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

3-2 MEMS: Applications 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 as plasma 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 vehi- cles [Wise, 1994]. A number of sensing approaches that offer different relative merits have evolved, and there 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 structural material of micromachined pressure sensors partly because its material properties are adequate, and there is significant manufacturing capacity and know-how that can be borrowed from the integrated circuit industry. The primary focus in this chapter is on schemes that use silicon as the structural material. The chapter is divided into six sections. The first section introduces structural and performance concepts that are common to a number of micromachined pressure sensors. The second and third sections focus in some detail on piezoresistive and capacitive pick-off schemes for detecting pressure. These two schemes form the basis of the vast majority of micromachined pressure sensors available commercially and studied by the MEMS research community. Fabrication, packaging, and calibration issues related to these devices are also addressed in these sections. The fourth section describes servo-controlled pressure sensors, which represent an emerging theme in research publications. The fifth section surveys alternative approaches and transduction schemes that may be suitable for selected applications. It includes a few schemes that have been explored with non-micromachined apparatus, but may be suitable for miniaturization in the future. The sixth section concludes the chapter. 3.2 Device Structure and Performance Measures The essential feature of most micromachined pressure sensors is an edge-supported diaphragm that deflects in response to a transverse pressure differential across it. This deformation is typically detected by measuring 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 dis- placement. The choice of silicon as a structural material is amenable to both approaches because it has a relatively large piezoresistive coefficient and because it can serve as an electrode for a capacitor as well. 3.2.1 Pressure on a Diaphragm The deflection of a diaphragm and the stresses associated with it can be calculated analytically in many cases. It is generally worthwhile to make some simplifying assumptions regarding the dimensions and boundary 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 rotational and longitudinal displacement. Mathematically, it permits the second derivative of the deflection to be zero at the edge of the diaphragm. However, the preferred assumption is that the edges of the diaphragm are rigidly affixed (built-in) to the support around its perimeter. Under this assumption the stress on the lower surface of a circular diaphragm can be expressed in polar coordinates by the equations: σ r ϭ [a 2 (1 ϩ v) Ϫ r 2 (3 ϩ v)] (3.1) σ t ϭ [a 2 (1 ϩ v) Ϫ r 2 (1 ϩ 3v)] (3.2) 3 и ∆P ᎏ 8h 2 3 и ∆P ᎏ 8h 2 © 2006 by Taylor & Francis Group, LLC 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 ratio shows 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 vary from 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. The deflection of a circular diaphragm under the stated assumptions is given by: d ϭ (3.3) 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, vary- 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 of grains of varying size and crystalline orientation can cause substantial variations. It is important to note that additional variations in mechanical properties may arise from crystal defects caused by doping and other disruptions of the lattice. Equation 3.3 indicates that the maximum deflection of a diaphragm is at its 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 these dimensions. 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 etchants that are selective to crystallographic planes, which results in square diaphragms. The deflection of a square diaphragm with built-in edges can be related to applied pressure by the following expression: ∆P ϭ ΄ 4.20 ϩ 1.58 ΅ (3.4) 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 Ͻ h, whereas the second term dominates for large deflections. For very large deflections, it approaches the deflection predicted for flexible membranes with a 13% error. 3.2.3 Residual Stress It should be noted that the analysis presented above assumes that the residual stress in the diaphragm is negligibly small. Although mathematically convenient, this is often not the case. In reality, a tensile stress w 3 c ᎏ h 3 w c ᎏ h Eh 4 ᎏᎏ (1 Ϫ v 2 )a 4 3 и ∆P(1 Ϫ ν 2 )(a 2 Ϫ r 2 ) 2 ᎏᎏᎏ 16Eh 3 Micromachined Pressure Sensors: Devices, Interface Circuits, and Performance Limits 3-3 ∆P a r d FIGURE 3.1 Deflection of a diaphragm under applied pressure. © 2006 by Taylor & Francis Group, LLC of 5–50MPa is not uncommon. This may significantly reduce the sensitivity of certain designs, particu- larly if the diaphragm is very thin. Following the treatment in Chau and Wise (1987) for a small deflection in a circular diaphragm with built-in edges, the governing differential equation is: ϩ Ϫ ΂ ϩ ΃ φ ϭ Ϫ (3.5) where σ i is the intrinsic or residual stress in the undeflected diaphragm, D ϭ Eh 3 /[12(1 Ϫ ν 2 )], and φ ϭϪdw/dr is the slope of the deformed diaphragm. The solution this differential equation provides w: w ϭ ∆P и a 4 ΄ I 0 ΂΃ Ϫ I 0 (k) ΅ ∆P и a 2 (a 2 Ϫ r 2 ) 2k 3 I 1 (k)D ϩ 4k 2 D (3.6) in tension, and w ϭ ∆P и a 4 ΄ J 0 ΂΃ Ϫ J 0 (k) ΅ ∆P и a 2 (a 2 Ϫ r 2 ) 2k 3 J 1 (k)D ϩ 4k 2 D (3.7) in compression. In these expressions, J n and I n are the Bessel function and the modified Bessel function of the first kind of order n,respectively. The term k is given by: k 2 ϭ ϭ (3.8) The maximum deflection (at the center of the diaphragm), normalized to the deflection in the absence of residual stress, is provided by: wЈ c ϭ (3.9) in tension, and wЈ c ϭ (3.10) in compression. It is instructive to evaluate the dependence of this normalized deflection to the dimen- sionless intrinsic stress, which is defined by: σ Ј i ϭ (3.11) As shown in Figure 3.2,residual stress can have a tremendous impact on deflection: atensile dimension- less stress of 1.3 diminishes the center deflection by 50%. For tensile (positive) values of σ Ј i exceeding 10, the center deflection (not normalized) can be approximated as for a membrane: w c ϭ (3.12) Returning to Figure 3.2, it is evident that the deflection can be increased by compressive stress. However, even relatively small values of compressive stress can result in buckling, so it is not generally perceived as a feature that can be reliably exploited. 3.2.4 Composite Diaphragms In micromachined pressure sensors, it is often the case that the diaphragm is fabricated not from a single material but from composite layers. For example, a silicon membrane can be covered by a layer of SiO 2 ∆P и a 2 ᎏ 4 σ i h (1 Ϫ ν 2 ) σ i a 2 ᎏᎏ Eh 2 16[2 Ϫ 2J 0 (k) ϩ kJ 1 (k)] ᎏᎏᎏ k 3 J 1 (k) 16[2 Ϫ 2I 0 (k) ϩ kI 1 (k)] ᎏᎏᎏ k 3 I 1 (k) 12(1 Ϫ v 2 ) σ i a 2 ᎏᎏ Eh 2  σ i a 2 h ᎏ D kr ᎏ a kr ᎏ a ∆P и r ᎏ 2D 1 ᎏ r 2 σ i h ᎏ D d φ ᎏ dr 1 ᎏ r d 2 φ ᎏ dr 2 3-4 MEMS: Applications © 2006 by Taylor & Francis Group, LLC or Si x N y for electrical isolation. In general, these films can be of comparable thickness, and have values of Young’s modulus and residual stress that are significantly different. The residual stress of a composite membrane is given by: σ c t c ϭ Α m σ m t m (3.13) where σ c and t c denote the composite stress and thickness, respectively, while σ m and t m denote the stress and thickness of individual films. Furthermore, if the Poisson’s ratio of all the layers in the membrane is comparable, the following approximation may be used for the Young’s modulus: E c t c ϭ Α m E m t m (3.14) 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 accomplish this 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 com- 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 H 2 0 at 4°C), and 760 Torr (or millimeters of Hg at 0°C). 3.2.6 Performance Criteria The performance criteria of primary interest in pressure sensors are sensitivity, dynamic range, full-scale output, linearity, and the temperature coefficients of sensitivity and offset. These characteristics depend Micromachined Pressure Sensors: Devices, Interface Circuits, and Performance Limits 3-5 0.1 0.01 0.1 1 10 100 1 10 100 Center deflection (Compression) Center deflection (Tension) Pressure sensitivity (Compression) Pressure sensitivity (Tension) Dimensionless stress (1−v 2 )  l a 2 /Eh 2 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.) © 2006 by Taylor & Francis Group, LLC on the device geometry, the mechanical and thermal properties of the structural and packaging materi- als, and selected sensing scheme. Sensitivity is defined as a normalized signal change per unit pressure change to reference signal: S ϭ (3.15) 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 by the saturation of the transduced output signal such as the piezoresistance or capacitance. It also may be limited 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 the device 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: TCS ϭ (3.16) where S is sensitivity. Another important parameter is the temperature coefficient of offset (TCO). The offset of a pressure sensor is the value of the output signal at a reference pressure, such as when ∆P ϭ 0. Consequently, the TCO is: TCO ϭ (3.17) where θ 0 is 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. 3.3 Piezoresistive Pressure Sensors The majority of commercially available micromachined pressure sensors are bulk micromachined piezore- 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 Ͻ100Ͼ oriented Si wafer with an anisotropic wet etchant such as potassium hydroxide (KOH). An electrochemical etch- stop, dopant-selective etch-stop, or a layer of buried oxide can be used to terminate the etch and control the thickness of the unetched diaphragm. This diaphragm is supported at its perimeter by a portion of the wafer that was not exposed to the etchant and remains at full thickness (Figure 3.1). The piezoresistors are fashioned by selectively doping portions of the diaphragm to form junction-isolated resistors. Although this form of isolation permits significant leakage current at elevated temperatures and the resistors present sheet resistance per unit length that depends on the local bias across the isolation diode, it allows the designer to exploit the substantial piezoresistive coefficient of silicon and locate the resistors at the points 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) was used 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 lim- ited to the thickness of the sacrificial layer, and can constrain the dynamic range. In Guckel et al. (1986), (reprinted in Microsensors (1990)) polysilicon was used to form both the diaphragm and the piezoresistors. ∂ θ 0 ᎏ ∂T 1 ᎏ θ 0 ∂S ᎏ ∂T 1 ᎏ S ∂ θ ᎏ ∂P 1 ᎏ θ 3-6 MEMS: Applications © 2006 by Taylor & Francis Group, LLC 3.3.1 Design Equations In an anisotropic material such as single crystal silicon, resistivity is defined by a tensor that relates the three directional components of the electric field to the three directional components of current flow. In general, the tensor has nine elements expressed in a 3 ϫ 3 matrix, but they reduce to six independent val- ues from symmetry considerations: ΄΅ ϭ ΄ ΅΄΅ (3.18) where ε i and j i represent electric field and current density components, and ρ i represent resistivity com- ponents. Following the treatment in Kloeck and de Rooij (1994) and Middleoek and Audet (1994), if the Cartesian axes are aligned to the Ͻ100Ͼ axes in a cubic crystal structure such as silicon, ρ 1 , ρ 2 , and ρ 3 will be equal because they all represent resistance along the Ͻ100Ͼ axes, and are denoted by ρ . The remain- ing components of the resistivity matrix, which represent cross-axis resistivities, will be zero because unstressed 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, ∆ ρ i , will be related to all the stress components. The stress can always be decomposed into three normal components ( σ i ), and three shear components ( τ i ), 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 a 36-element tensor. However, due to symmetry conditions, this tensor is populated by only three non- zero components, as shown: ΄ ΅ ϭ ΄ ΅ ϩ ΄ ΅ ; ΄ ΅ ϭ ΄ ΅΄ ΅ (3.19) σ 1 σ 2 σ 3 τ 1 τ 2 τ 3 0 0 0 0 0 π 44 0 0 0 0 π 44 0 0 0 0 π 44 0 0 π 12 π 12 π 11 0 0 0 π 12 π 11 π 12 0 0 0 π 11 π 12 π 12 0 0 0 ∆ ρ 1 ∆ ρ 2 ∆ ρ 3 ∆ ρ 4 ∆ ρ 5 ∆ ρ 6 1 ᎏ ρ ∆ ρ 1 ∆ ρ 2 ∆ ρ 3 ∆ ρ 4 ∆ ρ 5 ∆ ρ 6 ρ ρ ρ 0 0 0 ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 ρ 6 j 1 j 2 j 3 ρ 5 ρ 4 ρ 3 ρ 6 ρ 2 ρ 4 ρ 1 ρ 6 ρ 5 ε 1 ε 2 ε 3 Micromachined Pressure Sensors: Devices, Interface Circuits, and Performance Limits 3-7 X Y Z  1  3  2  2  3  3  1  1  2 FIGURE 3.3 Definition of normal and shear stresses. © 2006 by Taylor & Francis Group, LLC where the π ij coefficients, which have units of Pa Ϫ1 ,may be either positive or negative, and are sensitive to doping type, doping level, and operating temperature. It is evident that π 11 relates the resistivity in any direction to stress in the same direction, whereas π 12 and π 44 are cross-terms. Equation 3.19 was derived in the context of acoordinate system aligned to the Ͻ100Ͼ axes, and is not always convenient to apply. A preferred representation is to express the fractional change in an arbitrar- ily oriented diffused resistor by: ϭ π l σ l ϩ π t σ t (3.20) where π l and σ l are the piezoresistive coefficient and stress parallel to the direction of current flow in the resistor (i.e., parallel to its length), and π t and σ t are the values in the transverse direction. The piezore- sistive coefficients referenced to the direction of the resistor may be obtained from those referenced to the Ͻ100Ͼ axes in Equation 3.19 by using a transformation of the coordinate system. It can then be stated that: π l ϭ π 11 ϩ 2( π 44 ϩ π 12 Ϫ π 11 )(l 2 1 m 2 1 ϩ l 2 1 n 2 1 ϩ n 2 1 m 2 1 ) (3.21) π t ϭ π 12 Ϫ ( π 44 ϩ π 12 Ϫ π 11 )(l 2 1 l 2 2 ϩ m 2 1 m 2 2 ϩ n 2 1 n 2 2 ) (3.22) where l 1 ,m 1 , and n 1 are 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 l 2 ,m 2 , and n 2 are those for a unit length vector perpendicular to the resistor. Thus, l i 2 ϩ m i 2 ϩ n i 2 ϭ 1. As an example, for the Ͻ111Ͼ direction, in which projections to all three crystallographic axes are equal, l i 2 ϭ m i 2 ϭ n i 2 ϭ 1/3. A sample set of piezoresistive coefficients for Si is listed in Table 3.1.It is evident that π 44 dominates for p-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. It should be noted, however, that the piezoresistive coefficient can vary significantly with doping level and operating temperature of the resistor. A convenient way in whichtorepresent the changes is to normal- ize the piezoresistive coefficient to a value obtained at room temperature for weakly doped silicon [Kanda, 1982]: π (N,T) ϭ P(N,T) π ref . (3.23) Figure 3.4 shows the variation of parameter P for p-type and n-type Si, as N, the doping concentration, and T, the temperature, are varied. Figure 3.5 plots the longitudinal and transverse piezoresistive coefficients for resistor orientations on the surface of a (100) silicon wafer. Note that each figure is split into two halves, showing π l and π t simul- taneously for p-type Si in one case and n-type Si in the other case. Each curve would be reflected in the horizontal axis if drawn individually. Also note that for p-type Si, both π l and π t peak along Ͻ110Ͼ , whereas for n-type Si, they peak along Ͻ100Ͼ .Since anisotropic wet etchants make trenches aligned to Ͻ110Ͼ on these wafer surfaces, p-type piezoresistors, which can be conveniently aligned parallel or perpendicular to the etched pits, are favored. Consider two p-type resistors aligned to the Ͻ110Ͼ axes and near the perimeter of a circular diaphragm on a silicon wafer: assume that one resistor is parallel to the radius of the diaphragm, whereas the 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: ΂ ΃ ra ϭ Ϫ ΂ ΃ ta ϭϪ ∆P и (3.24) where the subscripts denote radial and tangentially oriented resistors. The complementary change in these resistors is well-suited to a bridge-type arrangement for readout, as shown in Figure 3.6. 3 π 44 a 2 (1 Ϫ v) ᎏᎏ 8h 2 ∆R ᎏ R ∆R ᎏ R ∆R ᎏ R 3-8 MEMS: Applications © 2006 by Taylor & Francis Group, LLC (The bridge-type readout arrangement is suitable for square diaphragms as well.) The output voltage in this case is given by: ∆ ∆ V 0 ϭ V s (3.25) ∆R ᎏ R Micromachined Pressure Sensors: Devices, Interface Circuits, and Performance Limits 3-9 TABLE 3.1 A Sample of Room Temperature Piezoresistive Coefficients in Si in 10 Ϫ11 Pa Ϫ1 . (Reprinted with Permission from Smith, C.S. [1954] “Piezoresistance Effect in Germanium and Silicon,” Physical Review 94, pp. 42–49 Resistivity π 11 π 12 π 44 7.8 ⍀ cm, p-type 6.6 Ϫ1.1 138.1 11.7 ⍀ cm, n-type Ϫ102.2 53.4 Ϫ13.6 0 0 0.5 1.0 1.5 10 16 10 16 10 17 10 18 10 19 10 20 10 17 10 18 10 19 10 20 0.5 1.0 1.5 N (cm −3 ) N (cm −3 ) P(N,T) P(N,T) p-Si n-Si T = −75°C T = −50°C T = −25°C T = 0°C T = 25°C T = 50°C T = 75°C T = 100°C T = 150°C T = 125°C T = 175°C T = −75°C T = −50°C T = −25°C T = 0°C T = 25°C T = 50°C T = 75°C T = 100°C T = 150°C T = 125°C T = 175°C 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.) © 2006 by Taylor & Francis Group, LLC 3-10 MEMS: Applications 0 0 90 90 100 100 110 110 120 120 130 130 140 140 150 150 160 160 170 170 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 20 30 40 50 60 70 80 90 100 110 20 30 40 50 60 70 80 90 100 110 −110 −100 −90 −80 −70 −60 −50 −40 −30 −110 −100−90 −80 −70 −60 −50 −40 −30 −20 −20 [010] [110] [100] [010]  l  t 0 0 90 90 1 0 0 1 0 0 110 110 120 120 130 130 140 140 150 150 1 6 0 1 6 0 170 170 1 0 1 0 20 20 30 30 40 40 50 50 6 0 6 0 70 70 80 80 20 30 40 50 60 70 80 90 100 110 20 30 40 50 60 70 80 90 100 110 − 110−100 −90 −80 − 70 −60 −50 −40 −30 −110 −90 −80 −70 −60 −50 −40 −30 −20 − 20 [010] [110] [100] [010] [110] [110]  l  t −100 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.) © 2006 by Taylor & Francis Group, LLC Since the output is proportional to the supply voltage V s , the output voltage of piezoresistive pressure sen- sors is generally presented as a fraction of the supply voltage per unit change in pressure. It is proportional to a 2 /h 2 , 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 of sensitivity, which is the fractional change in the sensitivity per unit change in temperature, is primarily governed by the temperature coefficient of π 44 . 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 the remainder of the sensing circuit to be located at some distance from the diaphragm, without deleterious effects from parasitic capacitance that may be incorporated. This stands in contrast to the output from capacitive 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 will have 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 the center. Since the maximum stresses occur at these locations, a resistor that extends between them will be subject 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 a non-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, including mechanical vibration of the diaphragm, electrical noise from the piezoresistors, and electrical noise from the interface circuit. Thermal energy in the form of Brownian motion of the gas molecules surrounding the diaphragm causes variations in its deflection that can be treated as though caused by an equivalent pressure. 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 Micromachined Pressure Sensors: Devices, Interface Circuits, and Performance Limits 3-11 R + ∆R V 0 Tangential resistor Radial resistor R + ∆R R − ∆R R − ∆R V s FIGURE 3.6 Schematic representation of Wheatstone bridge configuration (left) and placement of radial and tan- gential strain sensors on a circular diaphragm (right). © 2006 by Taylor & Francis Group, LLC [...]... 1 SЈ ϭ 19 2 ᎏ ϩ ᎏ Ϫ ᎏ ᎏ cap 4 2 k 8k 2k3 I1(k) © 2006 by Taylor & Francis Group, LLC ΅ in tension, and (3. 41) 3 -1 8 MEMS: Applications ΄ 1 J0(k) 1 1 SЈ ϭ 19 2 ᎏ Ϫ ᎏ Ϫ ᎏ ᎏ cap 4 2 k 8k 2k3 J1(k) ΅ in compression, (3. 42) where the variables are defined as for Equation 3. 6 These equations are plotted in Figure 3. 2, and as expected, the lines are very close to those of the normalized deflection at the center... (3. 59) Vo ϭ AZe(Fd Ϫ Fa) (3. 60) if Cref ϭ ε This signal is fed back to the electrostatic actuator, which generates force Fa: Fa ϭ ZaVo (3. 61) Substituting (3. 54) and (3. 61) into (3. 60), the output of the servo-controlled system is: Vo ϭ AZe(ZdP Ϫ ZaVo) (3. 62) Ze A Vo ϭ ᎏᎏ ZdP 1 ϩ Ze ZaA (3. 63) 1 Vo ϭ ᎏ ZdP as Ze A → ϱ Za (3. 64) The electrostatic force can be obtained by substituting (3. 63) into (3. 61) :... 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, vIN, which is provided by the output of the resistor bridge, determines the ratio of the currents in the emitter-coupled stage formed by Q3 and Q4: ΂ ΃ qνIN IC3 ᎏ ϭ exp ᎏ kT IC4 v1 (3. 33) + A1 − vo1 R4 R2 R3... Limits 3 -1 7 Diaphragm Electrode Substrate Dielectric layer FIGURE 3 .10 A touch-mode capacitive pressure sensor change in the capacitance per unit change in pressure, has the following proportionality [Chau and Wise, 19 87]: ∆C 1 Scap ϭ ᎏ ϭ ᎏ C и ∆P PAd 1 v a ͵ w dA ϭ 0.0746 ᎏ ᎏ E hd 2 4 3 A (3. 40) where d is the nominal gap between the diaphragm and the electrode, and the remaining variables represent the. .. is large, the same current also flows through the two resistors labeled R2, causing the voltage difference between the outputs of A1 and A2 to be: 2R2 (3. 31 ) vo1 Ϫ vo2 ϭ 1 ϩ ᎏ (v1 Ϫ v2) R1 ΂ ΃ 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 A3 and the surrounding resistors This causes the output... locating the pick-off capacitance outside the sealed cavity is illustrated in Figure 3 .15 [Park and Gianchandani, 2000] A skirt-shaped electrode extends outward from the periphery of the vacuum-sealed cavity, and serves as the element which deflects under pressure The stationary electrode is metal patterned on the substrate below this skirt As the external pressure © 2006 by Taylor & Francis Group, LLC 3- 2 2... Microelectromechanical Systems, pp 5 51 555.) R2 R1 T3 T1 T2 H Electrode Deformable skirt or flap (silicon) G1 G2 Substrate (glass) Sealed cavity (volume V) 2 Z Y X Vbias FIGURE 3 .15 Electrostatic attraction between the electrode and skirt opposes the deflection due to external pressure increases, the center of the diaphragm deflects downwards, and the periphery of the skirt rises, reducing the pick-off... smaller than the instability limit (i.e., the “pull-in” voltage at which the a diaphragm would collapse to the actuation electrode), the electrostatic pressure is relatively small The solutions that have been developed are described in the following case studies In the structure shown in Figure 3 .12 , the Si diaphragm is suspended between two glass wafers on which the sensing and feedback electrodes are... Figure 3. 6 would be connected to the two input terminals labeled 1 and ν2 in Figure 3. 7 The instrumentation amplifier has two stages [Sedra and Smith, 19 98] 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 R1 to be ( 1 Ϫ ν2)/R1 Since the input impedance of the op... has been shown to produce devices with low TCO 10 −6 8 Ni-Co-Fe alloy 7 Thermal expansion 6 W 5 4 3 Pyrex 2 Si 1 SiO2 200 400 600 800 T (K) FIGURE 3 .11 Thermal expansion coefficients of various materials as a function of temperature (Reprinted with permission from Greenwood, J.C [19 88] “Silicon in Mechanical Sensors,” J Phys E, Sci Instrum., 21, pp 11 14 11 28.) © 2006 by Taylor & Francis Group, LLC Micromachined . −60 −50 −40 30 11 0 10 0−90 −80 −70 −60 −50 −40 30 −20 −20 [ 010 ] [11 0] [10 0] [ 010 ]  l  t 0 0 90 90 1 0 0 1 0 0 11 0 11 0 12 0 12 0 13 0 13 0 14 0 14 0 15 0 15 0 1 6 0 1 6 0 17 0 17 0 1 0 1 0 20 20 30 30 40 40 50 50 6 0 6 0 70 70 80 80 20. Germanium and Silicon,” Physical Review 94, pp. 42–49 Resistivity π 11 π 12 π 44 7.8 ⍀ cm, p-type 6.6 1. 1 13 8 .1 11. 7 ⍀ cm, n-type 10 2.2 53. 4 13 .6 0 0 0.5 1. 0 1. 5 10 16 10 16 10 17 10 18 10 19 10 20 10 17 10 18 10 19 10 20 0.5 1. 0 1. 5 N. 0 90 90 10 0 10 0 11 0 11 0 12 0 12 0 13 0 13 0 14 0 14 0 15 0 15 0 16 0 16 0 17 0 17 0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 20 30 40 50 60 70 80 90 10 0 11 0 20 30 40 50 60 70 80 90 10 0 11 0 11 0 10 0 −90 −80 −70 −60 −50 −40 30 11 0