(21) |

this is in the form,

we can identify the constants as

(23) |

and,

(24) |

Equation 22 can be written as

(25) |

which can be integrated between the mesh point and to give

(26) |

rearranging gives,

(27) |

(28) |

where the definition has been used, then using the Bernoulli relation

(29) |

one can reformulate the equation as

(30) |

Care has to be taken to evaluating the Bernoulli expression about zero, so the Taylor expansion is used. Note, picking the right number of terms has a significant impact on numerical noise in the solver.

(31) |

The argument to the Bernoulli expression is is given as

(32) |

A thermal driving term can also be added to the expression, and should be reformulated as an effective mass when a temperature gradient is present in the device, due to it's thermal dependence.

(33) |

An analogous procedure can be used to derive the the hole current flux which will give

(34) |

(35) |

The standard finite difference scheme can then be used to rewrite the above equations ready for a computer to solve.