4.3 Quantization in UTB Films for Primed Subbands

Starting with the transformation to dimensionless form according to:

(4.28) |

and some further rearrangements

(4.29) |

the eigenvalue problem takes the following form:

Setting the determinant of (4.30) to zero allows to obatin as a function of

(4.31) |

Like before for the unprimed subbands, the obtained fourth order equation,

(4.32) |

can be reformulated into two second order equations:

The identities

(4.34) |

allow to introduce a dependence in (4.33) and formulate the problem as and vice versa

So as a function of is described by the following equation:

Substituting into (4.36) results in:

(4.37) |

and enables the derivation of as a function of

(4.38) |

In order to obtain the minus branch of (4.35) is used:

After analogous treatment of (4.39) the relation of as a function of is derived:

(4.40) |

The corresponding scaled energy dispersion relation is given by:

(4.41) |

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