3.6.3.2 The Quasi-Classical Formula

The calculation of the life times using the approaches shown so far is cumbersome and error-prone, since a precise value for the FWHM in regions where different QBS overlap is difficult to obtain. As an approximation the life time of a QBS can be computed from the quasi-classical formula [176]

$$\ensuremath{\tau_{\mathrm{q}}}= \frac{1}{TC(\ensuremath{{\mathcal... ...i}}-\ensuremath {{\mathcal{E}}_\mathrm{c}}(x)}} \,\ensuremath {\mathrm{d}}x \ ,$$ (3.116)

where $\ensuremath{{\mathcal{E}}_\mathrm{i}}$ is the resonance energy of the respective bound state and $x_i$ the classical turning point for this energy. The transmission coefficient $TC(\ensuremath{{\mathcal{E}}_\mathrm{i}})$ can be calculated by the transfer-matrix method or any other method that solves SCHRÖDINGER's equation.

A. Gehring: Simulation of Tunneling in Semiconductor Devices