One of the reasons causing convergence problems [256,257]
is the exponential dependence of the carrier concentration on the electrostatic
potential,
. A small potential variation
causes large variation in the carrier concentration
However, we show that an inappropriate energy grid for the discretization of the transport equations can be another reason of convergence problems in quantum transport simulations. It is demonstrated that with adaptive energy grids the iterative solution can converge very fast and the simulation time can decrease considerably.
In Section 4.8.2 it was shown that with a shift of an equidistant grid the
calculated carrier concentration can change sharply. This sensitivity
resulting from the numerical error causes convergence problems in the
self-consistent loop. In all non-adaptive methods some fixed energy grid is
adopted. In successive iterations of the POISSON and transport equations, the
electrostatic potential changes and this in turn affects the relative distance
between resonance energies and the energy grid points. As a result, the
evaluated carrier concentration can vary sharply in one iteration step, which
affects the calculated electrostatic potential for the next iteration. For a
quantitative analysis one can assume that the shift of energy grid is due to
the variation of the electrostatic potential,
.
The convergence of the self-consistent loop using the adaptive and non-adaptive
methods is studied. With the non-adaptive method energy grid points are
used. For the adaptive method, relative errors of
and
are assumed. Fig. 4.16 shows the infinity norm of the
potential update after each iteration. With the adaptive method the norm of
the potential update decreases exponentially and finally reaches a limit which
depends on the error tolerance of the integration. With the non-adaptive
method the norm of the potential update oscillates and no convergence is
achieved. Fig. 4.17 shows the calculated carrier concentration due to
several confined states, based on four successive iterations of the
non-adaptive method. From the first to the second iteration the carrier concentration
changes very sharply. Therefore, at the first iteration one is close to the
highly sensitive region (see Fig. 4.12). From the second to the third
iteration the carrier concentration changes only little, which can be mapped to
the low sensitive region. From the third to the forth iteration the variation
is large, which implies that we are again close to the highly sensitive
region. This sequence continues and prevents convergence. To avoid this problem
a fine grid spacing can be used, which decreases the sensitivity in all
regions. As it was shown in Fig. 4.10 the non-adaptive method requires a
grid spacing smaller than
for accurate result.
By reducing
to
in the adaptive method the
self-consistent iteration yields more accurate results, but the number of
required energy grid points increases, which increases the simulation time of
each iteration. Fig. 4.18 shows the infinity norm of the potential
update versus CPU-time. A suitable criterion for the termination of the
self-consistent loop was found as
. If the
maximum potential update in the device is much smaller than
,
the carrier concentration will change only weakly during the next iteration.
For most of the simulations performed such a criterion was satisfied for
.
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M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors