8.4.2 Drift Term

The discretization of the Wigner potential operator has already been implicitly discussed in the previous section on conservation of mass. We have shown, that one has to use an equispaced product grid to obtain conservation of mass in the discrete case.

Another undesirable feature of the potential operator is that we need the potential outside the simulation domain in the calculation of $V_{\mathrm{w}}(x,k)$ from 8.9, assuming that the integration domain is symmetric around $x$. Usually the potential is extended outside the domain with the values at the boundary which are fixed by the applied bias. With this we get that the integrand $V(x + y) - V(x - y)$ in the calculation of $V_{\mathrm{w}}(x,\cdot)$ is equal to the applied bias for $y \rightarrow \infty$ and, in general $\neq 0$. Consequently $V_{\mathrm{w}}(x,k)$ has a singularity at $k$ = 0. As $V_{\mathrm{w}}$ is odd, it is natural to set $V_{\mathrm{w}}(x,0) = 0$. The singularity can be calculated analytically, which reduces to the Fourier transform of a Heaviside function.