A. Derivation of the Impact-Ionization Integral

In the following the derivation of the impact-ionization integral from Section 5.1.2 will be given. First, the differntial equation

has to be solved. Using the simplified notation

with and one can derive the homogenous solution

(A.3) |

where C is the constant of integration. Applying the method of variation of constants, the ansatz of the particular solution is derived from the homogenous solution, using This ansatz function can be differentiated to

Comparison of the coefficients between equations (A.2) and (A.4) gives

(A.5) |

and evaluates to

(A.6) |

This leads to the particular solution

(A.7) |

and together with the homogenous solution to the solution of (A.2)

Using (A.8) our initial problem (A.1) solves together with the boundary conditions and to

To simplify the solution (A.9) the following relationship can be used. Considering

(A.10) |

the following simplification can be performed:

Making the relation (A.11) applicable, (A.9) can be rewritten at the position as

(A.12) |

and simplified to

(A.13) |

O. Triebl: Reliability Issues in High-Voltage Semiconductor Devices