressure-to-stress a4 relationship is proportional to P~-T' there must be a limit to the value of a to h compromise the sensitivity and the linearity, the two most important parameters for a pressure transducer. In searching for better diaphragm structures to simultaneously improve the sensitivity and linearity, back islands were introduced into the diaphragm structure by Whittier around 1980 [1]. This structure has been known as the twin-island structure and is schematically shown in Fig. 6.4. Using this clever design, good linearity can be achieved in very sensitive devices. The basic concept of this structure has been widely used until present in the designs of high performance pressure transducers, although modifications and new features have been added. The basic advantages of the twin-island structure are: (a) High linearity since the same type the resistors are stressed (lateral stress) and the balanced stress values for the two pairs of piezoresistors in the Wheatstone bridge. (b) High sensitivity due to the stress concentration in the regions where the piezoresistors are located.

248

Chapter 6. Piezoresistive pressure transducers

R1 ,,___

sr-

- ~9

R~

R,

(c)

(a)

rt Ft(b) ft Fig. 6.4. Schematic views of a twin-island structure (a) top view (b) cross-sectional view (c) Wheatstone bridge To achieve a better understanding of the basic features of the twin-island structure, we consider the region (CDEF in Fig. 6.4.) and a simplified structure shown in Fig. 6.5 is used for a quantitative analysis. The structure consists of two island regions and three gap regions. As the structure is symmetric, only half of the structure needs to be considered.

1 I. n': ..... O =

, a

bb

. . . .

x

Fig. 6.5. Simplified model for the twin-island structure The differential equation for the gap region I (x=- 0 to a) is: l pblX2 - E l l w" 1 ( x) = p c b l x - m o - -~

where I 1 -

2(1E

1

-v 2)

(6.13)

bl h3 is the moment of inertia for the region, Wl(X) the

displacement in the region, b1 is the width of the islands and mo is the restrictive bending moment at x=O. The differential equation for the island region II (x=a to b) is:

w

Pressure transducers with sculptured diaphragm structure

-Elzw" 2 (x) - p c b l x - m o --~1 pblX 2

249 (6.14)

E 2) blH 3 and Wz(X) is the displacement function in the where I 2 = 12(1-V region. The differential equation for the gap region III (x=b to c) is: -EI3w" 3 (x) - pCblX- m o --~1 pb 1x2

(6.15)

where 13=I1. The boundary conditions are:

WI(0 "- 0,

f

WI(0 ) -- 0

w~(a)- w~(a), w'~(a)- w'~(a) w2,(b) = w3(b ),

w'2 ( b ) - w'3 (b)

(6.16)

w3(c)-0

As H>>h, 12 is much larger than 11. Therefore, Eq. (6.14) can be simplified as: d2w2 - 0 dx 2

(6.17)

From Eqs. (6.13), (6.15), (6.17) and (6.16), we find:

3c(c 2 - b 2 + a 2 ) - ( c 3 - b 3 + a 3) m~ =

6 ( c - b + a)

blP

Using the notations of fi" = a , ~ = _b and c c f =

6(1-s

(6.18)

we have: m ~ - pblc2f The expression for stress in region I and III are: T(x)-6p-~

/

f

Xc

From Eq. (6.19), the stress at the edge (x=0) is:

(6.19)

Chapter 6. Piezoresistive pressure transducers

250

2 C

T( O) = 6P-hT f and the stress at the center (x=-c) is: - c2 ( l _ f ] r(c) = -6p-zL Note that T(0)is positive and T(c)is negative. Therefore, we can write: c

2

T( O) - T( c) - 3P-h-f

(6.20)

It is interesting that Eq. (6.20) is independent of the length of the islands (i.e., b - a ) and the equation applies even when the islands are removed (i.e., fiat diaphragm). Therefore, we come to the conclusion that the addition of the islands does not change the sensitivity of the piezoresistive bridge, but the sensitivities for the individual resistors forming the Wheatstone Bridge can be modified. For higher linearity, one design criteria is that the sensitivities of the two pairs of piezoresistors should be as balanced as possible. If the islands are properly located, this criterion can be very well met. For example, if c=lmm, a=0.1mm and b=0.9mm, we have ff = 0.1, b - 0.9 and f = 0.273. The stress Txx in regions I and III can be found by Eq. (6.19) C

2

and is shown in Fig. 6.6. The stress at x=-0 is T(0) - 1.64p~-~- and the stress at c2

c2

x=c is -1.36P-h-T . However, the average stress in region I is about 1.35p~a 2

and the average stress in region III is -1.35p-~-. The stresses in both regions I

I

V

are quite balanced. For a small a values and b = c, the maximum displacement of the structure can be approximated by:

Wmax - W'l (a ) 9 c -

pc4 f Eh 3

- - - a1_ +--a

2

1-2 6

ff

(6.21)

w

Pressure transducers with sculptured diaphragm structure 2.0

xx( p ce/~ )

1.0

-

J i

I :

; I

i

- 1 . 0

/

i i

a

251

b

1

C

t

'

I

i

|

a |

| |

|

|

i

|

i

-

-2.0

-

Fig. 6.6. Stress distribution in the diaphragm For the same design geometries used in the above, we have 0.27

pc 4 Eh 3

9

Wma x =

When compared with Eq. (2 158) for a flat long strip .

diaphragm in w the maximum displacement is reduced by a factor of 2 by the addition of the islands. According the above discussions, the sensitivity is not directly increased because of the introduction of the islands. However, the addition of the islands reduces the displacement significantly. Considering that there are large areas of thin diaphragm on both sides of the gap-island region, part of the pressure load on the thin diaphragm areas tends to be transferred onto the gap-island region (in fact, the gap regions) as the stiffness of the gap-island region is higher than that of the diaphragm. This effect increases the stress in the gap regions where the piezoresistors are located and, hence, increases the sensitivity of the devices. This effect is often referred to as a stress concentration effect. The stress concentration effect may increase the sensitivity of the pressure transducer by several ten percent, large enough to compensate for the stress losses due to the distance away from the diaphragm edges. Therefore, as a rough estimate, the output of the pressure transducer can be approximated by: C2 Vou t = 1.5p~-(1v) ~1 rt44V~

(6.22)

Another important feature of the design shown in Fig. 6.4 is that all four piezoresistors comprising the Wheatstone bridge are subjected to transverse stresses (two of them are positive and other two are negative). Therefore, the nonlinearities of the two pairs of piezoresistance have the same pattern and

252

Chapter 6. Piezoresistive pressure transducers

can be mostly canceled out by the Wheatstone Bridge. By the improvement of linearity, pressure transducers with higher sensitivity (or lower operation range) can be developed. On the nonlinearity of pressure transducers, readers are referred to w The above discussions on the twin-island structure are based on a simplified model. The results are approximate, but they are very useful for designing a twin-island structure pressure transducer. If quantitative results are required, CAD design tools such as ANSYS should be used.

I I I lpl

.......... ....

t t

,.

j

iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiijiiiiiiiiiiiiiiiiiiiiiiiiii] liiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii] Fig. 6.7. Over-range protection mechanism An additional advantage which could be provided by the islands is the over-range protection of the pressure transducer. Fig. 6.7 shows the overrange stop mechanism for front pressure loading. When the pressure is much higher than the nominal operation range, the displacement of the diaphragm is stopped when the islands contacts with the glass constraint. The original gap distance between an island and the glass constraint can be estimated by Eq. (6.21), where the p values used should be at least twice as large as the nominal operation range of the device. w

Beam-diaphragm structure

The twin-island structure described in w makes use of the large difference in flexure rigidity between the diaphragm areas and the island areas to adjust the stress distribution and stress concentration. However, the addition of islands on the back side imposes strict control requirements for the fabrication process because" (a) the dimensions of the gaps for the piezoresistors are related to the wafer thickness and the etching depth on the back side, and (b) the tolerance of double-sided alignment between the front side resistors and the etching mask on the back side is tight as the gaps are narrow for the designs. Generally, the thickness of the back island is close to the original wafer thickness (larger than 200 ktm) and the thickness of diaphragm can be below

w

Pressure transducers with sculptured diaphragm structure

253

20~tm. This difference is actually unnecessarily large for achieving all the advantages given by the twin-island structure. As the flexure rigidity of plate is proportional to the cube of its thickness, a difference in thickness by a factor of 2 to 3 gives enough difference in flexure rigidity for stress adjustment. For a high sensitivity pressure transducer, the thickness of diaphragm can be as thin as 10~tm or even lower. In this case, a thickness difference of about 20~tm can provide enough difference in flexure rigidity to adjust the stress distribution. To achieve excellent performances for high sensitivity pressure transducers without the drawbacks brought about by the thick islands on the back, a beam-diaphragm structure was proposed by the author in 1989 [2]. A typical design is shown in Fig. 6.8, where Fig. 6.8(a) is a top view and Fig. 6.8(b) is a cross sectional view. Instead of using back islands for stress adjustment, a dumbbell-shaped "beam" is formed by etching silicon to a depth of hb on the front side. The beam has three narrow regions with widths b1 and two wide regions with widths be . b1 should be small but still wide enough to accommodate the piezoresistors; b2 is always larger than b1 for effective stress concentration.

,

2b

*1

i. -f-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

L

U

+

J

{i ~

i 2a

,,L_q_~ ~ ~ ~ ,

i

beam area R,

A

I'

A'

'[

L .......................................................

:

-q

!

J

~'

b1

frame {a)

I /

f

hb

,l--J,

T"~

diap,hragm

/

,

,~rame beam-diaphragm

A

~

(b) Fig. 6.8. Schematic drawing of a beam-diaphragm structure (a) top view (b) cross sectional view

254

Chapter 6. Piezoresistive pressure transducers

A deep etching from the back is then made to form a flat bottom which covers the beam-diaphragm areas on the front (size of 2a x 2b ) as indicated by the dashed lines in Fig. 6.8.). The thickness of the diaphragm formed, h a , is usually very thin for a high sensitivity pressure transducer so that the thickness in the beam-diaphragm regions, h b + h d , is 2 to 3 times as large as

hd. The effect of stress concentration in the beam-diaphragm structure is twofold. The stress is first concentrated from the diaphragm to the beam due to the thickness difference. Then the stress is further concentrated at three narrow regions of the "beam" due to the difference in width. It is not difficult to place, with high accuracy, the resistors on the narrow regions of the beam as the alignment is single-sided. The tolerance for the double side alignment is large as the bottom of the back side etch is only required to cover the beam-diaphragm area on the front. As a matter of fact, back islands are still very beneficial if they are incorporated with the beam-diaphragm structure. If so, the bottom of the islands should be a little smaller than the wide regions of the beam so that they do not tighten the tolerance in double-sided alignment. The addition of the islands has two advantages. First, they increase the flexure rigidity of the wide regions of the beam to improve the linearity of the device when the thickness of beam regions, (h b + h d ), are not thick enough for a very sensitive pressure transducer. Secondly, they may provide over range stop for the pressure transducers. Some more structures with similar approaches but different etch patterns on the front side were proposed in the early 1990s for high sensitivity pressure transducers [3, 4]. Using these structures, the operation range of high sensitivity piezoresistive pressure transducer has been extended down to 1 kPa and even lower.

w

Design of polysilicon pressure transducer

The piezoresistors in a conventional pressure transducer are made by boron diffusion or ion implantation on an n-silicon substrate. Therefore, the piezoresistors are insulated by reverse biased p-n junctions. The leakage current of the junction should be small so that the device can work with high stability. It is well known that the leakage current of a p-n junction rises exponentially with temperature. Therefore, conventional pressure transducers work poorly at temperatures near or above 100~ For pressure transducers

w

Design of polysilicon pressure transducer

255

capable of working at high temperatures, various SOI materials have been used. As the resistors are dielectric-insulated, the pressure transducer can work at temperatures up to 200~ if conventional aluminum metalization is used. The working temperature can be higher than 300~ if appropriate materials are used. There are two categories of Si-based SO1 material: polysilicon SO1 and single crystal SO1. Polysilicon SO1 has been used for high temperature pressure transducer for many years. It features low cost and moderate sensitivity. A variety of single crystal SO1 materials (such as SOS, SIMOX, BESOI, Smart cut, etc.) have been used for high temperature pressure transducers in recent years. The single crystal SO1 pressure transducers have almost the same sensitivity as a conventional pressure transducer, but the cost is much higher. As the design of a single crystal SO1 pressure transducer is basically the same as that of a conventional pressure transducer, it is not discussed further. For polysilicon SO1 material, the orientations of polysilicon grains on top of SiO2/Si substrate are not related to the substrate under the SiO2, but, for anisotropic etching to form a diaphragm, the substrate is usually a (100) silicon wafer with a flat in < 110> direction. The design of polysilicon piezoresistor differs from that of single crystal silicon piezoresistor in that the longitudinal effect of piezoresistance is preferred for polysilicon as the longitudinal effect of piezoresistance is larger than the transverse effect by a factor of about 3 (see w Fig. 6.9 shows two typical designs for a polysilicon piezoresistive pressure transducer. edges of the dia hragm

edges of the

...................~ . .................. i i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i ..

R~ ! r---'-I

!

i ............

i

,

R~.. ...........

:i

! i

9.......................................................

Ca)

(b)

Fig. 6.9. Two typical designs of a polysilicon pressure transducer (a) with a flat diaphragm (b) with a twin-island structure

i

256

Chapter 6. Piezoresistive pressure transducers

Fig. 6.9(a) shows the design with a flat diaphragm. The diaphragm is rectangular and the piezoresistors are all perpendicular to the longer edge of the diaphragm with two resistors, R1 and R4, near the edge centers and the other two. resistors, R2 and R3, at the center of the diaphragm. With an applied pressure on the top, resistors R I and R4 are stretched and R2 and R 3 are compressed longitudinally. Fig. 6.9(b) shows a design with a twin-island structure. The structure is similar to the conventional design shown in Fig. 6.4, but the directions of the resistors are different so that the resistors are stretched or compressed longitudinally. As the resistors are in alignment with the islands, the gap between two islands and the gaps between an island and the frame have to be larger than those used in the conventional design. As will be discussed in ~6.4, the sensitivity of a polysilicon pressure transducer is only one forth to one fifth as large as a conventional pressure transducer with similar design parameters.

w w

Offset voltage and temperature coefficient of offset Offset voltage o f a pressure transducer

If the circuit of the Wheatstone bridge or the structure geometries of the four-terminal sensing element is ideally symmetric, the output of a pressure transducer should be zero for zero pressure input, and it remains zero with any temperature variations. However, there are always some non-ideal factors that cause a non-zero output voltage for a pressure transducer. This non-zero output at zero pressure input is referred to as the offset voltage of the pressure transducer. There are two main factors that cause offset voltage. The first factor is the deviation of the geometries of the sensing elements from their design. The second one is that the stresses in the chip may be caused by a mismatch of the coefficient of thermal expansion between silicon and the packaging materials. As the sensing elements are stress sensitive, an output voltage may appear even though the input pressure is zero. In this section, the offset voltage and its temperature coefficient will be discussed based on the assumption that it is caused by a geometry deviation in a Wheatstone bridge design. Consider a Wheatstone bridge consisting of four identical resistors of resistance RB, i.e., RI=R2=R3=R4=RB, according to the design. If each of the four resistors deviates from its design value by a specific small fraction due to process variations (such as photolithography, etc.), the four resistors are no

w

longer R4 -

Offset voltage and temperature coefficient of offset

identical.

If

R1 = RB(I+~, ),

R2 - R8(1+[32),

257

R3 - R B ( I + [ ~ 3 ) ,

R~(1 + ~4), as shown in Fig. 6.10(a), the offset voltage of the bridge is"

Vos = Vs

(Rl + R )(R3 +

Vs

[ 3 4 /[31 4 -, [33

(6.23)

The contribution by the deviation of a specific resistor Ri is one fourth of its relative deviation, with R2 and R3 in the positive direction and R~ and R4 in the negative direction. For the convenience of further discussion, let the circuit shown in Fig. 6.10(a) can be equivalent to the circuit shown in Fig. 6.10(b), where (6.24)

-- (~2 + ~3 -- ~1 -- ~4)

According to Eq. (6.23), 13is defined by the offset voltage as: 13- 4 V~

(6.25)

Vs Note that the offset voltage shown in Eq. (6.23) is not temperature dependent if all the resistors have the same temperature coefficient. This is indeed the case as the resistors are made by diffusion or ion implantation. However, if part of the offset voltage is caused by a thermal stress related to encapsulation, the offset voltage is generally temperature dependent.

RB(I+~1)

]RB(I+~3)

R~

Rs(1+13~)

] RB(I+I3 4 )

Rs(l+15 )

IG Ca)

IG (b)

Fig. 6.10. A piezoresistive bridge with deviations from design

w

Compensation of offset voltage

For many practical applications, the offset voltage of a pressure transducer has to be compensated for by some means so that further signal conditioning will be easier. The compensation of offset voltage is usually made by the use of external discrete resisters, either by parallel or by series connection.

Chapter 6. Piezoresistive pressure transducers

258

For parallel compensation, a discrete resistor, Re, is connected in parallel with resistor R 2 as shown in Fig. 6.11(a), supposing that [3 is positive. The condition for compensation is: Re 9Rs(I + 13) Re + R--BB(-1+~) = Ra

(6.26)

For small [3, the resistance of Re is: 1 Re =-~ R 8

(6.27)

For example, if R B = 5k$2 and 13= 0.02, we have Re = 250kg2.

R~

Rs

Rs

.Vol Rp1>

_.v~

[ RB(I+I3) RB ~G (a)

Rs

Ra(l+~ ) ~G (b)

Fig. 6.11. Compensation for offset voltage (a) using a parallel external resistor, (b) using a series external resistor Compensation can also be made using a series scheme. A small resistor R s = ~R 8 is connected in series with resistor R1, as shown in Fig. 6.11(b). For the same data in the example just mentioned, the resistance of the series resistor is: R s = 100-(2. For series compensation, the bridge has to be broken. For the convenience of offset compensation, the commercially available devices are usually a fiveterminal version or a twin half bridge version as shown in Fig. 6.12(a) and Fig. 6.12(b), respectively. The parallel and series schemes of offset compensation are effective for conventional applications where the environmental temperature does not change too much. However, these simple schemes are not effective if the pressure transducers are expected to operate in a larger temperature range. The reason is that the compensation schemes can null the offset voltage at a specific temperature but it causes temperature drift due to the different temperature coefficients between the bridge resistors and the discrete resistors used for compensation. Therefore, the offset appears again once the

w

Offset voltage and temperature coefficient of offset

259

temperature changes. The reappearing offset voltage can be quite large for large temperature variations.

lVs

Vola~R1

R3[ v ~

v~,

v~

R1

R3

Vo, -

-

v~

....

G (a)

G1

G2 (b)

Fig. 6.12. Typical leads in commercially available pressure transducers (a) a five-terminal version, (b) a dual half-bridge version

Assume that the temperature coefficient (TC) of the bridge resistors (TCRB) is ~b and the temperature coefficient of the discrete resistors is ~d. Usually, ~b is between +0.1% to +0.3% according to the doping level of the resistor and ~d is usually negative but small in value. For parallel compensation, if the compensation is made at a temperature to when the resistance of the bridge resistors is Rno and the compensation resistor is Rpo, the offset voltage at temperature t is:

Vos(t) =

1

-

(,- to)

Or, the temperature coefficient of offset after compensation is: TCO = - l (ctb - t~d )~V s 4

(6.28)

For example, if Ctb=+0.2%, C~d=0, [3=0.04, V s - 5 V , we have TCO=-IOOktV/~ For a temperature variation of 100~ we find Vos = - 10mV, a significant value for practical applications. For series compensation, the temperature coefficient of offset is: 1 TCO = ~(Ot, b -- (Zd )~V S

(6.29)

where a d is the temperature coefficient of the series resistor, R s . According to Eqs. (6.28) and (6.29), it is interesting that the temperature coefficient of offset caused by the compensation has the same magnitude for parallel compensation and for series compensation (provided that the

260

Chapter 6. Piezoresistive pressure transducers

temperature coefficient of the compensation resistors are the same) but in oppoSite signs. This i-emindS us that the temperature ~Coefficient of offset might be eliminated if the compensation is made partially by parallel compensation and partially by series compensation. The method will be discussed in the next section.

w

Compensation of temperature coefficient of offset

For certain applications, not only the offset voltage has to be compensated for but also the temperature coefficient of the offset voltage so that the offset voltage remains small in a large temperature range. As discussed in w the offset voltage can be compensated for by either a parallel or a series compensation scheme but the TCOs caused by these two compensation schemes are in opposite directions. Therefore, it is conceivable that if the offset voltage of a piezoresistive Wheatstone bridge is compensated for partially by parallel compensation and partially by series compensation as shown in Fig. 6.13 and the ratio is carefully selected, the resultant temperature coefficient of offset can be canceled out.

v RB(I+[~) /::i'6 [

Fig. 6.13. TCO compensation by a series-parallel compensation

According to Fig. 6.13 and supposing that c~d = 0, the conditions for simultaneous compensation of offset voltage and the temperature coefficient of offset are"

Rs + RB = (1 + [3)RBRp R,, + (1 +

(6.30)

and

1 Rs --~(X 4 RB

1 (1 +[3)RB b=

4

Re

(Xb

(6.31)

w

Offset voltage and temperature coefficient of offset

261

From Eq. (6.31), we have: Rs Re

= (1 + [3)RB2

(6.32)

From Eqs. (6.32) and (6.30), we find: (6.33) and Rp =

(1+ [3)(41 + [3 + 1)

R B -__

2R B

(6.34)

As was mentioned in w the offset voltage may also be caused by the residual stress arising from the mismatch of thermal expansion between silicon and the packaging materials. This factor also affects the TCO. If the residual stress plays an important role in the TCO, the results given in Eqs. (6.33) and (6.34) are no longer valid. Another source of trouble is that the temperature coefficient of the discrete resistors used for compensation is neither negligible nor constant. In these cases, the discussion above gives a useful rule of thumb: the T C O goes in the positive direction if the parallel resistance is reduced (or, if the series resistance is increased) and vice versa. Accurate compensation of the offset voltage and the T C O must be found by experiments using this guide. If the offset caused by packaging is negligible, a very effective compensation method can be used. This method uses on-chip series resistors for coarse compensation and uses an external discrete resistor connected to a small section of the on-chip resistor in parallel for fine adjustment. The scheme is illustrated by Fig. 6.14.

Vol

R~ as(1-o0

--m

Vt ~

RsCa+~) ~G Fig. 6.14. TCO compensation by partially parallel resistance

262

Chapter 6. Piezoresistive pressure transducers

By internal series compensation, [3 can be reduced to smaller than 0.005 and ~ + 13 can be about 0.005. If an external resistor, Rp, is selected to finetune the offset voltage to zero, Rp is given by:

Rp + (Cz + ~)R 8 = czR~ After compensation, the TCO of the bridge is found to be:

TCO=

4(cz+[3) (czb -~

For a specific value of (z + ~ = k, the maximum TCO, appearing for the condition of cz = [3, is: TCOmax = - 1---~ k (tXb -

ad)Vs

When C t b - C t d - 2 x l 0 - 3 / ~ TCOma x

w

=

(6.35) k=ct+[3=0.005

and Vs - 5 V ,

we have

3.1 ktV/~ C. Clearly, it has been significantly reduced.

Temperature coefficient of sensitivity

For a conventional silicon pressure transducer with piezoresistors, the signal output voltage is proportional to n44 of the resistor material. According to w ~44 is a function of the doping level. Meanwhile, ~44 is temperature dependent and the temperature coefficient of ~44 (TCrO is also a function of the doping level. TC~ is usually negative in sign. For a pressure transducer with a constant voltage supply, the signal output is: Vout =

1 a 2 -~ rtnnCp-h-fVs

(6.36)

where Vs is the supply voltage and c is a constant related to the structural design. Therefore, the temperature coefficient of sensitivity (TCS) of the pressure transducer is"

TCS=TCrt

(6.37)

For example, if TCr~= -0.2%, the sensitivity of the pressure transducer will be reduced by about 10% if the temperature is raised by 50~ As the temperature coefficient of sensitivity is significant, the compensation for TCS has to be considered for some applications.

w

Temperature coefficient of sensitivity

263

A widely used compensation scheme for TCS is to use a constant current supply instead of a constant voltage supply. If the supply current for a piezoresistive pressure transducer is I s , the output voltage becomes: 1 a2 Vout - -~ rtnnCp-h-f Is R B

(6.38)

where R8 is the bridge resistance. The temperature coefficient of resistance (TCR) is also a function of the doping level. As TCR has a positive temperature coefficient (i.e., TCR>O), the effect of TCR on TCS is opposite to that of TCrc. According to Eq. (6.38), we have: TCS = TCR + TC~

( 6.39)

The curves in Fig. 6.15 show TCrc and TCR as functions of the resistor material doping level. For convenience the absolute value of TCrt is used. As can be seen from Fig. 6.15, TCrc and TCR have the same value (but with opposite signs) at two critical doping levels: Nc~=2xlO18/cm 3 and Nc2-=5xl02~ 3. Therefore, if the doping level of the resistors is controlled to be equal to one of the two critical doping levels, the TCS of the pressure transducer will be zero due to the cancellation of TCR and TCrt.

TC(% /C) 0.35

0.31 0.25i

L

J

i

I

/

i i

I TCrti

0.2 i 0"151

1

i

0.1 0.05 +--

10

18

t

Nc

I

1

10

19

20

10

--{

Nc2

21

10

Ns

Fig. 6.15. TCR and TCrc as a function of Ns As exact cancellation of TCR and TCrc is not readily attainable, an adjustable approach is generally used for practical applications. The main feature of this approach is to control the doping level to be NsNc2 so that TCR > The TCS of the pressure transducer can thus be adjusted

bTCnl.

to zero by an external resistor, Re, in parallel with the bridge as shown in Fig. 6.16. According to Fig. 6.16, the output of the pressure transducer is:

Chapter 6. Piezoresistive pressure transducers

264

1 a2 ReR B V~ = -27r'44cP--h-f Is Rp + R 8

(6.40)

Therefore, the temperature coefficient of sensitivity is: TCS = TC~ +

RpTCR + Rsc~ p

(6.41) R B + Re

where tXp is the temperature coefficient of Re (i.e., Ctp =

1 dRe

). The

Re dT

condition for zero TCS can be found as: RB ( TCrc + (Xp ) Re = -

(6.42)

TCR + TCrc

As TCR>O and TCrc
R~(ITC~I-~) zce-Izc~l

(6.43)

Under the condition that TCR > ITCnI >> C~p, a reasonable value of Re for zero TCS can be found.

Under the condition of TCR > ITC~I, series compensation for TCS with constant voltage supply is also possible. The circuit for series compensation is shown in Fig. 6.17, where a constant supply voltage is used.

~I s

[ 9 You, -~

.vo,,, ~

1 , Fig. 6.16. Parallel compensation

Fig. 6.17. Series compensation

According to Fig. 6.17, the output of the pressure transducer is: 1

a2

R8

Vout - ~ rr44cp-~ Rs + R~, Vs

(6.44)

w

Nonlinearity

265

The T C S of the transducer is: T C S = TCrc + T C R -

gB Rs ~s- ~ TCR Rs + RB Rs + RB

S C R = TCrc + ~ (g T Rs + RB

- c~s)

(6.45)

where c~s is the temperature coefficient of Rs. The condition for zero T C S is" -RBTCrt R s = T C R + T C r c - c~s

(6.46)

Eq. (6.46) can also be written as: RBITCTcI gs

-

TCR

- ITC

(6.47)

I -

As ms is usually very small, Rs will be very large if T C R and ]TCrcI are close to each other. In this case, the effective voltage supply on the bridge is very small. Low sensitivity or a very high supply voltage must be tolerated for the series compensation scheme. In addition to the compensation schemes described above, there are many other compensation schemes using temperature sensitive components (such as thermistors, diodes, transistors) for T C S compensation. Sometimes, operational amplifiers are also used in the compensation circuits.

w

Nonlinearity

w

Definitions

The piezoresistive pressure transducer is categorized as a linear transducer, i.e., the output response of the transducer is expected to be in direct proportion to the input pressure measured. This is basically true for most piezoresistive pressure transducers within a tolerance of about one percent of the operation range (or, the full scale output). When the output-input relationship of a piezoresistive pressure transducer is calibrated with higher accuracy, the relationship is found to be a curve instead of a straight line. Therefore, the output-input relationship of a transducer is often referred to as the calibration curve of the transducer. For applications, the calibration curve of a transducer is often approximated by a specified straight line.

266

Chapter 6. Piezoresistivepressure transducers

The deviation of the specified straight line from the calibration curve of the transducer is characterized by a parameter called the nonlinearity. For each calibration point, there is a specific deviation. The nonlinearity error of a specific calibration point is defined as the deviation at this calibration point and is generally expressed as a percentage of the full scale output (FSO). The nonlinearity of a transducer is defined as the maximum deviation of all the calibration points, also expressed as a percentage of FSO. The nonlinearity of a pressure transducer is typically in the range of 0.5%--0.05%. There are quite a few methods for defining the specified straight line according to the calibration curve of a transducer. The straight line can be defined as the line connecting the two end points (at 0 and 100% pressure operation range). This line is called end-point straight line or terminal-based straight line. The end-point straight line can be shifted in a parallel direction to a certain extent to equalize the maximum deviations on both sides of the line so that the maximum value of deviation, i.e., the nonlinearity is minimized. Then, the line is called the best-fit straight line. Manufacturers like to use the best-fit straight lines as they give the best looking data. Sometimes, the best-fit straight line is based on the least squares method. As the end-point straight line method is the most straight forward, the most convenient and the most widely used method in practical applications, it is exclusively used in this book. As, in most practical applications, the pressure to be measured is directly read from the output of the pressure transducer based on the specified straight line instead of the calibration curve, the accuracy of the pressure measurement is significantly related to the nonlinearity of the pressure transducer used for the measurement. Therefore, the nonlinearity is one of the most important parameters for a pressure transducer in addition to its sensitivity, and TCO and TCS discussed in previous sections. The calibration curve of a pressure transducer is shown by the solid curve and the end-point straight line is shown by the dotted line in Fig. 6.18, where Pm denotes the maximum pressure input (operation range) and the corresponding output Vo(Pm) is the full scale output (FSO). Both the calibration curve and the straight line start from zero pressure input (lower limit of operation pressure range) as the offset voltage has been compensated for by some means. According to the definition described above, the nonlinearity at a specific pressure Pi is:

w Vo(Pi)- V~ Pm NL i = Vo(Pm)

Nonlinearity

267

Pi • 100%

(6.48)

According to Eq. (6.48), the nonlinearity can be either positive or negative for any calibration point. The nonlinearity of the pressure transducer is the maximum value of NL i .

/

. . . . . . . .

"~_...'"'" m)P ,," Vo(P v~l.~) P, 0

Pm

Fig. 6.18. Calibration curve and corresponding end-point straight line Mathematically, the Vo(p) relationship can be expressed as a series in power of p. Some typical conditions can be discussed as follows: (1) Vo(p) = ap, where a is the sensitivity of the pressure transducer. As the

calibration curve is a straight line, the nonlinearity is zero. (2) V o ( p ) = ap + bp 2 (generally a >> b). According to Eq. (6.48), we find: NL( p ) - ap + bp 2 - ( a + bPm ) p = b( p - Pm ) P ap m + bPm 2

-

ap m

The maximum nonlinearity value for the whole operation range appears at 1 P =-~ Pm and the value is:

NL-

ben

(6.49)

4a

If the constant, a, is positive, the sign of NL is dependent on the sign of b. For b > 0 , we have N L < O and, for b < 0 , we h a v e N L > 0 . The two situations are shown in Fig. 6.19. (3) V ( p ) = a p + b p 2 + c p pressure transducer is:

3, where a > > b > > c .

The nonlinearity of the

268

Chapter 6. Piezoresistive pressure transducers

b

c

N L ( p ) = ~pm ( p - p m ) p +

(p2 _ pm 2

)p

(6.50)

aPm

The maximum of NL(p) appears at:

Pl =

6c

Therefore, the nonlinearity of the transducer is" 2 2 N L - - b p - Pm Pl - c P m - P] p] aPm

(6.51)

aPm

4~

As a special case, if b = 0, we have Pl - - - ~ Pm

-

0577Pm and

N L = - ~.-~ C Pm 2 = --0.22 C Pm 2 8

a

(6.52)

a

Vo(P)

vo(p)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

iii i

....................iiii ..........i P~

0

(a)

Or.

p ~

0

(b)

Pm

Fig. 6.19. Calibration curves for (a) positive and (b) negative nonlinearity w

Nonlinearity o f a piezoresistive pressure transducer

Now let us consider a piezoresistive pressure transducer with a Wheatstone bridge on a silicon diaphragm. The output of the bridge is: _

V~

e2es-ele4 (R~ + R 2)(R 3 + R4) Vs

(6.53)

where V s is a constant supply voltage. Obviously, the nonlinearity of the pressure transducer is determined by the linearities of the four resistors. First, each resistance R i is a function of the applied pressure, p. According to Chapter 5, we have:

w

Nonlinearity

269

(6.54)

g i = Rio "k" Rio(~lTl dr 7[tT t "1"~ s T s )

Therefore, the nonlinearity of gi(p) is decided by two factors: (a) The nonlinear relationship between stress, T, and the applied pressure p. This is usually the most significant factor, especially for thin diaphragms. (b) The nonlinear relationship between piezoresistance, i.e., the nonlinear relationship between piezoresistive coefficient, rr, and stress T. The pressure dependence of piezoresistance can be generally expressed as:

R i = R i o ( l + a i p + b i p 2)

(6.55)

where subscript, i, denotes an individual resistor of the bridge. According to Eqs. (6.55) and (6.49), the nonlinearity of Ri(p)is: NL i =-

bi pm 4ai

(6.56)

According to Eq. (6.53), the nonlinearity of the pressure transducer can be determined by the nonlinearity of all the resistors. Furthermore, it may also be dependent on the sensitivities of the four individual resistors. By substituting Eq. (6.55) into Eq. (6.53) and supposing that R2oR3o = RloR4o, i.e., the offset voltage of the bridge is zero, we have: (1 + a z p + bzp2)(1 +a3P+ b3p2)-(1 + a l p + blp2)(1 + a 4 p + b4p 2)

Vo=Vs

[2 + (al + az)P + (hi +b2)pZ][2+(a3 +a4)p+(b3 +b4)p 2 ] (6.57)

If the terms related to the third power of p and up are neglected, we have:

a2 + a3 _ al _ a4

2(a2 + a3 _ al _ a4) p2 Vs (6.58)

We can find from Eq. (6.58) that the sensitivity of the pressure transducer is: 1

S = ~(a 2 + a 3 - a I - a 4)V s

(6.59)

For most designs, a 1 and a 4 have opposite signs to a 2 and a 3. Therefore, we have: S = ! ( a 2 + a 3 + la 11+ Ia 4[)Vs 4

(6.60)

Chapter 6. Piezoresistive pressure transducers

270

This means that the sensitivity of the pressure transducer is the average sensitivity of the four individual piezoresistors. According to Eq. (6.58), we can find that the nonlinearity of the pressure transducer is more complicated: a 2 + a~ - a 2 - a 2

NL = - I b2 + b3 - bl - b4 Pm+ 4 a2 + a3- aI - a4

Pm

(6.61)

8(a 2 + a 3 - a 1 - a4)

Assuming that a 2 and a 3 are positive while a 1 and a 4 are negative and defining d - (a 2 + a 3 - a 1 - a 4) / 4, we can write:

NL=

I [a2(NL2)+a3(NL3)-al(N~)-a4(NL4) ] 4~ + 1 (a 2 +a 2 _ a 2 - a ~ ) P m K

(6.62)

32~ 2 The first term in Eq. (6.62) is the weighted average of the nonlinearities of the four resistors. The factor, Pm~ , in the second term on the fight-hand side is the average piezoresistance of the full scale pressure, that is usually in the range of 0.01-0.05. 1 Now let us look at an example. Suppose that a 2 = a 3 and a I = a 4 = - ~ a 2 due to design and/or process reasons. If

pm'ff =

3 % , the nonlinearity caused

by the second term in Eq. (6.62) is NL = 0.25%. This is not negligible for many pressure transducers.

w

Nonlinearity caused by the "Balloon effect"

As mentioned in the previous section, there are three main factors that determine the nonlinearity of piezoresistive pressure transducers. They are: (1) the nonlinear relationship between the stress and the pressure applied, (2) the nonlinear relationship between the piezoresistive coefficient and the stress, and (3) the nonlinear output v o l t a g e due to the difference in piezoresistive sensitivity between bridge resistors. The third factor was thoroughly discussed in w The first factor will be discussed in this section in a qualitative way and the second will be discussed in w The discussion in w lead to the conclusion that the stress in a diaphragm is proportional to the applied pressure. The discussion is based on a linear theory. In the linear theory, it is assumed that the stress distribution is a result of pure bending, that is, the central plane of the diaphragm is not stretched or

w

271

Nonlinearity

compressed. This assumption means that the deflection of the diaphragm is small when compared to its thickness. If the deflection of the diaphragm is not small when compared to its thickness, the central plane of the diaphragm will be stretched like a balloon, hence the name of the " B a l l o o n e f f e c t ". When the stress caused by the stretch of the central plane is considered, the stress in the diaphragm consists of two parts; the first part, T b, caused by the bending of the diaphragm and the second part, TC, caused by the stretch of the central plane, i.e., (6.63)

T = Tb + Tc

When compared with the linear theory, the stress caused by bending, T b, is reduced in magnitude as the stretch of the diaphragm takes part of the pressure load. Since T b can be either positive or negative depending on the position in the diaphragm and the sign of the applied pressure, but TC is always positive, the Balloon effect on resistors on different surfaces is different. Let us consider a pressure transducer with resistors at the edge of the diaphragm with reference to Fig. 6.20. The bending stress at the resistors is positive when pressure is applied from the front surface while the bending stress at the resistors is negative when the pressure is applied from the back side of the diaphragm. Note that Tr is always positive. This means that both T b and T c are positive when the pressure is applied from the front side while T b is negative and Tc is positive when the pressure in applied from the back side. Therefore, when the pressure is applied from the front side the balloon effect is less significant than that when the pressure is applied on the back side. The nonlinearity caused by the Balloon effect is smaller for front pressure than for back pressure. r(p)

r(p)

sS S

linear theory.-" T

linear theory .-" s S

Irl s S

L o

(a)

0

(b)

Fig. 6.20. The influence of the Balloon effect on the stress at the diaphragm edges (a) for front pressure, (b) for back pressure

Chapter 6. Piezoresistive pressure transducers

272

As the Balloon effect is due to a stretching of the diaphragm, the Balloon effect is related to the displacement of the diaphragm. According to w the displacement at the center of a circular diaphragm is: w(0)=12(1-v2) 64E

a4

p-~ h

and the stress at the edge is:

Tr(a ) = 3a 2 Therefore, the larger the ratio of a / h the larger the nonlinearity of the pressure transducer. For high sensitivity pressure transducers, a / h is usually quite large. Therefore, the nonlinearity for these devices is a significant concern in the design. By using island-diaphragm designs, the displacement can be reduced for the same stress level. Therefore, island-diaphragm designs are often necessary for high sensitivity pressure transducers.

w

Nonlinearity of a piezoresistive effect

In Chapter 5, the piezoresistive effect of silicon was considered as linear, i.e., the piezoresistive coefficient of silicon was considered to be independent of stress. In fact, this is not true if it is examined with high accuracy. However, it is very difficult to investigate the higher order dependence of the piezoresistive coefficient on stress because there are too many components of the stress tensor and the measurement of higher order effects requires very high accuracy. Therefore, published data are scarce and incomplete, and verification of the data accuracy is difficult. According to the experimental results of Matsuda et al [5], for p-resistors in the <110> orientations with a doping level of 2xl0~8/cm 3, the stress dependence of nonlinearity of the piezoresistance is shown in Fig. 6.21. Based on Eq. (6.62) and the experimental results in Fig. 6.21, some considerations on the nonlinearity of the pressure transducer can be discussed. Let us consider a design with a square flat diaphragm and four resistors at the edge centers as shown in Fig. 6.22. Suppose that the pressure is positive on the front side and the stresses on resistors R1 and R4 are the same and can be approximated to be longitudinal stress, TI, only. On the other hand, the stresses on resistors R2 and R3 are the same transverse stress, Tt. Also, we assume that TI=Tt=T. This implies that the ai's in Eq. (6.62) have all the same

w

Nonlinearity

273

value. Therefore, the nonlinearity of the pressure transducer can be estimated by a simple equation: (6.64)

NL - I [ ( N L 2 ) h- (NL3) q- (NLI) -I- ( g t 4 )] NL %)

4 -200

NL(%)

-

4

- __-----___ ,,,..-"~ i ~ 100

- 1oo -4

T

I

-200

- 1oo

200

-

100

-

-

Compression (MPa)

~,-'i---7"

I

~ ,3 v

200

-4

Tension (aPa)

-

Compression (iPa)

(a)

Tension (iPa)

(b)

Fig. 6.21. Nonlinearity of a p-type piezoresistor in the <110> stress (doping level: 2xl018/cm 3) (a) longitudinal mode (b) transverse mode

According to Fig. 6.22, as R2 and R3 are stressed transversely, NL2 and NL3 are positive (as shown by the curve on the fight-hand side of Fig. 6.21(b) and NL~ and NL4 are also positive (as shown by the curve on the fight-hand side of Fig. 6.21(a). The nonlinearity of the pressure transducer is then the simple average of the four NLs of the resistors:

NL = I1(INZ

+

INL41 + INL=I + INL I)

(6.65) Vs

e d g e of the diaphragm I- . . . . . . . . . . . . . . .

i i

q~

R2

iR, !

i

R,I i

I

R~

R,

I

L.................

R,

J

(a)

(b)

Fig. 6.22. Square flat diaphragm with four resistors at edge centers (a) schematic layout (b) Wheatstone bridge

Now let us look at the twin-island design shown in Fig. 6.23. Assume that all four resistors are subjected to the same transverse stress value but with two

274

Chapter 6. Piezoresistive pressure transducers

different signs (say, the stresses on R~ and R4 are +T and the stresses on R2 and R3 are -T). According to Fig. 6.21 (b), NL2 and NL3 are negative (as shown by the curve on the left-hand side) and NL1 and NL4 are positive (as shown by the curve on the fight-hand side). As the nonlinearity of the pressure transducer is the simple average of the four NLs of the resistors, we have:

NL = Z(INZ I + INL41- INL [- INLI)

(6.66)

4

edge of the

.~ap._h.rags -

............

R1

R2

a3

l ..........

R4

............ islands

R1

l =

(a)

(b)

Fig. 6.23. Twin-island diaphragm pressure transducer with four resistors in parallel with each other (a) schematic layout (b) Wheatstone bridge

This implies that the nonlinearity values of the four resistors can be canceled out with each other to some degree. Therefore, a pressure transducer using only transverse piezoresistance has the advantage of lower nonlinearity. It was indeed found that the nonlinearity of pressure transducers with twinisland structures using only the transverse piezoresistive effect is much smaller than those using both the transverse and longitudinal piezoresistive effects.

w

Calibration of pressure transducers

For accurate pressure measurement, the calibration of the pressure transducer is extremely important. In fact, the accuracy of a pressure measurement is limited by the specifications of the pressure transducer used for the measurement. Therefore, a pressure transducer has to be calibrated either by the manufacturer or by the user before it can be used for pressure measurement. As the frequency bandwidth of pressure transducers is usually

w

Calibration of pressure transducers

275

much higher than the frequency of pressure signals in practical applications, the calibration of pressure transducers is usually static. As the calibration for a pressure transducer is very complicated and time consuming, the calibration cost presents a major part of the total cost of a pressure transducer. A typical calibration procedure for a pressure transducer can be described with reference to Figs. 6.24, 6.25 and Table 6.1.

Vo(P)

vR,, i vR'

V r. =VRn. VFml-- VRm1

'

01

Pi

Prn

"~ O

Fig. 6.24. Test circles for the calibration of pressure transducer

v~

(average) calibration curve

m

~p

Pl

Pi

Pr.

Fig. 6.25. Definition of some interim results

The calibration of a pressure transducer requires repeated measurements of a few selected standard pressures (pi, i= 1,2,... m, m > 5) using the pressure transducer to be calibrated. Each standard pressure is called a test point. The test points should be uniformly distributed over in the whole pressure operation range of the pressure transducer, including one test point at the lower limit of the operation range and another at the upper limit of operation. The measurements must be done in many cycles ( j = 1,2,..-n, n _>5 ). Each cycle consists of a forward excursion (k = F, forpl, P2,"" Pi,"" Pm) and a reverse excursion ( k = R, for Pm, Pm-1,"" Pi,"" Pl )" The output voltage of the

276

Chapter 6. Piezoresistive pressure transducers

pressure transducer for an excursion k ( k = F o r R ), testing point i, and the cycle number j, is denoted as Vk,i, j . Table 6.1. Example of test data sheet (nine test points) Vk, i,j

Test cycle j

Forward excursion Test points i (i=1,2 ..... 8)

End point i=m=9

Reverse excursion Test points i (i=8,7..... 1)

pF9 = pR9 VF91"- VR91

j=2 j=3

PF1 VFI1 VF12 VFI3

~176176 j=n

gFln

g~n

VF,I

VF,i

VF,8

VF,9 = VR, 9

SF, 1

SF,i

SF,8

SF,9 -" SR,9

j= 1

. . .

PFi ... pF8 VFil VF81 VFi2 VF82 gFi3 VF83

VF93-- VR93

pR8 ... pRi ... PR2 PR1 VR81 VRil VR21 VRll-" VFI2 VR82 VRi2 VR22 VR12"- VFI3 VR83 VRi3 VR23 VR13= VFI4

VF8n

VF9n= gR9n

gR8n gRin gR2n VR,8 VR,i VR,2

VF92-" VR92

gRln

Average m Vk,i

Standard deviation sk,i

SR,8 SR,i SR,2

VR,I SR, 1

Table 6.2. Interim data sheet

AverageVk,i

VF, 1

Standard deviation Sk,i

SF,1 ... SF,i ... SF,8

VF, i

VF,8

VF,9=VR,9 SF,9 = SR,9

VR,8 VR, i VR, 2

VR, 1

SR,8 SR,i SR,2 ... SR,1

The test results for a specific cycle, j, are filled into a row in Table 6.1. The test sequence for the first and second cycle are schematically shown in Fig. 6.24 (note: the differences of data for the same test pressures have been exaggerated for clarify). According to Table 6.1, the number of measurements for the calibration is 2 ( m - 1 ) n + 1. If m = 9 and n - 7, the total number is over one hundred. Once the measurement is completed and the data are listed in a table as shown in Table 6.1, the experimental data are processed as follows. (1) F i n d i n g the a v e r a g e f o r e a c h t e s t i n g p o i n t

For a test point ( k, i), the average output of the pressure transducer is found using: -

Vk i - Vk i j nj=l

( k - F , R; i - l, 2,. . . m )

The standard deviation of measurement for the test point is:

(6.67)

w

Ski =

I

n-l.=

ij

Calibration of pressure transducers

_)2

--Vki

(k = F,R; i = 1,2,---m)

277

(6.68)

The results of Eqs. (6.67) and (6.68) are jotted down in an interim table as shown by Table 6.2 and are schematically shown in Fig. 6.25 (note that the difference for the same test point but for different excursions has been exaggerated). (2) Finding the average for each test pressure The averages for each test pressure are found by the equation:

Vi -- -2 (VFi

-f" VRi )

(6.69)

The curve representing the V i --Pi relationship is referred to as the calibration curve of the pressure transducer, as shown by the dotted line in Fig. 6.25.

(3) Finding the overall standard deviation of the measurement The overall standard deviation, s, is found by the definition: s-

I

1 ~(S2iWS2i) 2 ( m - 1 ) i=l

(6.70)

Based on the results given in the above equations, some important parameters can be found:

(1) Full scale output (FSO) FSO = VFS -- g m - g 1

(6.71)

(2) Sensitivity (S) S=

VFs

(6.72)

Pm - Pl

As the sensitivity of a piezoresistive pressure transducer is proportional to the supply voltage of the Wheatstone bridge, the sensitivity of a pressure transducer is sometimes defined as:

S =

Vrs (Pm-Pl)Vs

where Vs is the supply voltage of the pressure transducer.

(6.73)

278

Chapter 6. Piezoresistive pressure transducers

(3) Hysteresis error (H) For each test pressure, the hysteresis error is defined as: !

H i = IVRi- VFi[x 100%

(6.74)

VFS The hysteresis error for the transducer is the maximum of His.

(4) Repeatability error (R) The repeatability error of the pressure transducers is defined as: R=

2s

VFS

•

(6.75)

for 95% confidence. Or, it is defined as: R=

3s

VFS

x 100%

(6.76)

for 99.73% confidence.

(5) Nonlinearity (NL) The definition of nonlinearity was given in w If the end-point straight line scheme is used, the points in the straight line are:

(6.77)

Vio - Vii + Vm - V1 (Pi - Pl ) Pm - Pl

The nonlinearity for each test pressure is:

NL i = Vi-V",o X 100%

(6.78)

VFS

and the nonlinearity for the pressure transducer is: N L "" ( N L i )max

(6.79)

(6) Nonlinearity and Hysteresis (NLH) The maximum deviation of

Vki

from the corresponding point in the straight

line, Vio, represents the error caused by the nonlinearity and hysteresis and is referred to as the nonlinearity and hysteresis error: I

NLH = I~ki - Vi~

VFS

x 100%

(6.80)

w 7. Calibration of pressure transducers

279

(7) C o m b i n e d error (or, accuracy, 6) There are two commonly used definitions for the combined error: (a) Definition I:

8 - +3/(NL) 2 + H 2 + R 2

(6.81)

(b) Definition II:

=---(ILHI + IRI)

(6.82)

As the value found using Eq. (6.82) is larger than that found using Eq. (6.81), the definition II is considered to be more strict than definition I. In the above discussions, the environmental temperature is assumed to be constant. If the temperature variation is considered, the situation is more complicated. In this case, the concept of an "error" band is used. If the calibration measurements are repeatedly made in a temperature range (say, 0 ~ --70~ o r - 2 0 ~ 1 7 6 etc.), the calibration curve will extend into a band with a finite width. The width of the band represents the m a x i m u m error the pressure measurement can give. An example of an error band is schematically shown in Fig. 6.26. Usually, the width of the error band is of the order of l%FS-3%FS.

eo(p)

: .P

Pl

Pr.

Fig. 6.26. An error band of pressure transducer

References [1] R.M. Whittier, Basic advantages of anisotropic etched transverse gauge pressure transducer, Endevco Tech paper, TP277 [2] M. Bao, L. Yu, Y. Wang, Micromechanical Beam-diaphragm structure improves performances of pressure transducers, Sensors and Actuators, A21-23 (1990) 137-141 [3] R. Johnson, S. Karbassi, U. Sridhar, B. Speldrich, A high-sensitivity ribbed and bossed pressure transducer, Sensors and Actuators, A35 (1992) 93-99 [4] S. Hein, V. Schlichting, E. Obermeier, Piezoresistive silicon sensor for very low pressure based on the concept of stress concentration, Transducers'93 (1993) 628-631

280

Chapter 6. Piezoresistive pressure transducers

[5] K. Matsuda, Y. Kanda, K. Yamamura, K. Suzuki, Nonlinearity of piezoresistive effect in n- and p-type silicon, Sensors and Actuators, Vol. A21-23 (1990) 45-48