2.4.2.1 The Drift-Diffusion Model

By multiplying (2.2) with the first two moments of the distribution function $\Phi_0=1$ and $\Phi_1=\hbar\mathbf{k}$, integration over $\mathbf{k}$ space, using a parabolic dispersion relation, and applying the macroscopic relaxation-time approximation (RTA) for the integral of the collision operator, the following equation system can be derived

$$\ensuremath{{\mathbf{\nabla}}}\cdot \ensuremath{{\mathbf{J_n}}} = \ensuremath {\mathrm{q}}R + \ensuremath {\mathrm{q}}\frac{\partial n}{\partial t}\ ,$$ (2.4)

$$\ensuremath{{\mathbf{\nabla}}}\cdot \ensuremath{{\mathbf{J_p}}} = - \ensuremath {\mathrm{q}}R - \ensuremath {\mathrm{q}}\frac{\partial p}{\partial t}\ ,$$ (2.5)

$$\ensuremath{{\mathbf{J_n}}} = \ensuremath {\mathrm{q}}n\mu_n \mathbf{E} + \ensuremath {\mathrm{q}}D_n \ensuremath{{\mathbf{\nabla}}}n\ ,$$ (2.6)

$$\ensuremath{{\mathbf{J_p}}} = \ensuremath {\mathrm{q}}p\mu_p \mathbf{E} - \ensuremath {\mathrm{q}}D_p \ensuremath{{\mathbf{\nabla}}}p\ .$$ (2.7)

In these equations $\mathbf{J}$ denotes the current density, $R$ the net recombination rate, $\mu$ the mobility, ${\mathbf{E}}$ the electric field, and $D$ the diffusion coefficient. Together with (2.1), these basic semiconductor equations form the drift-diffusion model which, due to its simplicity, is widely used for the simulation of semiconductor devices.

A. Gehring: Simulation of Tunneling in Semiconductor Devices