5.5 Contact Model

The generalized boundary condition is the constitutive relation for the contact potential $\psi_{C}^{}$ and reads

$$\psi_{C} \alpha \psi_{C}^{} \beta \gamma \delta$$ (5.20)

where Q_C is the contact charge and I_C = I_nC + I_pC + $\partial$Q_C/$\partial$t the contact current. It should be noted that all these quantities are solution variables, which simplifies the formulation of the contact models.

For the special case of a traditional voltage controlled contact $\alpha$ = 1, $\beta$ = $\gamma$ = 0, and $\delta$ = V₀ and (5.20) degenerates to

$$\psi_{C} \psi_{C}^{}$$ (5.21)

Modeling a series contact resistance using $\beta$ = R_C one gets

$$\psi_{C} \psi_{C}^{}$$ (5.22)

For a current controlled contact $\beta$ = 1, $\alpha$ = $\gamma$ = 0, and $\delta$ = I₀ and (5.20) degenerates to

$$\psi_{C}$$ (5.23)

For a charge controlled contact $\alpha$ = $\beta$ = 0, $\gamma$ = 1, and $\delta$ = Q₀ and (5.20) degenerates to

$$\psi_{C}$$ (5.24)

Using different units for the coefficients $\alpha$, $\beta$, $\gamma$, and $\delta$ (5.20) can be interpreted in different ways. These include a parallel conductance and capacitance to ground for a current controlled contact, and a series resistance and parallel capacitance to ground for a voltage controlled contact.

As (5.20) is normally not diagonal-dominant it is pre-eliminated. In (5.23) the main-diagonal entry is zero, therefore f_IC must be eliminated first in order to get the derivatives with respect to $\psi_{C}^{}$.