3.1.1 The Basic Semiconductor Equations

The basic equations solved in a device simulator are the Poisson equation and the continuity equations for electrons and holes.

$$\mathrm{div}( \varepsilon_{\mathrm{}}\cdot\mathrm{grad} \psi) = \mathrm{q}\cdot (n - p - C)$$ (3.1)

$$\mathrm{div} \mathbf{J}_n = \mathrm{q}\cdot\left(R+\frac{\partial n}{\partial t}\right)$$ (3.2)

$$\mathrm{div} \mathbf{J}_p = -\mathrm{q}\cdot\left(R+\frac{\partial p}{\partial t}\right)$$ (3.3)

The unknown quantities of this equation system are the electrostatic potential, $\psi$, and the electron and hole concentrations, $n$ and $p$, respectively. $C$ denotes the net concentration of the ionized dopants and other charged defects, $\varepsilon_{\mathrm{}}$ is the dielectric permittivity of the semiconductor, and $R$ is the net recombination rate.