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88                                                 Mechanical Transduction Techniques

                 the contribution to the gauge factor from the geometric effect is therefore between
                 1.4 and 1.6. Sensors that exhibit a change in resistance as a result of an applied strain
                 are generally termed strain gauges. Those in which the piezoresistive effect domi-
                 nates are often referred to as piezoresistors. As Table 5.1 shows, different materials
                 can have widely differing gauge factors.
                    So for a metal foil strain gauge or thin metal film, the geometric effect dominates
                 the piezoresistive effect; whereas for a semiconductor the converse is true.
                    Semiconductor strain gauges possess a very high gauge factor. P-type silicon has
                 a gauge factor up to +200, and n-type silicon has a negative gauge factor down to
                 –125. A negative polarity of gauge factor indicates that the resistance decreases with
                 increasing applied strain. In addition to exhibiting high strain sensitivity, semicon-
                 ductor strain gauges are also very sensitive to temperature. Compensation methods
                 must therefore be adopted when using semiconductor strain gauges.
                    A detailed account of the piezoresistive effect in silicon can be found in Middel-
                 hoek and Audet [1]; only a brief account will be given in this text. Essentially,
                 the effective mobilities of majority charge carriers are affected by the applied
                 stress. With p-type materials, the mobility of holes decreases and so the resistivity
                 increases. For n-type materials, the effective mobility of the electrons increases and
                 hence the resistivity decreases with applied stress. The effect is highly dependent on
                 the orientation. If the geometric effect in semiconductor strain gauges is neglected,
                 then the fractional change in resistivity is given by
                                           dρ
                                              = πσ +  πσ                              (5.10)
                                            p    l  l  t  t

                 where π and π are the longitudinal and transverse piezoresistive coefficients and σ
                        l     t                                                            l
                 and σ are the corresponding stresses. The longitudinal direction is defined as that
                      t
                 parallel to the current flow in the piezoresistor, while the transverse is orthogonal to
                 it. The two coefficients are dependent on the crystal orientation and doping (p-type
                 or n-type) and concentration. The temperature coefficient of piezoresistivity is
                 around 0.25 %/°C in both directions.
                    Polysilicon and amorphous silicon are also piezoresistive, but because they com-
                 prise crystallites, the net result is the average over all orientations. The temperature
                 coefficient of resistance (TCR), however, is significantly lower than that of single
                 crystal silicon and is generally less than 0.05%/°C. By carefully choosing the doping
                 levels, it is possible to reduce the TCR further.
                    Thin metal films behave in a similar manner to metal foil strain gauges and
                 hence it not surprising that their gauge factors are very similar. Such films can be
                 deposited directly onto the desired substrate (steel, ceramic, silicon) and therefore
                 become an integral part of the system, thus removing the need for adhesives as with


                                Table 5.1  Gauge Factors of Different Materials
                                Material                  Gauge Factor
                                Metal foil strain gauge   2–5
                                Thin-film metal           2
                                Single crystal silicon    –125 to +200
                                Polysilicon               ±30
                                Thick-film resistors      10
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