The MEMS Handbook MEMS Applications (2nd Ed) - M. Gad el Hak Episode 1 Part 3 pot

The deflection of asquare diaphragm with built-in edges can be related to applied pressure by the following expression: where a is half the length of one side of the diaphragm [Chau and

devices in catheters that can aid procedures such as angioplasty Many industrial applications exist that relate

to monitoring manufacturing processes In the semiconductor sector, for example, process steps such asplasma etching or deposition and chemical vapor deposition are very sensitive to operating pressures

In the long history of using micromachining technology for pressure sensors, device designs have evolved

as the technology has progressed, allowing pressure sensors to serve as technology demonstration cles [Wise, 1994] A number of sensing approaches that offer different relative merits have evolved, andthere has been a steady march toward improving performance parameters such as sensitivity, resolution,and dynamic range Although multiple options exist, silicon has been a popular choice for the structuralmaterial of micromachined pressure sensors partly because its material properties are adequate, and there

vehi-is significant manufacturing capacity and know-how that can be borrowed from the integrated circuitindustry The primary focus in this chapter is on schemes that use silicon as the structural material Thechapter is divided into six sections The first section introduces structural and performance concepts thatare common to a number of micromachined pressure sensors The second and third sections focus in somedetail on piezoresistive and capacitive pick-off schemes for detecting pressure These two schemes formthe basis of the vast majority of micromachined pressure sensors available commercially and studied bythe MEMS research community Fabrication, packaging, and calibration issues related to these devices arealso addressed in these sections The fourth section describes servo-controlled pressure sensors, whichrepresent an emerging theme in research publications The fifth section surveys alternative approachesand transduction schemes that may be suitable for selected applications It includes a few schemes thathave been explored with non-micromachined apparatus, but may be suitable for miniaturization in thefuture The sixth section concludes the chapter

The essential feature of most micromachined pressure sensors is an edge-supported diaphragm thatdeflects in response to a transverse pressure differential across it This deformation is typically detected bymeasuring the stresses in the diaphragm, or by measuring the displacement of the diaphragm An example

of the former approach is the piezoresistive pick-off, in which resistors are formed at specific locations of the diaphragm to measure the stress An example of the latter approach is the capacitive pick-off, in which

an electrode is located on a substrate some distance below the diaphragm to capacitively measure its placement The choice of silicon as a structural material is amenable to both approaches because it has arelatively large piezoresistive coefficient and because it can serve as an electrode for a capacitor as well

dis-3.2.1 Pressure on a Diaphragm

The deflection of a diaphragm and the stresses associated with it can be calculated analytically in manycases It is generally worthwhile to make some simplifying assumptions regarding the dimensions andboundary conditions One approach is to assume that the edges are simply supported This is a reason-

able approximation if the thickness of the diaphragm, h, is much smaller than its radius, a This

condi-tion prevents transverse displacement of the neutral surface at the perimeter, while allowing rotacondi-tionaland longitudinal displacement Mathematically, it permits the second derivative of the deflection to bezero at the edge of the diaphragm However, the preferred assumption is that the edges of the diaphragmare rigidly affixed (built-in) to the support around its perimeter Under this assumption the stress on thelower surface of a circular diaphragm can be expressed in polar coordinates by the equations:

where the former denotes the radial component and the latter the tangential component, a and h are the radius and thickness of the diaphragm, r is the radial co-ordinate, ∆P is the pressure applied to the upper

surface of the diaphragm, and ν is Poisson’s ratio (Figure 3.1) [Timoshenko and Woinowsky-Krieger,1959; Samaun et al., 1973; Middleoek and Audet, 1994] In the (100) plane of silicon, Poisson’s ratioshows four-fold symmetry, and varies from 0.066 in the [011] direction to 0.28 in the [001] direction[Evans and Evans, 1965’ Madou, 1997] These equations indicate that both components of stress varyfrom the same tensile maximum at the center of the diaphragm to different compressive maxima at its

periphery Both components are zero at separate values of r between zero and a In general, piezoresistors

located at the points of highest compressive and tensile stress will provide the largest responses Thedeflection of a circular diaphragm under the stated assumptions is given by:

where E is the Young’s modulus of the structural material This is valid for a thin diaphragm with simply

supported edges, assuming a small defection

Like Poisson’s ratio, the Young’s modulus for silicon shows four-fold symmetry in the (100) plane, ing from 168 GPa in the [011] direction to 129.5 GPa in the [100] direction [Greenwood, 1988; Madou,1997] When polycrystalline silicon (polysilicon) is used as the structural material, the composite effect

vary-of grains vary-of varying size and crystalline orientation can cause substantial variations It is important to notethat additional variations in mechanical properties may arise from crystal defects caused by doping andother disruptions of the lattice Equation 3.3 indicates that the maximum deflection of a diaphragm is atits center, which comes as no surprise More importantly, it is dependent on the radius to the fourth power,and on the thickness to the third power, making it extremely sensitive to inadvertent variations in thesedimensions This can be of some consequence in controlling the sensitivity of capacitive pressure sensors

3.2.2 Square Diaphragm

For pressure sensors that are micromachined from bulk Si, it is common to use anisotropic wet etchantsthat are selective to crystallographic planes, which results in square diaphragms The deflection of asquare diaphragm with built-in edges can be related to applied pressure by the following expression:

where a is half the length of one side of the diaphragm [Chau and Wise, 1987] This equation provides a

reasonable approximation of the maximum deflection over a wide range of pressures, and is not limited

to small deflections The first term within this equation dominates for small deflections, for which w c  h,

whereas the second term dominates for large deflections For very large deflections, it approaches thedeflection predicted for flexible membranes with a 13% error

FIGURE 3.1 Deflection of a diaphragm under applied pressure.

of 5–50 MPa is not uncommon This may significantly reduce the sensitivity of certain designs, larly if the diaphragm is very thin Following the treatment in Chau and Wise (1987) for a small deflection

particu-in a circular diaphragm with built-particu-in edges, the governparticu-ing differential equation is:

where σi is the intrinsic or residual stress in the undeflected diaphragm, D
Eh3/[12(1  ν2)], and

φ
dw/dr is the slope of the deformed diaphragm The solution this differential equation provides w:

in compression In these expressions, J n and I nare the Bessel function and the modified Bessel function

of the first kind of order n, respectively The term k is given by:

The maximum deflection (at the center of the diaphragm), normalized to the deflection in the absence

of residual stress, is provided by:

or SixNyfor electrical isolation In general, these films can be of comparable thickness, and have values ofYoung’s modulus and residual stress that are significantly different The residual stress of a compositemembrane is given by:

E c t c
冱

m

where the suffixes have the same meaning as in the preceding equation

3.2.5 Categories and Units

Pressure sensors are typically divided into three categories: absolute, gauge, and differential (relative)pressure sensors Absolute pressure sensors provide an output referenced to vacuum, and often accomplishthis by vacuum sealing a cavity underneath the diaphragm The output of a gauge pressure sensor is ref-erenced to atmospheric pressure A differential pressure sensor compares the pressure at two input ports,which typically transfer the pressure to different sides of the diaphragm

A number of different units are used to denote pressure, which can lead to some confusion when paring performance ratings One atmosphere of pressure is equivalent to 14.696 pounds per square inch (psi),101.33 kPa, 1.0133 bar (or centimeters of H20 at 4°C), and 760 Torr (or millimeters of Hg at 0°C)

com-3.2.6 Performance Criteria

The performance criteria of primary interest in pressure sensors are sensitivity, dynamic range, full-scaleoutput, linearity, and the temperature coefficients of sensitivity and offset These characteristics depend

0.1 0.01 0.1 1 10 100

Center deflection (Compression)

Center deflection (Tension)

Pressure sensitivity (Compression)

Pressure sensitivity (Tension)

Dimensionless stress (1−v2)
la2/ Eh2

Normalized pressure sensitivity and diaphragm center deflection

FIGURE 3.2 Normalized deflection of a circular diaphragm as a function of dimensionless stress (Reprinted with permission from Chau, H., and Wise, K [1987a] “Noise Due to Brownian Motion in Ultrasensitive Solid-State

Pressure Sensors,” IEEE Transactions on Electron Devices 34, pp 859–865.)

on the device geometry, the mechanical and thermal properties of the structural and packaging als, and selected sensing scheme Sensitivity is defined as a normalized signal change per unit pressurechange to reference signal:

where θ is output signal and ∂θ is the change in this pressure due to the applied pressure ∂P Dynamic

range is the pressure range over which the sensor can provide a meaningful output It may be limited bythe saturation of the transduced output signal such as the piezoresistance or capacitance It also may belimited by yield and failure of the pressure diaphragm The full-scale output (FSO) of a pressure sensor

is simply the algebraic difference in the end points of the output Linearity refers to the proximity of thedevice response to a specified straight line It is the maximum separation between the output and the line,expressed as a percentage of FSO Generally, capacitive pressure sensors provide highly non-linear outputs,and piezoresistive pressure sensors provide fairly linear output

The temperature sensitivity of a pressure sensor is an important performance metric The definition

of temperature coefficient of sensitivity (TCS) is:

where θ0is offset, and T is temperature Thermal stresses caused by differences in expansion coefficients

between the diaphragm and the substrate or packaging materials are some of the many possible contributors

to these temperature coefficients

The majority of commercially available micromachined pressure sensors are bulk micromachined sistive devices These devices are etched from single crystal silicon wafers, which have relatively well-controlled mechanical properties The diaphragm can be formed by etching the back of a 100 oriented

piezore-Si wafer with an anisotropic wet etchant such as potassium hydroxide (KOH) An electrochemical stop, dopant-selective etch-stop, or a layer of buried oxide can be used to terminate the etch and controlthe thickness of the unetched diaphragm This diaphragm is supported at its perimeter by a portion ofthe wafer that was not exposed to the etchant and remains at full thickness (Figure 3.1) The piezoresistorsare fashioned by selectively doping portions of the diaphragm to form junction-isolated resistors Althoughthis form of isolation permits significant leakage current at elevated temperatures and the resistors presentsheet resistance per unit length that depends on the local bias across the isolation diode, it allows thedesigner to exploit the substantial piezoresistive coefficient of silicon and locate the resistors at the points

etch-of maximum stress on the diaphragm

Surface micromachined piezoresistive pressure sensors have also been reported Sugiyama et al (1991)used silicon nitride as the structural material for the diaphragm Polycrystalline silicon (polysilicon) wasused both as a sacrificial material and to form the piezoresistors This approach permits the fabrication

of small devices with high packing density However, the maximum deflection of the diaphragm is ited to the thickness of the sacrificial layer, and can constrain the dynamic range In Guckel et al (1986),

lim-(reprinted in Microsensors (1990)) polysilicon was used to form both the diaphragm and the piezoresistors.

3.3.1 Design Equations

In an anisotropic material such as single crystal silicon, resistivity is defined by a tensor that relates thethree directional components of the electric field to the three directional components of current flow Ingeneral, the tensor has nine elements expressed in a 3 3 matrix, but they reduce to six independent val-ues from symmetry considerations:

where εi and j irepresent electric field and current density components, and ρirepresent resistivity ponents Following the treatment in Kloeck and de Rooij (1994) and Middleoek and Audet (1994), if theCartesian axes are aligned to the 100 axes in a cubic crystal structure such as silicon, ρ1, ρ2, and ρ3will

com-be equal com-because they all represent resistance along the 100 axes, and are denoted by ρ The ing components of the resistivity matrix, which represent cross-axis resistivities, will be zero becauseunstressed silicon is electrically isotropic When stress is applied to silicon, the components in the resis-tivity matrix change The change in each of the six independent components, ∆ρi, will be related to all thestress components The stress can always be decomposed into three normal components (σi), and threeshear components (τi), as shown in Figure 3.3 The change in the six components of the resistivity matrix(expressed as a fraction of the unstressed resistivity ρ) can then be related to the six stress components by

remain-a 36-element tensor However, due to symmetry conditions, this tensor is populremain-ated by only three zero components, as shown:

44

0000π

44

0

000π

44

00

where the πijcoefficients, which have units of Pa1, may be either positive or negative, and are sensitive

to doping type, doping level, and operating temperature It is evident that π11relates the resistivity in anydirection to stress in the same direction, whereas π12and π44are cross-terms

Equation 3.19 was derived in the context of a coordinate system aligned to the 100 axes, and is notalways convenient to apply A preferred representation is to express the fractional change in an arbitrar-ily oriented diffused resistor by:

where l1, m1, and n1are the direction cosines (with respect to the crystallographic axes) of a unit length

vector, which is parallel to the current flow in the resistor whereas l2, m2, and n2are those for a unit length

vector perpendicular to the resistor Thus, l i2 m i2 n i2
1 As an example, for the 111 direction, in

which projections to all three crystallographic axes are equal, l i2
m i2
n2i
1/3.

A sample set of piezoresistive coefficients for Si is listed in Table 3.1 It is evident that π44dominates forp-type Si, with a value that is more than 20 times larger than the other coefficients By using the domi-nant coefficient and neglecting the smaller ones, Equations 3.21 and 3.22 can be further simplified Itshould be noted, however, that the piezoresistive coefficient can vary significantly with doping level andoperating temperature of the resistor A convenient way in which to represent the changes is to normal-ize the piezoresistive coefficient to a value obtained at room temperature for weakly doped silicon[Kanda, 1982]:

and T, the temperature, are varied.

the surface of a (100) silicon wafer Note that each figure is split into two halves, showing πland πttaneously for p-type Si in one case and n-type Si in the other case Each curve would be reflected in thehorizontal axis if drawn individually Also note that for p-type Si, both πl and πt peak along 110 ,whereas for n-type Si, they peak along 100 Since anisotropic wet etchants make trenches aligned

simul-to 110 on these wafer surfaces, p-type piezoresissimul-tors, which can be conveniently aligned parallel orperpendicular to the etched pits, are favored

Consider two p-type resistors aligned to the 110 axes and near the perimeter of a circulardiaphragm on a silicon wafer: assume that one resistor is parallel to the radius of the diaphragm, whereasthe other is perpendicular to it Using the equations presented previously, it can be shown that as pres-sure is applied, the fractional change in these resistors is equal and opposite:

(The bridge-type readout arrangement is suitable for square diaphragms as well.) The output voltage inthis case is given by:

R

TABLE 3.1 A Sample of Room Temperature Piezoresistive Coefficients in Si in 10 11 Pa 1 (Reprinted with Permission from Smith, C.S [1954] “Piezoresistance Effect in Germanium

and Silicon,” Physical Review 94, pp 42–49

1016

0.5 1.0 1.5

FIGURE 3.4 Variation of piezoresistive coefficient for n-type and p-type Si (Reprinted with permission from

Kanda, Y [1982] “A Graphical Representation of the Piezoresistive Coefficients in Si,” IEEE Transactions on Electron

Devices 29, pp 64–70.)

1 0

1 0

16 0

17 0

17 0

1 0

16 0

16 0

17 0

17 0

FIGURE 3.5 Longitudinal and transverse piezoresistive coefficients for n-type (upper) and p-type (lower) resistors

on the surface of a (100) Si wafer (Reprinted with permission from Kanda, Y [1982] “A Graphical Representation of

the Piezoresistive Coefficients in Si,” IEEE Transactions on Electron Devices 29, pp 64–70.)

Since the output is proportional to the supply voltage V s, the output voltage of piezoresistive pressure sors is generally presented as a fraction of the supply voltage per unit change in pressure It is proportional

sen-to a2/h2, and is typically on the order of 100 ppm/Torr The maximum fractional change in a piezoresistor

is in the order of 1% to 2% It is evident from Equations 3.24 and 3.25 that the temperature coefficient ofsensitivity, which is the fractional change in the sensitivity per unit change in temperature, is primarilygoverned by the temperature coefficient of π44 A typical value for this is in the range of 1000–5000 ppm/K.While this is large, its dependence on π makes it relatively repeatable and predictable, and permits it to

be compensated

A valuable feature of the resistor bridge is the relatively low impedance that it presents This permits theremainder of the sensing circuit to be located at some distance from the diaphragm, without deleteriouseffects from parasitic capacitance that may be incorporated This stands in contrast to the output fromcapacitive pickoff pressure sensors, for which the high output impedance creates significant challenges

3.3.2 Scaling

The resistors present a scaling limitation for the pressure sensors As the length of a resistor is decreased,the resistance decreases and the power consumption rises, which is not favorable As the width is decreased,minute variations that may occur because of non-ideal lithography or other processing limitations willhave a more significant impact on the resistance These issues constrain how small a resistor can be made

As the size of the diaphragm is reduced, the resistors will span a larger area between its perimeter and thecenter Since the maximum stresses occur at these locations, a resistor that extends between them will besubject to stress averaging, and the sensitivity of the readout will be compromised In addition, if the nom-inal values of the resistors vary, the two legs of the bridge become unbalanced, and the circuit presents anon-zero signal even when the diaphragm is undeflected This offset varies with temperature and cannot

be easily compensated because it is in general unsystematic

3.3.3 Noise

There are three general sources of noise that must be evaluated for piezoresistive pressure sensors, includingmechanical vibration of the diaphragm, electrical noise from the piezoresistors, and electrical noise fromthe interface circuit Thermal energy in the form of Brownian motion of the gas molecules surroundingthe diaphragm causes variations in its deflection that can be treated as though caused by an equivalentpressure The treatment in Chau and Wise (1987) and Chau and Wise (1987a) provides a solution for rar-efied gas environments in which the mean free path between collisions of the gas molecules is much larger

R + ∆R

V0

Tangential resistor

Radial resistor

tan-than the dimensions of the diaphragm For a circular diaphragm with built-in edges, the noise equivalentmean square pressure at low frequencies (i.e in a bandwidth limited by the pressure sensor), is given by:

p

where B is the system bandwidth, T is absolute temperature, k is Boltzmann’s constant, m1and m2are the

masses of gas molecules, P1and P2, are the respective pressures on the two sides of the diaphragm, and α

is 1.7 It is worth noting that this treatment of intrinsic diaphragm noise is for relatively low pressures, atwhich the levels gas noise is relatively small It ignores the thermal vibrations within the structural materialbecause these are typically even smaller However, this is not necessarily true in very high vacuum envi-ronments, which may permit noise from the thermal vibrations of the structural materials to dominate

At higher pressures, for which viscous damping dominates, the mean square noise pressure on the face of the diaphragm is given by:

sur-p

where R a is the mechanical resistance (damping) coefficient per unit active area, i.e., R/A, where R is the total damping coefficient and A is the area of the diaphragm [Gabrielson, 1993] The term R may also be stated as

mωo/Q, in which m is the effective mass per unit area of the diaphragm, ωois the resonant frequency, and

Q is the mechanical quality factor of the diaphragm resonance.

Squeeze film damping between the diaphragm and any opposing surface can be a very significant tributor to sensor noise Since this is more important in capacitive pressure sensors because of the relativelysmall gap between the diaphragm and the sensing counter-electrode, it is discussed in the next section

con-In addition, thermally generated acoustic waves also deserve attention at higher pressures The imate two-sided power spectrum is given by Chau and Wise (1987):

where S a is the input noise pressure, ρ is the density, f is the frequency, and c is the speed of sound in the

fluid surrounding the pressure sensor This is evidently significant at higher frequencies, and is typicallysignificant above 10 KHz

The electrical noise from the piezoresistor, identified above as the second of three components, is alsoBrownian in origin The equivalent noise pressure from this source can be presented as:

p

where V s is the voltage of the power supply used for the resistor bridge, and S p is the sensitivity of the

piezoresistive pressure sensor [Chau and Wise, 1987] It was noted following Equation 3.25 that S pis

pro-portional to (a2/h2), which makes this component of mean square noise pressure proportional to (h4/a4)

In many cases, the dominant noise source is not the sensor itself but the interface circuit If ∆V minresents the minimum voltage difference that can be resolved by the interface circuit, then the pressure res-olution limit becomes:

rep-Γ

which is proportional to (h2/a2) [Chau and Wise, 1987]

In comparing the noise sources identified for piezoresistive pressure sensors, the following examples areuseful [Chau and Wise, 1987] For a small device with diaphragm length of 100 µm and thickness 1 µm,the Brownian noise in a 100 Hz bandwidth and 760 Torr on both sides of the diaphragm is 105Torr

The piezoresistor noise, assuming a 2 kΩ bridge resistance and a 5 V supply, is about 0.11 mTorr, and the cuit noise is 1.5 mTorr assuming that the minimum resolvable voltage is 0.5 µV In contrast, a 1 mm longdiaphragm of 1 µm thickness has 4 106mTorr Brownian noise, 1.1 103mTorr piezoresistor noise,and 9.5 103mTorr circuit noise In all of these calculations, the component identified in Equation3.27 is ignored.

cir-3.3.4 Interface Circuits for Piezoresistive Pressure Sensors

An instrumentation amplifier is a natural choice for interfacing with a piezoresistive full-bridge In such

an arrangement, the differential output labeled V0inFigure 3.6would be connected to the two input minals labeled ν1and ν2 in Figure 3.7 The instrumentation amplifier has two stages [Sedra and Smith,

ter-1998] In the first stage, which is formed by operational amplifiers A1 and A2, the virtual short circuits

between the two inputs of each of these amplifiers force the current flow in the resistor R1to be (ν1 ν2)/R1.Since the input impedance of the op amps is large, the same current also flows through the two resistors

labeled R2, causing the voltage difference between the outputs of A1and A2to be:

Thus, the gain provided by this first stage can be changed by varying the value of R1 The second stage is

simply a difference amplifier formed by op amp A3and the surrounding resistors This causes the outputvoltage to be:

A separate gain stage can be added to the circuit past the instrumentation amplifier This may also be used

to subtract the temperature dependence of the signal that can be generated as reference voltage from thetop of a current-driven bridge

Spencer et al (1985) described another type of circuit that can be used to read out a resistor bridge, which

is a voltage controlled duty-cycle oscillator (VCDCO) As shown in Figure 3.8, this circuit incorporates a

cross-coupled multi-vibrator formed essentially by Q1, Q2, and the surrounding passive elements The

input voltage, v IN, which is provided by the output of the resistor bridge, determines the ratio of the

cur-rents in the emitter-coupled stage formed by Q3and Q4:

FIGURE 3.7 Schematic of an instrumentation amplifier used to read out a resistor bridge circuit (Reprinted with

per-mission from Sedra, A.S., and Smith, K.C [1998] Microelectronic Circuits, 4th ed., Oxford University Press, Oxford.)

The output of the circuit, as illustrated in Figure 3.8, is a square wave, which shows a dependence on theinput voltage in both the duty cycle and period To understand the operation of the circuit and obtain a sim-ple quantitative model for its behavior, one may begin with the assumption that the output has just switched

low, causing Q1to turn off In terms of the timing diagram, this is the beginning of t2 The entire tail current

I EE , then, is provided by Q2, and neglects the base current of Q2, v z ⬇ VCC, which causes v yto eventually settle

at approximately V CC  0.7 V; i.e., below v z by the drop across the base-to-emitter diode of Q2 This condition

also sets v OUT at V CC  R.I EE With Q1turned off, Q3continues to extract I C3 from C until v x drops below v OUT

by 0.7 V, which causes Q1to turn on, drop v z , and thereby turn off Q2 This in turn causes v OUTto rise, ending

the time period t2 Thus, at the end of t2, the voltage across C, v x – v y ⬇ (V CC  R.I EE  0.7) –

(V CC  0.7 V)
R.I EE Similarly, at the end of t1, this is R.I EE The change in voltage across C, in either

t1or t2, is therefore 2R.I EE The resulting change in charge must be balanced by the current flow in the giventime interval, thus:

permis-Oscillator: Basis for a New A-to-D Conversion Technique,” Rec of the Third International Conference On Solid-State

Sensors and Actuators, pp 49–52.)

v IN
冢 冣ln冢 冣 (3.36)

This equation can be used to determine the input voltage from a measurement of t1and t2 An important

result is that any ratio that involves t1and t2is independent of all circuit variables; drifts in their valuesover time will not affect the accuracy and resolution of the readout An exception to this immunity is the

input offset of Q3and Q4, which can affect the signal It is also worth noting that the output is digital, in that it is discretized in voltage, although not in time However, the time durations can be easilyclocked, making the circuit conducive to analog-to-digital conversion of the output signal

pseudo-3.3.5 Calibration and Compensation

Regardless of the choice of interface circuit, it is generally necessary to calibrate the overall output Forthe circuit in Figure 3.8, the input voltage can be expressed as the sum of two components: one whichcarries the pressure information as represented in Equation 3.25, and a non-ideal offset term which waspreviously ignored:

Use of this equation requires the determination of S, v OFF, and their temperature coefficients The minimumcalibration, therefore, requires the application of two known pressures at two known temperatures Using thismethod, 10-bit resolution has been achieved [Spencer et al., 1985] The conversion time was 3 ms, and theLSB was 26 µV with an input noise of 690 µVRMSover a 100 KHz measurement bandwidth The linear pro-

portionality of v IN to ln(t1/t2), as anticipated in Equation 3.28, held within 5 µV from 20 mV to 20 mV.Like the compensation of temperature dependencies, non-linearity in sensor response (at a single tem-perature) can also be compensated Linearity is expressed as a percentage of the full-scale output Withoutcompensation, for piezoresistive pressure sensors this is typically around 0.5% (8 bits) As will be dis-cussed later, it can be substantially poorer for capacitive sensors because of their inherently non-linearresponse Digital compensation will be addressed within that context

Capacitive pressure sensors were developed in the late 1970s and early 1980s The flexible diaphragm in thesedevices serves as one electrode of a capacitor, whereas the other electrode is located on a substrate beneath

it As the diaphragm deflects in response to applied pressure, the average gap between the electrodes changes,leading to a change in the capacitance The concept is illustrated in Figure 3.9

Although many fabrication schemes can be conceived, Chau and Wise (1988) described an attractivebulk micromachining approach Areas of a silicon wafer were first recessed, leaving plateaus to serve asanchors for the diaphragm Boron diffusion was then used to define an etch stop in the regions that wouldeventually form the structure The top surface of the silicon wafer was then anodically bonded to a glasswafer that had been inlaid with a thin film of metal, which served as the stationary electrode and providedlead transfer to circuitry The undoped silicon was finally dissolved in a dopant-selective etchant such as

ethylene diamine pyrocatechol (EDP) In order to maintain a low profile and small size, the interface circuitwas hybrid-packaged into a recess in the same glass substrate, creating a transducer that was small enough

to be located within a cardiovascular catheter of 0.5 mm outer diameter Using a 2 µm thick, 560 280 µm2

diaphragm with a 2 µm capacitor gap, and a circuit chip 350 µm wide, 1.4 mm long, and 100 µm thick,pressure resolution 2 mmHg was achieved [Ji et al., 1992]

The capacitive sensing scheme circumvents some of the limitations of piezoresistive sensing For example,since resistors do not have to be fabricated on the diaphragm, scaling down the device dimensions is eas-ier because concerns about stress averaging and resistor tolerance are eliminated In addition, the largestcontributor to the temperature coefficient of sensitivity — the variation of π44— is also eliminated Thefull-scale output swing can be 100% or more, in comparison to about 2% for piezoresistive sensing There

is virtually no power consumption in the sense element as the DC current component is zero However,capacitive sensing presents other limitations: (1) the capacitance changes non-linearly with diaphragmdisplacement and applied pressure; (2) even though the fractional change in the sense capacitance may

be large, the absolute change is small and considerable caution must be exercised in designing the sensecircuit; (3) the output impedance of the device is large, which affects the interface circuit design; and (4)the parasitic capacitance between the interface circuit and the device output can have a significant negativeimpact on the readout, which means that the circuit must be placed in close proximity to the device in

a hybrid or monolithic implementation An additional concern is related to lead transfer and packaging

In the case of absolute pressure sensors the cavity beneath the diaphragm must be sealed in vacuum.Transferring the signal at the counter electrode out of the cavity in a manner that retains the hermeticseal can present a substantial manufacturing challenge

to compute the response of a capacitive pressure sensor However, it can be shown that for deflectionsthat are small compared to the thickness of the diaphragm, the sensitivity, which is defined as the fractional