For a distribution function where is the position, is the wave vector and is the time the Boltzmann equation for parabolic energy bands reads

(2.1) |

Here is the given energy band diagram, is the electrostatic potential and is the carrier charge (negative for electrons). The independent variables are , , .

We define the group velocity

(2.2) |

For now we restrict ourselves to parabolic bands. Then

(2.3) |

with effective mass and

(2.4) |

The momentum is defined as

(2.5) |

Using the group velocity the Boltzmann equation becomes

(2.6) |

where we introduced the electrical field .

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R. Kosik: Numerical Challenges on the Road to NanoTCAD