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Single-Crystal Silicon Carbide MEMS: Fabrication, Characterization, and Reliability 7-29

7.6.2 Transducer Parametric Analysis

The resistors are arranged in a Wheatstone bridge configuration, which, ideally, will have a pressure off-
set voltage, V , of zero if all resistors are exactly the same. Deviation from perfect resistor symmetry results
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in a non-zero V across the bridge that is expressed as:
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V R R R R
3
1
4
in
2
V (7.30)
oz 2 R R R R
1 2 3 4
where R , R , R , and R are the bridge resistor elements (Ω) and V is the input voltage (V). Another
1 2 3 4 in
influence on V is the thermomechanical package-induced stress discussed above. The relative change in
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each resistor element due to externally applied and residual strain leads to the modification of equation
(7.1) to be expressed as:
∆R R(ε ε )GF (7.31)
δ
The TCR of a resistor approximates the resistor value via the expression:

R(T) R (1 β∆T) (7.32)
o
where R is the resistance at the reference temperature (room temperature), and ∆T (°C) is the temperature
o
change from the reference temperature. The temperature effect of the gauge factor is similarly modeled by:

G(T) GF(1 γ∆T) (7.33)

It is important to note that under an unstrained condition (i.e., no applied pressure and sensor not pack-
aged), the bridge output is governed only by Equations (7.30) and (7.32). With the sensor attached to the
package at elevated temperature and cooled down to room temperature, it experiences a package-induced
residual strain, as expressed in Equation (7.29), thus creating a change in the resistance and V . When
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pressure and temperature are applied simultaneously, each resistor element experiences a relative change,
which is expressed by combining Equations (7.29) and (7.31–7.33) as:

2
∆R(T) R G [1 (β γ)∆T γβ(∆T) ](ε ε ) (7.34)
o
o
δ
Therefore, with the sensor fully packaged and no applied pressure (ε 0), the change in resistance with
temperature results in change in V . As the operating temperature increases, ε will decrease correspond-
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δ
ingly (relaxation), and the resistance change becomes largely governed by β and less by γ parameters.
Equation (7.34) is particularly important for understanding the overall behavior of the transducer, especially
with regard to temperature compensation and transducer calibration. It also reveals the effect of ther-
momechanically induced stress on the system and its deleterious effect on the long-term output stability.

7.6.3 AST Protocol

The AST protocol used is shown in Figure 7.22.We evaluated the stability and reliability of 6H-SiC pressure
transducers at temperatures starting from 25°C up to the temperature of instability onset, T .This is
unstable
the temperature at which the V fluctuates such that no reliable pressure measurement can be performed.
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The initial pressure tests at room temperature were performed in Step 1, where 20 cycles of 0 psi →
P max → 0 psi were applied at steps of 10%P max and held for ten seconds per step. The pressure, P max , was
predetermined to be 100psi from previously validated finite element modeling and burst pressure analysis.
The failing transducers after this step are removed for failure analysis (FA) to determine the failure mech-
anisms. The passing transducers after Step 1 were thus validated to handle the maximum pressure.
In Step 2,the transducers were heated from room temperature to 100°C and cycled 20 times between 0 and
100psi. This 20-cycle pressurization was repeated at increments of 100°C following one-hour, zero-pressure

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