B. Driving Force Discretization

TO IMPLEMENT the discretization scheme described in Chapter 3 into the device simulator MINIMOS-NT an expression for the driving force was required. The driving force is defined by

(B.1) |

To obtain the discrete driving force the discretized current density eqns. (3.63) and (3.64)

(B.2) | ||

(B.3) |

must therefore be divided in some way by the electron concentration . Thus, an average carrier concentration is introduced via the following definition

By comparing the coefficients of eqn. (B.4) with those from eqn. (B.2)

(B.5) | ||

(B.6) |

and using the identity

(B.7) |

the new argument of the BERNOULLI function can be calculated

(B.8) | ||

(B.9) |

and the average carrier concentration is finally found to be

(B.10) |

Applying the identity

(B.11) |

to eqn. (B.4) yields

(B.12) |

After inserting from eqn. (2.188)

(B.13) |

the expression for the discretized driving force can easily be obtained

(B.14) |

The consistency of the discretization can be checked by calculating the driving force in the limit of

(B.15) | ||

(B.16) |

where the abbreviations for and have been expanded. Using the total derivative yields

(B.17) | ||

(B.18) | ||

(B.19) |

which is the one-dimensional projection of the driving force

(B.20) |