3.1.1 Basic Semiconductor Equations

The basic semiconductor equations which are derived from Maxwell's equations determine a relation among the electric and magnetic field, the space charge density and the current density given by [104]

$$\mathrm{rot}\,{{\mathbf{E}}} = - \partial_{t}{{\mathbf{B}}} \ ,$$ (3.3)

$$\mathrm{div}\,{{\mathbf{B}}} = 0 \ ,$$ (3.4)

$$\mathrm{rot}\,{{\mathbf{H}}} = {\mathbf{J}} + \partial_{t}{{\mathbf{D}}} \ ,$$ (3.5)

$$\mathrm{div}\,{{\mathbf{D}}} = \rho \ ,$$ (3.6)

where ${\mathbf{E}}$ and ${\mathbf{H}}$ are the electric and magnetic fields, and ${\mathbf{B}}$ and ${\mathbf{D}}$ are the induction and displacement vectors, respectively. Furthermore, $\rho$ is the space charge density, and ${\mathbf{J}}$ the conduction current density.

The expressions (3.3) through (3.6) are linked by the relations

$${\mathbf{D}} = \varepsilon \cdot{\mathbf{E}} \ ,$$ (3.7)

$${\mathbf{B}} = \mu \cdot {\mathbf{H}} \ ,$$ (3.8)

where $\varepsilon$ and $\mu$ are the permittivity and permeability tensors, respectively. Equations (3.7) and (3.8) are valid only in materials where no piezoelectric or ferroelectric phenomena occur. The frequency dependence of $\varepsilon$ and $\mu$ can be neglected.

Combining the fourth MAXWELL equation (3.6) with (3.7) and (3.8) to make it applicable for semiconductor problems, and substituting the space charge density $\rho$ by the algebraic sum of the charge carrier densities and the ionized impurity concentrations

$$\rho = {\mathrm{q}}\cdot(n-p+N^+_D-N^-_A),$$ (3.9)

one arrives at the POISSON equation for anisotropic materials, determining the electrostatic potential $\psi$, given by

$$\mathrm{div}(\varepsilon\cdot\mathrm{grad}\,\psi) = {\mathrm{q}}\cdot(n-p+N^+_D-N^-_A),$$ (3.10)

where ${\mathrm{q}}$ is the elementary charge, $n$($p$) is the charge carrier density of electrons (holes), and $N^+_\mathrm{D}$ ( $N^-_\mathrm{A}$) is the concentration of ionized donors (acceptors). The permittivity tensor $\varepsilon$ has the same form as the representative tensor $\sigma$ in (3.2).

The continuity equations (3.11) and (3.12) are the conservation laws for the carriers. They are derived from the third MAXWELL equation (3.5) for the flow of electrons and holes, and maintain their usual form in anisotropic materials as well.

$$\mathrm{div}\,{\mathbf{J}}_n = {\mathrm{q}}\cdot\left(R+\frac{\partial n}{\partial t}\right)$$ (3.11)

$$\mathrm{div}\,{\mathbf{J}}_p = -{\mathrm{q}}\cdot\left(R +\frac{\partial p}{\partial t}\right)$$ (3.12)

The quantity R describes the net generation or recombination rate of electrons and holes and is modeled explicitly in Section 3.5.

T. Ayalew: SiC Semiconductor Devices Technology, Modeling, and Simulation