this .. sensors reference pressurereggiani/old-files/clark1979.pdf · the low-cost high-performance...

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 12, DECEMBER 1979 P 887

P.ressure Sensitivity in Anisotropically Etched Thin-Diaphragm Pressur.e- Sensors . ':

. .

REFERENCE PRESSURE , -

DIFFUSED PIEZORESISTORS IN A THIN DIAPHRAGM

Aktruct-In this paper the differential equations governing thin, square-diaphragm silicon pressure sensors are developed and solved using finitedifference numerical methods. Diaphragm deflection and stress patterns are presented in a normalized form applicable to diaphragms of arbitrary thickness and size. For l-mm2 10ym-thick diaphragms in (100) silicon, the calculated pressure sensitivity of full-bridge piezoresistive sensors is about 68 pV per volt supply per mmHg, which agrees well with experimental data. Variations in the pressure sensitivity of piezoresistive and capacitive structures due to process-induced variations in diaphragm thickness, size, taper, alignment, and resistor size are evaluated. Sensitivity is influenced most strongly by diaphragm thickness, with alignment an important secondary consideration in resistive devices.

T INTRODUCTION

HE DRAMATIC PROGRESS achieved in digital electronics over the past decade is well known. Component

density has doubled virtually every year with a corresponding decrease in functional cost. Several recent studies [l] , (21 have indicated that this progress may well continue at least another decade, with a factor of 1000 in component density and 50 in speed yet to be achieved. The rapidly improving performance and decreasing cost of digital electronics is caus- ing electronic instrumentation and control to pervade many new areas, including health care, industrial process control, and transportation.

As the cost of microcomputer control decreases, the emphasis in many applications is shifting to the system periphery, where sensors and actuators are needed to couple electronic and nonelectronic systems. In many cases, sensors are the most important consideration in assessing the feasibility of ex- tending microprocessor control into a new area. During the past few years, there has been continuing interest in realizing the needed sensors in silicon so that the batch-processing techniques developed for integrated circuits can be used.

One of the most needed sensors for next-generation electronic systems is the silicon pressure sensor. Applications range from health care to transportation. More than a dozen applications for pressure transducers on the automobile have been identified [3] and silicon thin-diaphragm piezoresistive sensors appear very promising for many of these system needs. Although silicon pressure sensors have existed for many years,

Manuscript received July 19, 1978; revised May 4, 1979. This work was supported by the National Science Foundation under Grant

S . K. Clark is with the Department of Applied Mechanics and Engi-

K. D. Wise is with the Department of Electrical and Computer Engi-

ENG-7648297.

neering Science, University of Michigan, Ann Arbor, MI 48109.

neering, University of Michigan, Ann Arbor, MI 48109.

i;

(a) . _

EXTERNAL P,RESSURE

METALIZED REFERENCE CAVITY THIN DIAPHRAGM (METALIZED)

1 SEAL

SILICON SUPPORT CHIP , (REFERENCE PLATE I

I \

\REFERENCE PRESSURE INLET

cb) Fig. 1. Thin-diaphragm silicon pressure sensing structures. (a) FuB-

bridge piezoresistive device. (b) Variable-gap capacitive transducer,

they have generally been expensive and prone to temperature ..

drift as well as other instabilities. The pervasiveness of the microcomputer is creating high-volume markets for such sensors and may at last provide sufficient motivation to develop the low-cost high-performance devices needed in future systems.

Silicon-based pressure sensors have made substantial progress in recent years, partly as a result of advances in integrated circuit technology. Two basic types of silicon devices exist. Piezo-junction devices, which utilize pressure-induced changes in reverse leakage current, are very pressure sensitive but require stress levels approaching the fracture stress of silicon [4] , [5] and are subject to a variety of instability mechanisms. Piezoresistive sensors employing a thin silicon diaphragm oper- ate at lower stress levels, are more stable, and are more suitable €or batch fabrication [ 6 ] . Capacitive transducers which use a thin silicon diaphragm as the movable plate of a capacitor have also been described 171 . The thin-diaphragm pressure sensors are shown in Fig. 1 . Most recent progress has come through the development of improved etching technologies for forming the thin silicon diaphragm. The use of selective, anisotropic etching, in particular, has made diaphragm formation a relatively high-yield batch process [8].

In the design of thin-diaphragm pressure sensors, a number

0018-9383/79/1200-1887$00.75 0 1979 IEEE

of problems are-encountered. The applicable mechanical and electromechanical constants for silicon are widely scattered in the literature and frequently do not correspond to present crystal orientations and processes. The relationships between stress, diaphragm deflection, and resistance change with pressure are known in closed form for circular diaphragms but for square or rectangular diaphragms they are not, and no pub- lished numerical solutions are known. Without a knowledge of these stress distributions, the sensitivity of a given design to effects such as diaphragm thickness taper and finite resistor size (which averages stress over the resistor area) cannot be determined.

This paper first reviews the basic relationships between prels- sure, stress, deflection, and resistance change for thin silicon diaphragms. Numerical constants are summarized for the present wafer orientations and processes, including new data on the piezoresistive coefficient and fracture stress in silicon. The results of a finite-difference computer simulation of square diaphragms are then presented, and these stress distributions are used as the basis of pressure sensitivity com- parisons between square and circular piezoresistive sensors and between piezoresistive and capacitive sensing structures. 13f- fects such as diaphragm thickness taper, diaphragm-resistox misaJignmerit, and finite resistor size are also considered.

MECHANICAL PRQPERTIES OF THIN SILICON DIAPHRAGMS Consider a thin silicon plate subjected to an applied pressure

p . If the deflection in response to pressure, w(x, y) is small compared with the plate thickness h, then we can assume that the JSirchoff-Love hypothesis is valid [9] . Then, in Cartesian coordinates, the equilibrium equation relating the bending moments M to the pressure can be written as

m e plate is in the x-y plane, and the bending moments au? re- lated to the deflection by

M y = - [D, $+Do *] ax2

and

Here, the bending rigidities D are functions of x and y since the plate thickness h in general varies with X and y. The rigidities are defined as

(3)

and

G,, h3 Dry = - 12

E, and E, are Young's moduli in the x and y directions, respectively, and G,, is the shear modulus. Poisson's ratio in the x and y directions, v,, and v,,,, measures expansion in one direction in response to a compression in the other direction.

Substituting (2) into (l), the differential equation relating plate deflection to applied pressure is of the form

where Lo, L 1 , and Lz are operators involving the third- and fourth-order spatial derivatives (of w ) with the rigidities D , and their first- and second-order spatial derivatives as coefficients. Once the plate deflection w is known, the stress components can be found from

I where a, and u, are the stress components in the x and y directions, respectively, T,, is the shear stress, and we have assumed v,,, = vyx = v.

These differential equations are difficult to solve analytically, especially when the diaphragm thickness h is not constant. For square diaphragms, a numerical solution is appropriate and can allow the simulation of several physically significant effects.

The finite-difference method can be applied by modeling the plate at a number of discrete points forming a uniform mesh. Each mesh point can be identified in x and y by subscripts i and j , as shown in Fig. 2. Equation (4) can be expressed at these mesh points by an equation of form

where { w } is a matrix whose elements are the deflections at all points (i, j ) and. { B } is a vector describing the load at all points. The matrix { A } contains the coefficients of the fmite- difference equations, approximating the needed derivatives. Identifying mesh points surrounding point (i, j ) by numbers 1 through 16 as in Fig. 2, the finte difference equation can be expressed as

16 Akw(k)=AZ

k=O ,

where A reflects the mesh size and w(8) = w(i, j - 1). The fmite-difference coefficients are given as follows:

A . = 8D0(0) + D,(2) + 4D,(O) + D,(6) + Dy(8)

CLARK AND WISE: PRESSURE SENSITIVITY IN THIN-DIAPHRAGM PRESSURE SENSORS 1889

0

Fig. 2. Mesh arrangement used in the finitedifference numerical solution of the diaphragm deflection and stress equations.

A1 = Do(2) +Do@) &=Do@- l)+Do(i+ l), i = 3 , 5 , 7

Ai = -2 [D,(i) t Dx(0) t Do(i) t Do(O)] , i = 2 , 6

Ai = -2 [D,,(i) t D,,(O) t Do(i) + Do(O)], i = 4,8

Ai = $D,,,(i - S), i = 9,11,13,15

Ai = Dx(i - 8) - 4 [Dxy(i - 9) + Dxy(i - 7), i = 10,14

A12 = Dy(4) - + W x y ( 3 ) + DXY(5)I

and

A16 =Dy(8) - $ ' DX.Y(7)1.

Equations ( 5 ) can be written in finite difference form as

ux(i, 13 = - (iy ') { E , [w(2) - 2w(O) + w(6)I 2A2 (1 - v')

+E,V [w(4) - 2w(O) + ~ ( 8 ) ] }

-t vEX [ ~ ( 2 ) - 2140) + w(6)] }

and

It should be noted that Chugh and Gesund [lo] obtained the matrix { A 3. for an isotropic case and showed it to be very stable in numerical computations.

In the solution of plate equations, either simply supported or built-in edges are typically assumed. The boundary condi- tions at the edge are

w=-- - 0 (simply supported) ax2

or

w=-= aw 0 (built-in edges). ax

For a square plate of uniform thickness having simply supported edges, the deflection has been expressed in analytical form [I 11 ; however, for sensors formed by selective etching, built-in edges are appropriate.

In order to evaluate (7) and (8), Young's moduli, the shear modulus, and Poisson's ratio must be known for the wafer orientation used. For the cubic silicon crystal, the basic elastic constants are [12]

ell = 1.674 X 10" dyn/cm2

cl2 = 0.652 X 10'' dynlcm'

and

cM = 0.796 X 10" dynlcm'.

These coefficients relate stress and strain in the crystal as described in [12] . Subscripts refer to directions along the crystal axes. The compliance coefficients are derived from the elastic constants and are found to be [ 121 , [13]

sll = 0.764 X cm2/dyn

s12 = -0.214 X lo-'' cm2/dyn

and

sM = 1.256 X lo-'* cd ldyn .

Based on wafer availability and the anisotropy of known silicon etchants, the (100) wafer orientation is the most viable for present production sensors. As shown elsewhere [14] , the maximum pressure sensitivity in these diaphragms using p-type sensing resistors requires that the diaphragm edge be aligned with the (110) direction, i.e., with the wafer flat. Since the diaphragm edges are at an angle 0 = 45" to the basic crystal directions, we must reorient the Compliance coefficients as [13]

si1 = sll (COS 4e + sin 4e) t (2s12 t s4) COS 28 sin 2e = $(2Sll + 2s12 +sa)

= $(2s12 + 2Sll - s a )

si2 = s12 (COS 4e + sin 4e) + (2sll - sa) COS 2e sin 2e

and

SL = S- (COS 4e + sin 4e) + 4 ( S l l - s12) COS 2 e sin 2e = $ ( S M + 2s11 - 2s12).

We can now determine the required elastic constants as

E, = Ey = [sil] -' = 1.698 X 10" dynlcm'

C,, = [ s z ] -' = 0.622 X 10" dynlcm'

and t

- v x y - v y x - v = - - - - s12 - 0.066 4 1

where x and y are oriented along the (110) crystal directions. This value of v is in agreement with [ 151 and is a more rapidly varying function of direction than has sometimes been assumed [6].

In order to make the finite-difference solutions of general applicability, it is useful to put them in dimensionless form,

1890 TSEE TRANSAC'T'EONS ON ELECTRON DEVICES, VOL. ED-26, NO. 12, DECEMBER 1979

DISTANCE, X = x f l

Fig. 4. Dimensionless displacement of a square silicon 'diaphragm having built-in edges as a function of position on the diaphragm.

04 - ~xhz /po12

(b) Fig. 3. A typical squarediaphragm piezoresistive sensor similar to that

shown in Fig. l(a). Top views shown in (a) top illumination only, and (b) top and bottom illumination (showing the thin silicon diaphragm). Chip size is 2 mm X 2 mm with a l-mm-wide lO-pm-.thick silicon diaphragm.

dividing every distance by the length I of the diaphragm edge, and dividing every stress or modulus term by the applied 'pressure. Thus a normalized dimensionless deflection parameter can be expressed in terms of the actual deflection as

- - Dxw P14

w D = -

and the normalized dimensionless stress is

- ox h2 ux = -.

PI2

Fig. 3 shows several views of a square-diaphragm piezoresistive pressure sensor such as that shown in Fig. l(a). This device consists of a full bridge of diffused sensing resistors which are located near the edge of a thin diaphragm forme:d by selective anisotropic etching. The diaphragm (and resistors) are aligned with the (110) directions in the (100) crys- tallographic plane. The diaphragm' measures 1 mm across and is approximately 10 pm thick. This geometry will illus- trate the application of the finite-different solutions.

Fig. 4 shows the dimensionless i7Bx for one quadrant of a square diaphragm having built-in edges. Since there is sym- metry about the x and y axes, it is convenient to choose the origin at the center of the diaphragm and display the fiiite- difference solutions for only one quadrant. Dimensionless diaphragm deflection is shown as a function of x for contours

-0.2 I I I I 0 01 0 :2 0.3 0.4 0.5

DISTANCE. X = x / l

Fig. 5. Dimensionless stress distributions on a silicon diaphragm having built-in edges.

at several values of y. At the center, the dimensionless deflection is a maximum, where

using the elastic properties previously quoted for silicon. This result may be compared with the known solution for a square isotropic plate with built-in edges and the same elastic constants which is known [21] to produce a value of 1.51 X for (1 1). For the specific case of a square silicon plate having a length of 1 mm and a thickness of 10 pm the deflection at the center is calculated from (1 1) to be w, = 1.278 pm at p = 100 mmHg. This result is reasonable, since for a circular diaphragm having a 1-mm diameter and built-in edges the calculated [6], [8] deflection is 0.92 pm at 100 mmHg. For a diameter of 1.4 mm (the diagonal of the square diaphragm), the deflection is calculated to be 3.68 pm. The square- diaphragm deflection should certainly fall between these two extremes. This deflection magnitude also agrees with ob- servations made on actual transducers.

Fig. 5 shows the stress distributions for one quadrant of a square plate having built-in edges. The results are similar to those obtained analytically for circular diaphragms. Stress is a


slowly varying function of position over most of the plate; however, the stress component perpendicular to the diaphragm edge passes through zero about two-thirds of the way to the edge and then increases rapidly to its maximum value at the edge. Near the edge, pressure sensitivity can, therefore, be expected to be a sensitive function of resistor placement of the diaphragm.

PIEZORESISTIVE COEFFICIENTS The piezoresistive coefficient relates the fractional change in

resistance to the applied stress. For a diffused resistor subjected to parallel and perpendicular stress components ull and ul, respectively, the resistance change is

where rill and nl are the piezoresistive coefficients parallel and perpendicular to the resistor length [6] . . The presently ac- cepted explanation for the piezoresistive effect is based on Herring's [16] formulation of the many-valley conduction/ valence-band model. The effect of lattice strain is to scatter carriers from one valley to another. This scattering is thought to perturb the averaging over longitudinal and transverse effective masses Which, an unstressed lattice, leads to an isotropic overall effective mass. The effect can either increase or decrease the resistivity depending on whether carriers are scattered to states of higher or lower effective mass.

In a cubic semiconductor, the matrix of piezoresistive coefficients contains only three independent values, convention- ally labeled nll , n12, and 7 ~ ~ ~ . The coefficients nll and nl can be derived for any arbitrary direction in the crystal from these three coefficients [4]. The dominant piezoresistive coefficients in n- and p-type silicon are nll and na, respectively. Conventional processing is more compatible with p-type diffused resistors, and this case will be analyzed here. Maximum pressure sensitivity is achieved for this case using two parallel and two perpendicular resistors in the [110] direction, all located near the edge of the diaphragm [8] . The diaphragm is accordingly oriented in the [110] direction as well.

The piezoresistive coefficients are sensitive to several quan- tities in addition to conductivity type and orientation, including temperature and doping level [ 171 , [ 181 . The temperature coefficient of n-, as measured in thin-diaphragm structures, varies between -2700 and -4500 ppm/"0 for p-type Gaussian resistors having junction depths of 1.5 pm and surface concentrations between 10" and 1019 ~ m - ~ . The magnitude of 1~- varies from 138 X cm2/dyn in 8-52. cm bulk material to 90 X cm2/dyn for diffused erfc layers having a surface concentration of lOI9 ~ m - ~ . We have measured n4 at 79 X cm2/dyn for Gaussian resistors having surface concentrations in the range of 1.5 to 4 X 10" and this value is used in this paper. Commonly used values for nl1 and 7r12 are 3 X cm2/dyn and -1 X lo-'' cm'/

and

nl =fl12 +(%l - 7112 - n44>($> = -38.5 X cm2/dyn. (1 3)

Hence, for resistors oriented parallel and perpendicular to a (110) diaphragm edge (Fig. 3), the resistance changes will be opposite and nearly equal. Using a full bridge, the temperature coefficients of the individual resistors are eliminated in the differential bridge output voltage to the degree the resistors track each other.

Referring to Fig. 5, the stresses at the edges of the diaphragm are

U~ = U, (0.5,O) ='0.294

and

so that for I,= 1 mm and h = 10 pm

and

At 100 mmHg, the perpendicular resistance change is about 1 percent while the maximum diaphragm stress is about 4 X lo+' dyn/cm2. The measured rupture stress in (100) silicon is about 3.6 X lo9 dyn/cm2 [ H I .

For a full bridge having a differential output voltage A V and an applied supply voltage of V,,, the fractional bridge output is

A V = 94.5 pV/V * m H g .

The error in A V if rll and r12 are neglected is insignificant. The stress magnitudes predicted in Fig. 5 are reasonable in view of analytical expressions derived [14] for circular diaphragms having built-in edges. For circular plates having a diameter equal to the side dimension of the square diaphragm, the stress levels (and pressure sensitivity) of the square plate is greater by about 50 percent, while for a diameter equal to the diagonal of the square, the circular plate is 28 percent more sensitive. For most applications, square diaphragms are preferable from the standpoints of both sensitivity and fabrication ease. In situations requiring minimum overall dimensions (as in catheter-tip transducers for cardiovascular studies) the circular sensor yields higher sensitivity.

dyn, respectively.

and perpendicular piezoresistive coefficients are given by It can be shown that in the 11101 directions, the parallel

PROCESS SENSITIVITIES Effect o f Resistor Size

The foregoing sensitivity calculations assume that the resistors act as point transducers at the edge of the diaphragm. In any real device, resistor size represents a tradeoff between

1892 IEEE TRANS.4CTTOMS ON ELECTRON DEVICES, VOL. ED-26, NO. 12, DECEMBER 1979

0.3 0 005 0.10 0.15 0.20 0.25

DIMENSIONLESS RESISTOR LENGTH, A4/4

Fig. 6. Stress averaging due to finite resistor dimensions for resistors oriented normal to the diaphragm edge.

reproducibility, which improves as the resistor size increases, and pressure sensitivity, which decreases with resistor size due to stress averaging over the finite resistor dimensions. Stress averaging effects in the parallel resistors are small and car1 be estimated readily using Fig. 5. For perpendicular resistors near the diaphragm edge, the average stress can be estimated using Fig. 6, which shows the average perpendicular stress as a fcmc- tion of resistor length. For a perpendicular resistor extendling in 100 pm from the edge of a 1-mm diaphragm, the average stress is only 60 percent of the peak stress at the edge. Stress in the parallel resistors is reduced due to the necessity of placing them in from the diaphragm edges by an amount sufficient to guard against alignment errors and ensure the resistor is actually realized on the diaphragm.

For a transducer having resistors 200 pm long and 20 [rm wide with parallel resistors placed 50 pm inside the diaphragm edge and perpendicular resistors forming a pi having legs 100 pm apart (e.g., Fig. 3), the calculated pressure sensitivity is 46 pV/V * mmHg. This is less than half the sensitivity calculated for point sensors at the edges and illustrates the significance of the rapid stress gradients near the edges. If the resistor dimensions are reduced to 100 pm by 10 pm, the sensitivity increases to 57 pV/V * mmHg and if, in addition, the resistor separation from the diaphragm edge is decreased to 25 pm, the sensitivity rises to 68 pVIV * mmHg. This is probably close to the maximum practical sensitivity for such a sensor. Fig. 7 summarizes this sensitivity as a function of diaphragm thickness. Pressure sensitivity data obtained from sensors similar to that shown in Fig. 3 have been found to agree closely with these calculated values. For example, five sensom from a recent processing run having diaphragm thicknesses between 9 and l l pm, exhibited pressure sensitivities of from 48 to 55 pV/V * mmHg, while four devices from another run wit'h diaphragm thicknesses in the 14- to 16-pm range had sensitivities between 20.8 and 25.8 pV/V. mmHg. The sensor design corresponded to the lower curve in Fig. ?.

DIAPHRAGM LENGTH-TO-THICKNESS RATIO. f / h

2 5 0 167 125 100 83 71 63 56 I o3 I I I I I I

10 SQUARESIRESISTOR

POINT RESISTORS AT EDGES

Z =25 pm, IO pm FEATURE __ 2.50 pin, IO p m FEATURE

Z = 50 r m , 20 gm FEATURE

t 1 4 4 6 8 IO 12 14 16 18

DIAPHRAGM THICKNESS h (pm) FOR 1.1 mm

Fig. 7. Pressure sensitivity in piezoresistive transducers as a function of diaphragm thickness.

Diaphragm Taper Nonuniform diaphragm thickness (taper) frequently occurs

during processing, and it is useful to estimate the significance of this effect using fmite-difference analysis. A linear taper across the full diaphragm is assumed such that for a I-mm-wide nominally IO-pm-thick diaphragm, the actual thickness varies from 11 pm at one edge to 9 pm at the other. In our experi- ence, this degree of taper represents a near worst case condi- tion for anisotropic etching. Fig. 8 compares the displacement of this tapered plate with the results for a uniform diaphragm, while Fig. 9 shows the dimensionless stress distributions for the tapered plate. Although the point of maximum deflection is shifted off center by the taper, neither the maximum deflection nor the volume displaced by the plate are significantly altered. The effect of taper on the pressure sensitivity of a full piezoresistive bridge can be evaluated using Fig. 9. For small sensors with the direction of taper aligned with the perpendicular bridge resistors the bridge output changes from 94.5 pV/V . mmHg to 90.4 or 98.7 pV/V . mmHg, depending on the bridge connections. For a taper aligned normal to the resistors, the calculated pressure sensitivity is 90.8 or 98.2 pV/V * mmHg. Hence a taper of 2 pm across a 10-pm-thick 1-mm-wide diaphragm results in a change in the bridge output of less than k5 percent. Diaphragm taper should not have a major effect on the pressure sensitivity of production sensors for devices having diaphragms thicker than 10 pm.

Diaphragm Size Using anisotropic etching to form the diaphragm, changes in

wafer thickness will result in variations in diaphragm size (I), since for a given mask opening ( M ) and wafer thickness (T),


"-bo 20- - a - N x 1.8-

_ _ _ - TAPERED PLATE UNIFORM THICKNESS PLATE

'3 10-

I I I I 5 - 0 5 -04 -03 - 0 2 -01 0 0.1 0 2 0 3 0 4 0.5 0 6 I a x 11

Fig. 8. Dimensionless displacement of a tapered silicon diaphragm having built-in edges.

0.5 I I I I I I I I I I - 0 . Fz(Y=O1 FOR UNIFORM PLATE

- Zx( j=Ol FOR TAPERED PLATE

0 3

0.2 z.JY=ol

-0.2 I I I I I I I I -0.5 -0.4 -0.3 -0 2 -01 0 0.1 0.2 0 3 0.4 0 5

COORDINATES. P = x / l

Fig. 9. Dimensionless stress distributions on a tapered silicon plate. The dashed and solid lines represent stress components for the tapered plate, while points of comparison are shown for the corresponding uniform plate.

the diaphragm size is

I = M - @ ( T - h). (16)

If we assume a fixed mask size and a wafer thickness which is subject to variations of +5 pm (e.g., 195 to 205 pm) then the resulting variation in diaphragm size is about 14 pm (+7 pm). Using Fig. 5 to estimate the effect on wafer stresses and the bridge output voltage, the result for minimum resistor features of 10 pm indicate a variation in A V/V,, of only about 1 percent from this source. The bridge output here is relatively in- sensitive to compensating effects. For a smaller than expected diaphragm, the effect in the normalizing factor decreases the pressure sensitivity as the square of the diaphragm size. How- ever, the closer positioning of the resistors to the diaphragm edge increases their sensitivity so that there is little net effect on the bridge output. For a diaphragm larger than expected, a similar compensation occurs in the opposite direction.

Diaphragm Alignment Since the diaphragm is formed by selective anisotropic etch-

ing from the back side of the wafer, registration of the back- side etch mask with the front side resistor pattern is required.

This alignment can be done using infrared optics or with a double-side photolithography in which alignment marks are placed on both sides of the wafer as a first step in processing. A 10-pm alignment error is estimated as worst case for a production system; 5 pm would likely be more typical. For 10- pm resistor features and a nominal parallel resistor location 25 pm inside the diaphragm edge, a 10-pm misalignment of the diaphragm produces .a variation in pressure sensitivity from 71.9 to 62.6 pV/V - mmHg, when shifts parallel and perpendicular to the resistor directions are both considered. The fractional change in output is thus +7 percent for this geometry, making alignment of the resistors and diaphragm a relatively important consideration. For a 5-pm misalignment, the calculated sensitivity variation is about +4 percent.

Since the alignment tolerance is independent of structural feature size, there are several ways to reduce the importance of alignment in pressure sensitivity. Increasing the diaphragm size decreases the relative misalignment if the resistors are also scaled and produces a pressure output which increases as I * . The percentage change in A V due to diaphragm misalignment decreases with 1. Alternatively, the resistors can be positioned in regions having lower stress gradients so that exact positioning of the diaphragm is less critical. This generally means moving the resistors away from .the diaphragm edges toward the center. However, the pressure sensitivity can be greatly reduced for such configurations. Individual stress components are reduced by nearly one-third, but, more importantly, the difference between uII and ul is also reduced so that the fractional resistance change with pressure may decrease an order of magnitude or more. For a full bridge with resistors 10 pm wide by 100 pm long positioned at (kO.1, +0.1), near the center, the calculated bridge output is only 8.8 pV/V . mmHg with a fractional sensitivity to misalignment by 10 pm of about 10 percent. Sensitivity to wafer thickness is also increased.

Gzpacitive Sensors For a capacitive sensor as shown in Fig. 1@), the capaci-

tance between the plates is

where eo is the free-space dielectric constant and S is the zero- pressure plate separation. If we define an effective plate deflection as

where A is the diaphragm area, then the change in capacitance with pressure is

AC= Co (L) -Co (%) S - d

where Co is the zero-pressure capacitance. For deflections which are small compared with h, the effective deflection d is about one-fourth of the center deflection we, and the small-

1894 IEEE TRANSACTIOIUS ON ELECTRON DEVICES, vaL. ED-26, NO. J .Z , DECEMBER 1979

\. IO2 I I I I I

2 4 6 8 IO 12 14 16 DIAPHRAGM THICKNESS, pm

Fig. 10. Pressure sensitivity for unloaded thin-diaphragm capacitive transducers as a function of diaphragm thickness and a plate separation.

signal pressure sensitivity can be calculated from I , h , and S . Fig. 10 shows this sensitivity for square silicon diaphragms having I = 1 mm as a function of h for several values of S . Since the sensitivity is proportional to deflection, it is a strong function of I and h, varying as the fourth and third powers of these variables, respectively.

Comparing Figs. 7 and 10, the intrinsic (unloaded) sensitivity of the capacitive device is approximately an order of magnitude greater than for a piezoresistive device having the same diaphragm thickness. The capacitive structure is also f ee from some of the thermal effects present in piezoresistive devices and is a candidate for some high-temperature applications. Since the zero-pressure capacitance of a 1-mm2 device is small, this device requires some form of on-chip interface circuitry if a significant reduction in sensitivity due to packag- ing and lead capacitance is to be avoided. Such circuitry com- plicates the fabrication and compensation of the capacitive device. Output voltage can be in the form of an analog signal as with a piezoresistive bridge or can be frequency modulated if the interface includes a currentcontrolled oscillator.

Diaphragm taper has very little effect on the sensitivity of capacitive devices since the capacitance is a function of the integrated displacement. Variations in wafer thickness (hence diaphragm size) are more important since the diaphragm deflection is proportional to Z 4 . A thickness variation of 200 k5 pm would change the unloaded pressure sensitivity by +3 percent. The maximum allowed pressure for planar structures is that which produces a center deflection equal to the plate separation S . For thin diaphragms and narrow plate separation, the maximum pressure becomes very small (about 20 mmHg for I = 1 mm, h = 4 pm, S = 2 pm) so that the high intrinsic sensitivity of the capacitive device may not be useful in some applications.

An important concern in both resistive and capacitive structures is the reference cavity between the plates. For absolute

sensors, a major temperature coefficient arises from gas expansion in the reference cavity unless the cavity is sealed in vacuum. If sealed in vacuum, however, the large offset pressure can take a sensitive gauge out of its linear response range. For a relative sensor, the collection of moisture between the plates is a serious concern in capacitive devices.

DISCUSSION Both piezoresistive and capacitive sensors utilizing thin sili-

con diaphragms offer relatively high pressure sensitivity over wide dynamic ranges. Full-bridge piezoresistive devices are probably the more easily used, although appropriate signal detection circuits are also available for capacitive transducers [19] , which are particularly attractive for low-power applications requiring an FM output format. Both approaches utilize integrated-circuit batch-processing techniques and are poten- tial solutions to high-volume low-cost applications in many areas. For both types of devices, diaphragm thickness is the structural parameter requiring the greatest control.

Several approaches are possible for monitoring and controlling diaphragm thickness, including V-groove [8] , infrared transmittance, and buried-layer (boron etch-stop) [20], [21] techniques. In the first approach, a series of V-grooves having different termination depths is etched from the front wafer surface as the diaphragm etch proceeds from the back side. As the diaphragm thickness reaches the range of V-groove termination depths, a simple visual indication of diaphragm thickness is provided, allowing diaphragm thickness control to within about 1 pm. Infrared transmittance is more difficult to imple- ment and requires some compromise between area and spot monitoring. A major source of sensor-to-sensor variability in diaphragm thickness is taper in wafer thickness, which is moni- tored but not corrected for by these first two techniques. The use of a boron buried layer or simple diffusion' to slow or stop the etch makes diaphragm thickness control a matter of controlling epitaxial growth and diffusion, both of which are relatively accurate and reproducible. The effects of wafer taper are almost completely suppressed. The use of such diffused layers for etch termination is limited to diaphragms of moderate thickness due to outdiffusion from the buried layer and the high doping levels required (5 X 1019 cm-'). For diaphragms thicker than about 15 pm, wafer taper and absolute diaphragm thickness variations are relatively less important, and the V-groove approach is satisfactory.

Table I summarizes the sensitivity of piezoresistive and capacitive sensors to a variety of process variations for both V-groove and diffused diaphragm thickness control. In capacitive devices, both diaphragm thickness and plate separation must be closely controlled, while piezoresistive sensors are most sensitive to diaphragm alignment and thickness. Align- ment errors could probably be cut in two over the values listed using infrared alignment optics.

With careful process control and appropriate sensor design,

'The results presented in this paper do not include the effects of possible built-in stress in the silicon lattice due to heavy boron concentrations. These possible effects are being studied. Anisotropic etch stops at p-n junctions are also possible using electrochemical etching techniques.


TABLE I CALCULATED VARIATION IN THE PRESSURE SENSITIVITY OF PIEZORESISTIVE AND CAPACITIVE SENSORS

DUE TO PROCESS-INDUCED DIAPHRAGM VARIATIONS

Variation (Percent) Diaphragm Parameter piezoresistive* Capacitive**

Thickness: 10 vm f 1 v m (f 0.2 pm)

Taper: 9 um to 11 um: 2 v d m m (<O .1 vm/mm)

size: 1 mm f 7 vm

Alignment: f 10 vm

Separation: 5 pm f 0.2 um

+30 (f6)

<fl ( < C f l )

f3

- f4

* a = 1 mm, h = 10 pm, 10 pm features, 20 vm edge separation, 68 vu/v-mmHg. Values in parentheses assume the use of a buried-layer etch stop.

* * a = 1 nun, h = 10 vm, 5 vm separation, 640 pv/v-mmHg unloaded.

it should be possible to batch fabricate these sensors at low cost, realizing a high yield with production spreads in pressure sensitivity of 10 percent or less. To achieve 1-percent tolerance levels, however, some form of individual calibration of both the zero-pressure offset and the pressure sensitivity will be necessary. The automated laser trimming of compensating resistors or capacitors fabricated on the same monolithic die as the active diaphragm is one viable approach as fully monolithic sensors containing interface circuitry are evolved.

CONCLUSIONS This paper has reviewed the electromechanical properties of

thin-diaphragm silicon pressure sensors. The equations governing diaphragm deflection, stress levels, and piezoresistance are presented and quantified for the most common diaphragm orientation. Solutions to these equations were derived using fmite-difference computer simulations and were presented in dimensionless form applicable to anisotropically etched square diaphragms of arbitrary. size and thickness. Using these equations, the pressure sensitivities of piezoresistive and capacitive sensors and their dependence on a number of processing variables were determined.

The stress distributions illustrated in this paper provide a basis for understanding present sensors and the tradeoffs necessary in their design and processing. Further, they offer a starting point for the creation of new designs. The stress distributions for diaphragms of constant or gradually tapered thickness are clearly not optimum from a device standpoint. From a device view, the desired piezoresistive structure would be one producing relatively high stress levels per unit pressure, a large difference between longitudinal and transverse stress levels, and relatively small spatial stress gradients over the regions for transducer placement. This arrangement would maximize pressure sensitivity while minimizing the effects of diaphragm thickness variations and misalignment. Altering the planar nature of the diaphragm through the use of multiple boron diffusions may provide one means of achieving these goals and investigations into a number of structures are being pursued. Several temperature-induced effects are also being incorporated into the basic simulation program to allow device simulation as a function of both pressure and temperature.

ACKNOWLEDGMENT The authors gratefully acknowledge the contributions of

S. Amada in the computational portions of the work reported in this paper and the assistance of S. C. Kim in the fabrication of the piezoresistive sensors.

REFERENCES R N. Noyce, “Microelectronics,” in Sci. Amer., vol. 237, pp. 63- 69, Sept. 1977. K. D. Wise, K. Chen, K. Zinn, and R. E. Yokely, “The impact of microcomputers on aviation: A technology forecasthg and assess- ment study” (vol. l), U.S. Department of Transportation, Fed- eral Aviation Administration, DOT Rep. FA76WAI-609, Sept. 1977. R. B. Hood, “Sensors, displays, and signal conditioning,” in SAEIZEEE Conx on Automotive Electronics (Detroit, MI, Feb. 1974). J. J. Wortman et al., “Effect of mechanical stress on p-n junction device characteristics,” J. Appl. Phys., vol. 37, pp. 3527-3530, Aug. 1966. J. J. Wortman and L. K. Monteith, “Semiconductor mechanical sensors,” ZEEE Trans. Electron Devices, vol. ED-16, pp. 855- 860, Oct. 1969. 0. N. Tufte, P. W. Chapman, and D. Long, “Silicon diffused- element piezoresistive diaphragms,” J. Appl. Phys., vol. 33, pp.

W. D. Frobenius, A. C. Sanderson, and H. C. Nathanson, “A microminiature solid-state capacitive blood pressure transducer with improved sensitivity,” ZEEE Trans. Biomed. Eng., vol. BME-

Samaun, K. D. Wise, and J. B. Angell, “An IC piezoresistive pressure sensor for biomedical instrumentation,” ZEEE Trans. Biomed. Eng., vol. BME-20, pp. 101-109, Mar. 1973. Y. C. Fung, Foundations of Solid Mechanics. Englewood Cliffs, NJ: Prentice Hall, 1965. A. K. Chugh and H. Gesund, “Difference operator for variable stiffness plates,” Znt. J. Numerical Methods Eng., vol. 9, pp.

S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells. New York: McCraw-Hill, 1959. W. R. Runyan, Silicon Semiconductor Technology. New York: McGraw-Hill, 1965. R. F. S. Hearmon, Applied Anisotropic Elasticity. London: Ox- ford Univ. Press, 196 1. Samaun, “An integrated circuit piezoresistive pressure sensor for biomedical instrumentation,!’ Ph.D. dissertation, Stanford UN- versity, Stanford, CA, Aug. 1971.

nology, vol. V. ASD-TDE-63-316, Research Triangle Institute, Research Triangle Institute, Integrated Silicon Device Tech-

Durham, NC, 1964.

3322-3327, NOV. 1962.

20, pp. 312-314, July 1973.

701-709,1975.

I. 896 IEEE TRANSACTXXVS ON ELECTRON DEVICES, VOL. ED-26, NO. 12, DECEMBER 1979

[16] C. Herring, “Transport properties of a many-valley semiconductor,” Bell Syst. Tech. J., vol. 34, p. 237, Mar. 1955.

[17] 0. N. Tufte and E. L. Stelzer, “Piezoresistive properties of silicon diffused layers,”J. Appl. Phys., vol. 34, p. 313, Feb. 1963.

[ 181 J. M. Borky, “Silicon diaphragm pressure sensors with integrated electronics,” Ph.D. dissertation, University of Michigan, Arm Arbor, 1977.

[19] T. B. Fryer, “Capacitance pressure transducers,” in ZndweZling and Implantable Pressure Transducers. D. G . Fleming, W. H. KO,

and M. R. Neuman, Eds. Cleveland, OH: CRC Press, 1977. [ 201 A. Bohg, “Ethylene diamine-pyrocatechol-water mixture shows

etching anomaly in boron-doped silicon,” J. Electrochem. Soc., vol. 112,pp. 401402, Feb. 1971.

[21] K. D. Wise and S. K. Clark, “Diaphragm formation and pressure sensitivity in batch-fabricated silicon pressure sensors,” in ZEDM Dig. Tech. Papers, pp. 96-99, Dec. 1978.

[22] R. J. Roark, Formulas for Stressandstrain. New York: McGraw- Hill, 1965.

Development of a Miniature Pressure Transducer for BiomediclaI Applications

Abstract-Miniature absolute pressure transducers are needed for biomedical implant applications. The design parameters are: 1) sensitivityr, 2) size, 3) hysteresis, 4) temperature coefficient, 5) long-term stability, and 6) biocompatibdity.

The present devices cannot meet all the requirements, particularly in size, long-term stability, and biocompatibdity. The results of a development program on implantable and indwelling pressure transduce~rs are reported with emphasis on the study of long-term stability and n?- producibility associated with reduced size.

The sensor is a resistive bridge diffused on a (100) oriented, I- to 4-Kt . cm silicon wafer; the chip is 1.25 X 3.75 X 0.2 mm in size. The wafer is preferentially etched to produce a 20---thick rectangular diaphragm with the sensor at its center. By proper selection of the dnsor and diaphragm geometry, the bridge output is linear with pressure and is more reproducible than previous designs. By eutectically sealing two chips together back-to-back, under vacuum, with a 78-22- percent AuSn alloy, an absolute pressure transducer is formed.

With the present assembly method, after aging, devices can reach a stability of * 1 mmHg per month at 300 mmHg full scale and a sensitivity of 10 pV/mmHg. V. A summary of the evaluation results cb different assembly structures and techniques on the performance of the pressure transducer is presented, with suggestions on the design direction for miniature pressure transducers that require long-term stability.

An interface circuit that provides stable amplification and temperature compensation was designed using the pressure bridge elements to sense the temperature. The interface circuit output will be 5 mV/mmHg with a temperature coefficient of 0.2 mmHgf‘C and near-zero hysteresis.

P INTRODUCTION

RESSURE is a common parameter used in biomedical research as well as clinical care; these measurements are

essential in many patient management situations, e.g., intrai-

Manuscript received July 16,1979; revised September 4,1979. W. H. KO and S. F. Boettcher are with Case Western Reserve Univer-

J. Hynecek is with Texas Instruments, Central Research Laborato- sity, Cleveland, OH 44106.

ries, Dallas, TX 75265.

cranial pressure in neurosurgery, blood pressure in surgery and intensive care, air pressure in respiratory diseases, intrauterine pressure in obstetrics, abdominal and urinary pressure for diagnosis of respective disorders, etc. Besides external and catheter tip measurements, it is frequently desirable to use implanted pressure measurement systems for long-term monitoring. Although there are various indwelling and implantable pressure transducers available [l] - [7] there are still factors which have not been satisfactorily resolved and which restrict the use of these transducers. These factors are size, long- term stability, temperature sensitivity, small pressure sensitivity, pressure and temperature hysteresis, poor resistance to the corrosive body ambients, poor biocompatibility, and, finally, high cost.

This article summarizes the results of research carried out at the Engineering Design Center of Case Western Reserve University, Cleveland, OH, for the, purpose of studying the causes of and developing the solutions to the above problems.

The development of a design for a biomedical pressure transducer was aimed at:

1) Measurement of absolute pressure in the body with a sealed reference chamber.

2) Long-term baseline stability (better than 0.5 percent/ month).

3) Low pressure and temperature hysteresis (less than 0.3- percent full operating range).

4) Low temperature coefficient (after compensation less than 0.5 percent/’C in the 25OC-45”C temperature range).

5) Material that is biocompatible with the body. 6) Small size. 7) Pressure range from 0 to k300 mmHg. Unfortunately, not all the problems have been resolved.

At the present time, the biocompatible coating used to pro- tect the lead wire connection is not able to withstand pres-

0018-9383/79/1200-1896$00.75 @ 1979 IEEE

this .. sensors reference pressurereggiani/old-files/clark1979.pdf · the low-cost high-performance...

Documents