3.5.5 Comparison

Fig. 3.12 shows the transmission coefficient for the described methods for a triangular energy barrier (left) and a two-step non-linear energy barrier (right). The inset shows the energy barrier and the values of $\vert\Psi\vert^2$ for an energy of 2.8eV on a logarithmic scale. The dotted lines refer to the constant-potential transfer-matrix method. In the left figure the numerical instability of the transfer-matrix method leads to an increasing transmission coefficient for energies below 1eV. These numerical problems occur for both the constant-potential and the linear-potential approaches.

The GUNDLACH and analytical WKB methods deliver similar results for the triangular barrier. For the stacked dielectric shown in the right figure, the analytical WKB and GUNDLACH methods cannot be used. The numerical WKB, transfer-matrix, and QTB methods deliver similar results, however, the WKB method does not resolve oscillations in the transmission coefficient.

It can be concluded that for a single-layer dielectric, the analytical WKB method yields reasonable accuracy as compared to the other, computationally more expensive methods. For stacked dielectrics, however, only the numerical WKB, transfer-matrix, or QTB methods can be used in the first place. Since transfer-matrix based methods exhibit problems regarding numerical stability, only the QTBM and the numerical WKB methods remain. Since the numerical WKB method also needs a numerical integration, its advantage in terms of computational effort is not high enough to rule out the QTBM. Furthermore, if resonance effects -- such as in dielectrics with quantum wells, see Section 5.2.2.3 -- have to be taken into account, the QTBM remains as the method of choice for a reliable transmission coefficient estimation.

Figure 3.12: The transmission coefficient using different methods for a dielectric consisting of a single layer (left) and for a dielectric consisting of two layers (right). The shape of the energy barrier and the wave function at 2.8 eV is shown in the inset.

A. Gehring: Simulation of Tunneling in Semiconductor Devices