As discussed in Section 6.3, (
)-curves can show resonant tunneling at low temperatures and for small drain voltages, caused by defects, dopants, impurities, oxide variability or any other fluctuation in the potential which can lead to confinement and
thus the formation of a quantum dot (QD). As can be seen in the
(
) curve in Fig. 6.9 recorded on SmartArray B using Tech. A with
at 4.2 K, there are devices which show an active RTN defect in the same region where the resonance occurs, see Fig. 7.15. This allows to study the interaction between the active RTN defect and the resonance caused by a QD.
Figure 7.15: Resonant tunneling measured at cryogenic temperatures on an nMOS of Tech. A with dimensions . For the two selected drain voltages
and
the impact of the resonance on an RTN signal characterized in the same voltage region is studied.
Between and
RTN traces with lengths of 10 s have been recorded using a step size of
. This was done for two drain voltage conditions
and
. The steps have been detected using Otsu’s method for every recorded trace and the distribution of the capture time
, the emission time
and the step heights
were extracted, as can be seen for a few examples in Fig. 7.16 for
with
. The extracted distributions allow to calculate the mean values of
,
and
for every
in the measured range.
Figure 7.16: An exemplary 10 s RTN trace recorded on Tech. A at with
. Using Otsu’s method, steps can be detected which allow to extract the distributions of
,
and
. Thus, a mean step height and mean capture and emission times for measurement conditions
can be computed.
The calculated mean capture and emission times across the characterized region can be seen in Fig. 7.17. With increasing
,
decreases while
is constant. This asymmetry between
and
is well known from various technologies and operational conditions and does not contribute to the resonance in the
(
) curve [114]. Also the exponential behavior of the time constants on the gate voltage was shown for various technologies [MJC4, 174, 114] under various measurement conditions. Contrary to previous
reports [224], the exponential dependence of the charge transition times is not affected by the resonance, which occurs in the same gate voltage region.
This exponential dependence of the charge transition times on the gate voltage agrees with the models derived in NMP theory which are discussed in detail Chapter 3. According to the classical limit of NMP theory, the charge transition rates depend exponentially on the classical transition barrier between neutral and charged state
of a trap. When applying to the device, the trap level is shifted. These shifts result in a proportional shift of the transition barrier and an exponential
-dependence. Describing the
-dependence with the quantum mechanical picture is more complex because it involves the vibrational wavefunctions. A shift in
leads to a change in the overlap of the vibrational wavefunctions resulting in a complex relation between the transition rate and
. However, according to a simplified idea of nuclear tunneling, a quantum mechanical tunneling rate also depends exponentially on the classical barrier. And this classical barrier shows a linear dependence on
, resulting in an exponential dependence on
. Additionally, Fig. 3.9 shows the dependence of the lineshape function on the energetic difference between the initial
and the final state. This energetic offset is proportional to
and looks qualitatively the same at cryogenic temperatures as for room temperature, which additionally indicates an exponential dependence of the transition rates on
, even at cryogenic temperatures.
Based on NMP theory, the transition rates and the corresponding transition times depend on the lineshape function (which depends on the relative positions of the trap levels) and on the carrier density of states . While the energy levels are not affected by a QD, the
would be dramatically different inside of a QD and therefore directly affect the charge transition times. Since this is not the case in the measured mean capture and emission times, it can be concluded that the defect is
located outside of the QD as indicated in Fig. 7.18.
Figure 7.18: Local disorders in the potential cause the formation of a quantum dot (QD) which is responsible for the resonance in the (
) curve. A trap can be located anywhere in the oxide. If it is located close to the QD, the changing charge carrier density is expected to have a strong impact on the trap. Charge transition times of a trap located far
away from the QD as indicated in this figure are not expected to be affected by the QD.
While the exponential dependence of the transition times on can be explained within NMP theory, the relation between the step heights
and the QD is more complex. As can be seen in Fig. 7.19 for
and
the step height
(blue) is clearly affected by the QD and is thus related to the resonance in the
(
) curve (red). Often,
is normalized to
, as can be seen in the central plots. These normalized step heights look similar to data in previous studies [240], however, a direct comparison is not possible because
does not increase monotonically with
. Using the initially measured
(
) curves,
can be mapped from the current domain to an equivalent threshold voltage
as can be seen in Fig. 7.19 (lower panels). This representation shows that
does not increase or decrease directly proportionally with
, which renders a simple explanation of the relation between
and
impossible.
Figure 7.19: The extracted step heights in the measured currents for both
and
(upper panels) are strongly impacted by the QD, which causes resonant tunneling (red lines). While this dependence is less clear when normalized to
as is done frequently in literature [240] (central panels) the strong coupling to the QD can be seen in the mapped equivalent voltage shift step height.
A strong impact of the QD on has been reported before in [224, 242]. In these studies a uniform potential fluctuation model was used for describing the dependence of
on
According to this simple model, RTN should vanish at the turning points where , which is in contradiction to our measurements. Therefore, a more complex model is needed to describe the relation between source-drain current and
. For this, a precise description of the device electrostatics and transport phenomena is needed, including confinement and the impact of single dopants or other possible sources of the resonance. It might even be
necessary to include specific percolation paths in the drain-source current to obtain a conclusive model of the dependence of
on
. Developing such a model, however, exceeds the scope of this thesis but might be investigated in future studies.