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5.1 Piezoresistivity 87
If it is assumed that the block is made of a resistive material, then its resistance,
R, is given by
l ρ
R = (5.3)
A
where ρ is the bulk resistivity of the material (Ωcm), l is the length, and A is the
cross-sectional area (i.e., the product of width w and thickness t).
Hence,
l ρ
R = (5.4)
wt
Differentiating the equation for resistance gives
l ρ l ρ l ρ
dR = d + dl − dw − dt (5.5)
ρ
2
wt wt wt wt 2
and hence
dR dρ dl dw dt
= + − − (5.6)
R ρ l w t
By definition, ε = dl/l, so the following equations apply on the assumption
l
that we are dealing with small changes, and hence dl = ∆l, dw = ∆w, and dt = ∆t:
dw dt
=ε =−νε and =ε =−νε (5.7)
w w l t t l
where ν is Poisson’s ratio. Note the minus signs, indicating that the width and thick-
ness both experience compression and hence shrink. It is worth noting that the
above example illustrates a positive Poisson’s ratio. 1
Therefore, from (5.6) and (5.7) we have
dR dρ
= +ε +νε +νε (5.8)
R ρ l l l
From (5.1) the gauge factor is therefore
dR R dρρ
GF = = + ( + ν12 ) (5.9)
ε ε
l l
Equation (5.9) indicates clearly that there are two distinct effects that contribute
to the gauge factor. The first term is the piezoresistive effect ((dρ/ρ)/ε) and the sec-
l
ond is the geometric effect (1 + 2). As Poisson’s ratio is usually between 0.2 and 0.3,
1. Materials having a negative Poisson’s ratio do exist. That is to say, as you stretch them, the width and thick-
ness actually increase. Examples of such materials include special foams and polymers such as polyte-
trafluoroethylene (PTFE).