Piezoresistive Materials
A piezoresistive material responds to an applied stress with a change in its resistivity. Normally piezoresistive materials are semiconductors, and the effect is associated with band structure changes that alter the carrier mobility and number density much more significantly than an applied stress would change, for example, the resistance of a similar metal. The equivalent elastoresistance effect relates the strain in the material to a change in resistivity. Note that the dependence between the structural mechanics and the electrical properties of the material is unidirectional, so that applied currents do not induce changes in the material stress or strain (assuming that other effects, such as heating of the material, are negligible).
Within a piezoresistive material, the relation between the electric field, E, and the current, J, becomes:
(6-1)
where ρ is the resistivity and Δρ is the induced change in the resistivity. In the general case both ρ and Δρ are rank 2 tensors (matrices). The change in resistance is related to the stress, σ (for the piezoresistance form of the equations), or the strain, ε (for the elastoresistance form of the equations), by the constitutive relationship:
(6-2)
where Π is the piezoresistance tensor (SI unit: Pa−1Ωm) and M is the elastoresistance tensor (SI unit: Ωm). Both of these quantities are material properties. Π and M are in this case rank 4 tensors; however, they can be represented as matrices if the resistivity, stress, and strain are converted to vectors within a reduced subscript notation.
Tensor Versus Matrix Formulations