Electric Currents Formulation
The flow of electric current through a conducting medium in the quasistatic case is governed by the equation of continuity which can be written in the form:
where J is the induced electric current, Je includes external current sources that do not result from the material conductivity (such as thermoelectric currents), D is the electric displacement field, and Qj is a current source term. Both Qj and Je are usually zero for piezoresistive devices.
In the stationary or quasistatic case, the electric current is related to the electric field, E, by the constitutive relation:
(6-3)
where σ(c) is the conductivity tensor.
Relating E and D to the electric potential (E=-V, D=ε0ε(r)E, where ε0 is the permittivity of free space and ε(r) is the relative permittivity tensor) gives:
(6-4)
If current flow is restricted to a thin layer on the surface of a material this equation takes the form:
(6-5)
where the operator t represents the tangential derivative along the thin layer surface. Equation 6-4 and Equation 6-5 form the basis of the Electric Currents interface.
The constitutive relationship described by Equation 6-3 uses the conductivity form rather than the resistivity form used in Equation 6-1.
Since the equations of piezoresistivity are by convention formulated in the resistivity form, Equation 6-1 must be inverted to give:
(6-6)
Although this expression appears cumbersome, it is required in this form for the general case where the conductivity and change in resistivity are nonsymmetric tensor quantities. For convenience, the tensor σ(c, eff) is defined as:
(6-7)
The Piezoresistivity, Domain Currents Interface solves a modified form of Equation 6-4 on the domain level:
while The Piezoresistivity, Boundary Currents Interface and The Piezoresistivity, Shell Interface solve a modified form of equation Equation 6-5 on the boundary level:
where σ(c, eff) is computed from Equation 6-7 and from the definition of Δρ given in Equation 6-2.