4.8.2 Non-adaptive Energy Grid

One can straightforwardly divide the integration domain into $N_\mathrm{E}$ equidistant intervals $\Delta E =(E_\mathrm{max}-E_\mathrm{min})/N_\mathrm{E}$. A disadvantage of this method is that the numerical error can not be pre-defined. This problem is more pronounced when the integrand is not smooth. To evaluate (4.60) numerically a trapezoidal rule and an equidistant grid spacing are used. The dependence of the accuracy on the following two parameters is studied, namely the grid spacing, $\Delta E$, and the relative distance between the peak and the nearest grid point, $\delta E$. These parameters are normalized as $\alpha =\Delta E/\Gamma$ and $\beta =\delta E/\Gamma$. The relative error in calculating the carrier concentration, $(n-\tilde{n}(\alpha))/n$, as a function of grid spacing is shown in Fig. 4.10. Here, $n$ is the analytically exact value of the carrier concentration (4.60) and $\tilde{n}$ refers to the numerically calculated carrier concentration as a function of $\alpha$ and $\beta$.

The variation of the calculated carrier concentration $(\tilde{n}(0)-\tilde{n}(\beta))/\tilde{n}(0)$ with respect to the shift of energy points is shown in Fig. 4.11. The reference $\beta =0$ implies that one of the grid points aligns with the peak of the resonance. The oscillatory behavior depends on the grid spacing. A shift equal to the grid spacing gives the same result. As a measure of the sensitivity of the calculated carrier concentration with respect to grid positions $\partial \tilde{n}/\tilde{n} \partial \beta$ is shown in Fig. 4.12. To reduce this sensitivity, a very fine grid spacing has to be adopted. This quantity is characteristic of the numerical error, and needs to be controlled to avoid convergence problem in the self-consistent iteration loop (see Section 4.9.2).

Figure 4.10: The relative error in evaluating the carrier concentration with respect to the grid spacing is shown. The inset shows the normalized LORENTZian shape of the density of states of a bound state. The peak of the resonance is shifted to the zero point. At E= $\pm \Gamma /2$ the function is half of its maximum. The solid line shows the exact function and the dashed curve shows the approximation of the function based on the Trapezoidal rule. The grid spacing is $\Delta E$ and the shift of energy grids from the reference point is $\delta E$. These parameters are normalized as $\alpha =\Delta E/\Gamma$ and $\beta =\delta E/\Gamma$. The reference $\beta =0$ implies that the one of the grid points aligns with the peak of the resonance. The parameters in this figure are $\alpha =1/3$ and $\beta =0$.

Figure 4.11: The relative variation of the calculated carrier concentration with respect to the normalized position $\beta$ of energy grid points.

Figure 4.12: The relative sensitivity of the calculated carrier concentration with respect to the position of energy grid points. This term originates from the numerical error in the evaluation of the carrier concentration. For coarse grid spacing $\alpha >1$, this quantity increases considerably.

In summary, the accuracy of the non-adaptive method strongly depends on the grid spacing and the position of grid points. If the grid spacing is sufficiently fine, $\alpha<1$, the numerical error is small, but it increases considerably for coarser grid spacing, $\alpha >1$. For accurate results a grid spacing smaller than $\Gamma$ has to be employed. For example, to resolve a resonance of $\Gamma\approx \mathrm{1~\mu eV}$ width in an energy range of $\mathrm{1~eV}$ more than $\mathrm{10^{6}}$ energy grid points are required, which would severely increase the computational cost. For even narrower resonances, (eg. $\Gamma\approx \mathrm{1~n eV}$), an equidistant grid is no longer feasible. To avoid these problems an adaptive method needs to be employed. M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors