D. Relation Between Charge Density and Potential Drop in the MOSFET Channel

(7.29) |

Since, the overall charge has to be zero, must be fullfilled. Substituting by the relation before the following expression is derived:

(7.30) |

Now, grouping the expressions containing and :

(7.31) |

followed by the relation and the assumption that , (p-doped), we get:

The equation connecting the potential and the doping can now be deduced. The identity:

is introduced in order to rewrite (7.32) as a differential equation of first order:

(7.34) |

The boundary conditions are set to at the surface, and 0 for :

(7.35) |

After the integration the following relation is gained:

(7.36) |

Since the electric field is related to the potential via , the derived expression describes the dependence of the electric field on the surface potential:

(7.37) |

Applying Gauß's law:

(7.38) |

leads to the desired formultaion, connecting the surface charge density with the surface potential :

T. Windbacher: Engineering Gate Stacks for Field-Effect Transistors