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Single-Crystal Silicon Carbide MEMS: Fabrication, Characterization, and Reliability 7-23

If this reaction proceeds to the right, the limiting reactant would be silicon, since it is fully consumed
by platinum. This proposition is made more likely, given that the heat of formation of Pt Si
2
1
1
( ∆H 10kcalmol ) is less than that of TaSi ( ∆H 28kcalmol ) [Andrews and Phillips, 1975].
f 2 f
It is important to examine the oxygen concentrations in Figures 7.15(a) and 7.15(b). Both parts of the
figure show very little migration of oxygen into the metallization. This finding implies a much more sta-
ble contact structure than the Ti/TiN/Pt approach [Okojie et al., 1999], which was affected by significant
oxidation of the metals in the contact. Various groups have extensively studied the oxidation kinetics of
the silicides. Murarka (1988) concluded that the oxidation mechanisms for most silicides essentially have
the same heats of formation as those of normal oxides. Lie (1984) and Razouk (1982) confirmed that the
oxidation kinetics of tantalum disilicide has a parabolic rate. This result implies that it would take an
appreciable length of time for oxygen to diffuse through the entire contact.

7.5 Sensor Characteristics

Generally, the design of devices that sense physical phenomena and provide electrical readouts calls for
the interpolation of two or more kinds of mathematical relationships. In the design of high-temperature
pressure sensors, the Equations that describe the physical phenomena (i.e., pressure and diaphragm deflec-
tion) are interpolated with the electrical Equations that express the resulting output voltage.
The Equations that model deflecting diaphragms are classified into two main categories. One category
models maximum diaphragm deflections that are less than the diaphragm thickness (linear case),
whereas the other supports diaphragms with maximum deflections greater than the diaphragm thickness
(nonlinear case). If the maximum deflection of the membrane is less than its thickness, as occurs in appli-
cations of short-range pressure measurement, there is generally a reasonable degree of linearity of
diaphragm deflection in response to applied pressure. For larger pressure measurements where deflection
is equal to or greater than the thickness of the diaphragm, the deflection of the diaphragm in response to
applied pressure is no longer linear [Timoshenko and Woinowsky-Krieger, 1959]. For the device to be
used continuously over a long period of time, the membrane must be capable of repeatedly deflecting
under applied pressure with precision and little hysteresis. To achieve this aim, the membrane must retain
its elastic property after it is subjected to maximum applied pressures. To that effect, there is a need to
choose materials with an appreciable linear region on the Stress-Strain curve. For the diaphragm to retain
its elastic integrity, the stress induced by pressure must not exceed the yield or fracture point. In essence,
the maximum operating stress should be at a point below the yield and fracture stress limit. If the oper-
ating stress reaches the elastic or fracture limit, there is a very strong likelihood that the diaphragm would
lose its elasticity, become permanently deformed, and possibly fracture.
Aresistor on a circular diaphragm arranged tangentially, with current flowing parallel to the resistor,
will experience a longitudinal stress induced by the tangential strain component. It will also experience a
transverse (radial) strain component (strain perpendicular to resistor length), which usually inserts a
negative piezoresistance coefficient. On the other hand, as depicted in Figure 7.3(b), the radially oriented
resistor, with current flowing parallel along its length, will be dominated by the stress induced by the lon-
gitudinal strain component. The transverse effect is introduced via the tangential stress, with its corre-
sponding negative piezoresistance coefficient. These findings have been extensively verified and used in
silicon. The output of the sensor is strongly affected by the orientation of the resistors. Therefore, the
resistor geometry and orientation should be such that only one strain component exists and the other is
suppressed.
When the maximum deflection, w, of a clamped circular plate is less than its thickness, the equation
that describes it is expressed as [Timoshenko and Woinowsky-Krieger, 1959]:

Pa 4
w 0.89 (7.20)
64D

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