Performance Limits
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 rotacondi-tional 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:
σr [a2(1v)r2(3v)] (3.1)
σt 3∆P[a2(1v)r2(13v)] (3.2)
8h2 3∆P 8h2
where the former denotes the radial component and the latter the tangential component,aandhare the radius and thickness of the diaphragm,ris the radial co-ordinate,∆Pis 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 ofrbetween 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 (3.3)
where Eis 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 冤4.20 1.58 冥 (3.4)
where ais 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 wch, 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
w3c h3 wc
h Eh4
(1v2)a4
3∆P(1ν2)(a2r2)2 16Eh3
∆P
a r d
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, 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:
冢 冣φ (3.5)
where σ
i is the intrinsic or residual stress in the undeflected diaphragm, DEh3/[12(1ν2)], and φ dw/dris the slope of the deformed diaphragm. The solution this differential equation providesw:
w
in compression. In these expressions,JnandInare the Bessel function and the modified Bessel function of the first kind of ordern, respectively. The termkis given by:
k2 (3.8)
The maximum deflection (at the center of the diaphragm), normalized to the deflection in the absence of residual stress, is provided by:
wc (3.9)
in tension, and
wc (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:
σi (3.11)
As shown in Figure 3.2, residual stress can have a tremendous impact on deflection: a tensile dimension-less stress of 1.3 diminishes the center deflection by 50%. For tensile (positive) values ofσiexceeding 10, the center deflection (not normalized) can be approximated as for a membrane:
wc (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 SiO2
∆Pa2
or SixNyfor 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:
σctc冱
m
σmtm (3.13)
where σ
candtcdenote the composite stress and thickness, respectively, while σ
mandtmdenote 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:
Ectc冱
m
Emtm (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 H20 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
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−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 materi-als, and selected sensing scheme. Sensitivity is defined as a normalized signal change per unit pressure change to reference signal:
S (3.15)
whereθis output signal and ∂θis 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 (3.16)
whereSis 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 ∆P0.
Consequently, the TCO is:
TCO (3.17)
whereθ
0is offset, andTis 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.