10.2.2 Mesh Size

The mesh size is critical because of the sharp resonances in the transmission probabilities. Figure 10.5 depicts the transmitted current density over the wave vector $k$ as calculated by the QTBM. The GaAs/AlGaAs RTD is in equilibrium, particles are injected from the left and the right with a Fermi-Dirac distribution. In the example shown the energy corresponding to the highest $k$-value is $0.6 \mathrm{eV}$.

Figure 10.5: Spikes in the transmitted current density resulting from sharp resonances

For the QTBM the $k$-mesh used for the injected particles is refined in the region of the resonance. The necessary resolution in $k$ to resolve the resonances is not in the numerically feasible domain for the finite difference Wigner function method. The needed number of points in $k$, $N_k$, is too high and the used equi-spaced mesh does not resolve the sharp peaks. In a comparison between WFM and QTBM the problem is aggravated as also the limit $L_{\mathrm{coh}} \rightarrow \infty$ has to be performed.

The used mesh sizes in the Wigner method are too small to ensure an accurate solution for coherent transport. The choice of a very crude mesh is sometimes justified by out of place physical arguments (``atomic monolayer spacing'') or even by a reference to the uncertainty relation. As the finite difference Wigner method is a dense discretization in $k$, the mesh sizes which are feasible in practice are only limited. Much of the work on Wigner simulation was done 10 to 15 years ago and small mesh sizes like this were used.

In defense of the Wigner method we note that scattering is usually included in the WFM model. In this case the transmission peaks are naturally broadened and meshing constraints are less stringent.