- 2.1.1 Maxwell's Equations
- 2.1.2 Poisson's Equation
- 2.1.3 Continuity Equations
- 2.1.4 Current Relations

- 2.1.5 The Semiconductor Equations

2.1 Classical Semiconductor Device Equations

Here, is the electric field, the magnetic field, the displacement vector, and the magnetic flux density vector. denotes the conduction current density, is the electric charge density, and is the partial derivative with respect to time.

Equation (2.1) expresses the generation of an electric field due to a changing magnetic field (Faraday's law of induction), (2.2) predicts the absence of magnetic monopoles (magnetic sources or sinks), (2.3) reflects how an electric current and the change in the electric field produce a magnetic field (Ampere-Maxwell law), and finally (2.4) correlates the creation of an electric field due to the presence of electric charges (Gauss' law).

We are using the Maxwell's equations to derive parts of the semiconductor device equations, namely the Poisson equation and the continuity equations.

2.1.2 Poisson's Equation

where is the permittivity tensor. This relation is valid for materials with time independent permittivity. As materials used in semiconductor devices normally do not show significant anisotropy of the permittivity, can be considered as a scalar quantity in device simulation. The total permittivity is obtained from the relative and the vacuum permittivity as . Table 2.1 gives an overview of relative permittivity constants for some materials commonly used in semiconductor devices.

As
for the stationary case can be expressed as a
gradient field of a scalar potential field

Substituting (2.5) and (2.6) in (2.4) we get

(2.7) |

The space charge density consists of

(2.10) |

Together (2.8) and (2.9) lead to the form of Poisson's
equation commonly used for semiconductor device simulation

(2.12) |

When we consider the charged impurities as time invariant and introduce a quantity to split up (2.13) into separate equations for electrons and holes, we get

The quantity gives the net recombination rate for electrons and holes. A positive value means recombination, a negative value means generation of carriers. Models for are presented in Section 2.3.

For low electric fields, the drift component of the electric current can be
expressed in terms of Ohm's law as

(2.16) |

(2.17) |

(2.18) |

(2.19) |

(2.20) |

(2.21) |

(2.22) |

(2.23) |

(2.26) |

(2.27) |

(2.28) |

(2.29) |

(2.30) |

This set of equations is widely used in numerical device simulators and provides only the basics for device simulation. In modern simulators they are accompanied by higher order current relation equations like the hydrodynamic, six-, or eight-moments models. There are models for the carrier mobility, the carrier generation and recombination (Section 2.3), for quantum effects like quantum mechanical tunneling (Section 5.3) or quantum confinement (Section 2.4.1) and of course for modeling of device degradation, as negative bias temperature instability (Chapter 6).

R. Entner: Modeling and Simulation of Negative Bias Temperature Instability