A. Re-expressing X

(7.1) |

The energy dispersion then takes the following form:

(7.2) |

Setting the determinant of the dimensionless Hamiltonian to zero allows to express as a function of energy :

Re-expressing the fourth order equation (7.3) as a second order equation by

(7.4) |

we find the solution,

(7.5) |

This formulation preserves all four solutions for . Embracing two sets of values in two separate equations leads to the following expressions:

(7.6) |

Using the following identities:

(7.7) |

leads to the desired expressions and .

The transformation to dimensionless units of the corresponding expression for is:

(7.9) |

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