2.4.1 Hierarchy of Semiconductor Device Simulation Models

Models of increasing sophistication can be derived for the simulation of charge transport in semiconductor devices, as shown in Fig. 2.11. The most important equation, which all models have in common, is POISSON's equation to determine the electrostatic potential

$$\ensuremath{{\mathbf{\nabla}}}\cdot (\kappa \ensuremath{{\mathbf{\nabla}}}\phi) = \ensuremath {\mathrm{q}}(n - p - C) \ ,$$ (2.1)

where $\phi$ denotes the electrostatic potential, $\kappa$ the dielectric permittivity, $n$ and $p$ the electron and hole concentration, and $C = \ensuremath {N_\mathrm{D}}- \ensuremath {N_\mathrm{A}}$ the net concentration of impurities. The transport of carriers is described by the BOLTZMANN transport equation (BTE) which is a semi-classical formulation of charge transport.

Quantum-mechanical effects are described by the SCHRÖDINGER equation. To incorporate quantum-mechanical effects into classical device simulation, BOLTZMANN's transport equation can be coupled to the SCHRÖDINGER equation, or the WIGNER equation can be applied [39,40,41,42]. Transport models based on solutions of the BOLTZMANN transport equation can be derived using the method of moments [43,44,45] which yields the drift-diffusion model [46], the energy-transport or hydrodynamic model [47], or higher-order transport models [48]. Furthermore, an approximate solution can be obtained by expressing the distribution function as a series expansion which leads to the spherical harmonics approach [49,50,51,52,53].

A. Gehring: Simulation of Tunneling in Semiconductor Devices