3.6.3.3 The Eigenvalues of the Non-HERMITian HAMILTONian

For open-boundary conditions, the system is described by a HAMILTONian which is not HERMITian and admits complex eigenvalues. The most straightforward way to calculate the life times is to directly find the complex eigenvalues of the system HAMILTONian. This, however, is not easily possible because the eigenvalue problem is nonlinear: The matrix elements depend on the eigenvalue [177]. The numerical implementation of this method will be described in Section 4.3.3.

The complex eigenvalues have been used to calculate the life times of the structure shown in the left part of Fig. 3.14. The complex energies and life times found are shown in Table 3.2. The values perfectly agree with the values found using the method based on the evaluation of the reflection-coefficient.

The life times of the first and second QBS have been evaluated as a function of the gate bias and the thickness of the dielectric layer as shown in the left part of Fig. 3.15. The life time decreases with increasing gate bias which is due to the higher penetrability of the energy barrier. The results of the gate current density (3.92) is shown in the right part of Fig. 3.15, where the TSU-ESAKI tunneling current was not considered.

This method, however, is by far the most computationally demanding one and it has not been implemented in MINIMOS-NT since problems regarding the stability of the underlying algorithms have been observed.

Table 3.2: Eigenvalues found by using a resonance-finding algorithm based on the determinant of the open-boundary HAMILTONian.

$\ensuremath{{\mathcal{E}}_\mathrm{i}}$	$\ensuremath {{\mathcal{E}}_\mathrm{re}}$	$\ensuremath {{\mathcal{E}}_\mathrm{im}}$	$\ensuremath{\tau_{\mathrm{q}}}$
	[eV]	[eV]	[s]
1	0.2695	$1.503 \times 10^{-20}$	$4.376 \times 10^{4}$
2	0.4695	$1.830 \times 10^{-19}$	$3.594 \times 10^{3}$
3	0.6256	$5.285 \times 10^{-15}$	$1.244 \times 10^{-1}$
4	0.7549	$2.794 \times 10^{-11}$	$2.354 \times 10^{-4}$
5	0.8629	$4.231 \times 10^{-8}$	$1.555 \times 10^{-8}$
6	0.9503	$2.005 \times 10^{-5}$	$3.281 \times 10^{-11}$

Figure 3.15: The life time of the first and second QBS for different gate dielectric thicknesses and gate voltages (left) and the resulting gate current density considering the first three quasi-bound states (right).

A. Gehring: Simulation of Tunneling in Semiconductor Devices