Page 47 - An Introduction to Microelectromechanical Systems Engineering

26 Materials for MEMS

way to incorporate stress-independent diffused temperature sensors. The crystal-
orientation-dependence of the piezoresistive coefficients takes a more complex func-
tion for piezoresistors diffused in {110} wafers, but this dependence fortuitously dis-
appears in {111} wafers. More descriptive details of the underlying physics of
piezoresistivity and dependence on crystal orientation can be found in [20, 21].
If we consider p-type piezoresistors diffused in {100} wafers and oriented in the
<110> direction (parallel or perpendicular to the flat), it is apparent from the posi-
tive sign of π in Table 2.4 that the resistance increases with tensile stress applied in
//
the parallel direction, σ , as if the piezoresistor itself is being elongated. Further-
//
more, the negative sign of π implies a decrease in resistance with tensile stress
⊥
orthogonal to the resistor, as if its width is being stretched. In actuality, the stretch-
ing or contraction of the resistor are not the cause of the piezoresistive effect, but
they make a fortuitous analogy to readily visualize the effect of stress on resistance.
This analogy breaks down for n-type piezoresistors.
Like many other physical effects, piezoresistivity is a strong function of tempera-
ture. For lightly doped silicon (n-or p-type, 10 cm ), the temperature coefficient of
18
-3
π and π is approximately –0.3% per degree Celsius. It decreases with dopant con-
// ⊥
-3
19
centration to about –0.1% per degree Celsius at8×10 cm .
Polysilicon and amorphous silicon also exhibit a strong piezoresistive effect. A
wide variety of sensors using polysilicon piezoresistive sense elements have been
demonstrated. Clearly, piezoresistive coefficients lose their sensitivity to crystalline
direction and become an average over all orientations. Instead, the gauge factor, K,
relating the fractional change in resistance to strain is often used. Gauge factors in
polysilicon and amorphous silicon range typically between –30 and +40, about a
third that of single-crystal silicon. The gauge factor decreases quickly as doping con-
centration exceeds 10 cm . However, one advantage of polysilicon over crystal-
−3
19
line silicon is its reduced TCR. At doping levels approaching 10 cm , the TCR for
20
−3
polycrystalline silicon is approximately 0.04% per degree Celsius compared to
0.14% per degree Celsius for crystalline silicon. The deposition process and the
dopant species have been found to even alter the sign of the TCR. For example,
emitter-type polysilicon (a special process for depositing heavily doped polysilicon
to be used as emitter for bipolar transistors) has a TCR of –0.045% per degree Cel-
sius. Resistors with positive TCR are particularly useful in compensating the nega-
tive temperature dependence of piezoresistive sensors.

Piezoelectricity

Certain classes of crystals exhibit the peculiar property of producing an electric field
when subjected to an external force. Conversely, they expand or contract in response

Table 2.4 Piezoresistive Coefficients for n- and p-Type {100}
18
Wafers and Doping Levels Below 10 cm -3
π // π ⊥
2
2
(10 -11 m /N) (10 -11 m /N)
p-type –107 ––1 In <100> direction
–172 –66 In <110> direction
n-type –102 –53 In <100> direction
––31 –18 In <110> direction
Note: The values decrease precipitously at higher doping concentrations.