4.8.3 Adaptive Energy Grid

There is a variety of methods available for numerical adaptive integration [247]. Adaptive strategies divide the integration interval into sub-intervals and, typically, employ a progressive formula in each sub-interval with some fixed upper limit on the number of points. If the required accuracy is not achieved by the progressive formula, the sub-interval is bisected and a similar procedure carried out on each half. This sub-division process is carried out recursively until the desired accuracy is achieved. An obvious way to obtain an error estimate is based on the comparison between two quadrature approximations [248]. However, due to the dependence of such procedures on the underlying integration formulae, this method may not be reliable [249]. Error estimation with sequences of null rules has been proposed as a simple solution [250]. In adaptive quadrature algorithms the error estimate governs the decision on whether to accept the current approximation and terminate or to continue. Therefore, both the efficiency and the reliability depend on the error estimation algorithm. The decision to further subdivide a region may be based on either local or global information. Local information refers only to the region being currently processed, while global information refers obviously to data concerning all regions. Integration programs based on global subdivision strategies are more efficient and reliable [251].

In this work a global error estimator based on the null rules method has been employed [249]. The efficiency of this method is studied for the LORENTZian function (4.60). Figure 4.13-a shows the number of required energy grid points for an interval $\mathrm{[-1,1]~eV}$ versus the relative error $\epsilon$ of the integration. The required number of energy grid points versus the width of the resonance, $\Gamma$, is shown Fig. 4.13-b. To resolve a very narrow resonance ( $\Gamma\approx10^{-9}~\mathrm{eV}$) with a very high accuracy ( $\epsilon=10^{-6}$) only a few hundred grid points ( $N_\mathrm{E}\approx\mathrm{500}$) are required.

Figure 4.13: a) shows the number of required energy grid points versus the maximum desired relative error, $\epsilon$, in the adaptive integration method. b) shows the number of required energy grid points versus the width of the resonance, $\Gamma$. The LORENTZian function (4.60) is used as a reference.

Figure 4.14: The left figure shows the normalized spectrum of the carrier concentration in a Schottky type CNT-FET. The right figure shows the spectrum of the carrier concentration in the middle of the device for the energy range shown by the arrow. The results achieved from the adaptive and non-adaptive method are compared. With the aid of the adaptive method narrow and close resonances are resolved with a total number of $N_\textrm {E}\approx$1000 energy grid points, whereas the non-adaptive method misses some resonances with the same number of energy grid points.

Figure 4.14 shows the normalized spectrum of the carrier concentration in a Schottky type CNT-FET (see Section 2.8.2). The length of the device is $\mathrm{50~nm}$. Energy barriers at the metal-CNT interfaces cause longitudinal confinement in the tube. Since the device is quite long, the spacing between confined states is very small. In CNTs the electron-phonon interaction is rather weak and the confining SCHOTTKY barriers are thick, such that resonances are only weakly broadened, (see Chapter 5). Due to phonon absorption and emission processes there will be more resonances compared to the ballistic case. In this case, if a non-adaptive method is employed the numerical error in the calculation of the carrier concentration can be large. The right part of Fig. 4.14 compares the results achieved from the adaptive and non-adaptive methods. The relative error in the electron density of the non-adaptive method reaches up to $53\%$ in the middle of the device.

In [252] the resonant states have been determined by an eigenvalue solver for finding the roots of the characteristic equation. However, this method has several drawbacks. Due to the non-linearity introduced by the self-energies, a non-linear eigenvalue solver has to be employed. Usually non-linear solvers are based on NEWTON's method. Using a non-linear solver for each iteration can increase the simulation time severely and introduce additional convergence problems. For example, most solvers fail to find narrow resonances located closely to each other. The output of the solver is the energy position and the width of the resonance, but not any information about the shape of the resonance. In general the shape of resonances deviates from the ideal LORENTZian shape. The grid has to be allocated based on an initial guess. This implies that the accuracy of the calculated carrier concentration can not be predefined and strongly depends on the how close the initial guess is to the actual solution. With the adaptive method the discussed problems do not occur.

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors