Charge Trapping and Variability in CMOS Technologiesat Cryogenic Temperatures

7.6 RTN in the Resonant Tunneling Region

As discussed in Section 6.3, (math image)((math image))-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 (math image)((math image)) curve in Fig. 6.9 recorded on SmartArray B using Tech. A with \( W\times L = \SI {100}{\nano \meter }\times \SI {70}{\nano \meter } \) 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 \( W\times L = \SI {100}{\nano \meter }\times \SI {70}{\nano \meter } \). For the two selected drain voltages \( \vd =\SI {10}{\milli \volt } \) and \( \vd =\SI {25}{\milli \volt } \) the impact of the resonance on an RTN signal characterized in the same voltage region is studied.

Between \( \vg =\SI {0.3}{\volt } \) and \( \vg =\SI {0.4}{\volt } \) RTN traces with lengths of 10 s have been recorded using a step size of \( \Delta \vg =\SI {1}{\milli \volt } \). This was done for two drain voltage conditions \( \vd =\SI {10}{\milli \volt } \) and \( \vd =\SI {25}{\milli \volt } \). The steps have been detected using Otsu’s method for every recorded trace and the distribution of the capture time (math image), the emission time (math image) and the step heights \( \eta \) were extracted, as can be seen for a few examples in Fig. 7.16 for \( \vg =\SI {0.366}{\volt } \) with \( \vd =\SI {25}{\milli \volt } \). The extracted distributions allow to calculate the mean values of (math image), (math image) and \( \eta \) for every (math image) in the measured range.


Figure 7.16: An exemplary 10 s RTN trace recorded on Tech. A at \( \vg =\SI {0.336}{\volt } \) with \( \vd =\SI {25}{\milli \volt } \). Using Otsu’s method, steps can be detected which allow to extract the distributions of (math image) , (math image) and \( \eta    \). Thus, a mean step height and mean capture and emission times for measurement conditions \( (\vg ,\vd ,T) \) can be computed.

The calculated mean capture and emission times across the characterized (math image) region can be seen in Fig. 7.17. With increasing (math image), (math image) decreases while (math image) is constant. This asymmetry between (math image) and (math image) is well known from various technologies and operational conditions and does not contribute to the resonance in the (math image)((math image)) 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.

(-tikz- diagram)

Figure 7.17: The charge transition times of an RTN signal in the resonant tunneling region show an exponential dependence on (math image) for both \( \vd =\SI {10}{\milli \volt } \) and \( \vd =\SI {25}{\milli \volt } \) (left). This can be explained within NMP theory by the linear dependence of the classical barrier on (math image) as can be seen in the right figure.

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 (math image) to the device, the trap level is shifted. These shifts result in a proportional shift of the transition barrier and an exponential (math image)-dependence. Describing the (math image)-dependence with the quantum mechanical picture is more complex because it involves the vibrational wavefunctions. A shift in (math image) leads to a change in the overlap of the vibrational wavefunctions resulting in a complex relation between the transition rate and (math image). 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 (math image), resulting in an exponential dependence on (math image). 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 (math image) and looks qualitatively the same at cryogenic temperatures as for room temperature, which additionally indicates an exponential dependence of the transition rates on (math image), 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 \( D_{n} \). While the energy levels are not affected by a QD, the \( D_{n} \) 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.

(-tikz- diagram)

Figure 7.18: Local disorders in the potential cause the formation of a quantum dot (QD) which is responsible for the resonance in the (math image)((math image)) 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 (math image) can be explained within NMP theory, the relation between the step heights \( \eta \) and the QD is more complex. As can be seen in Fig. 7.19 for \( \vd =\SI {10}{\milli \volt } \) and \( \vd =\SI {25}{\milli \volt } \) the step height \( \eta \) (blue) is clearly affected by the QD and is thus related to the resonance in the (math image)((math image)) curve (red). Often, \( \eta \) is normalized to (math image), 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 (math image) does not increase monotonically with (math image). Using the initially measured (math image)((math image)) curves, \( \eta \) can be mapped from the current domain to an equivalent threshold voltage (math image) as can be seen in Fig. 7.19 (lower panels). This representation shows that \( \eta \) does not increase or decrease directly proportionally with (math image), which renders a simple explanation of the relation between \( \eta \) and \( \vg \) impossible.


Figure 7.19: The extracted step heights \( \eta    \) in the measured currents for both \( \vd =\SI {10}{\milli \volt } \) and \( \vd =\SI {25}{\milli \volt } \) (upper panels) are strongly impacted by the QD, which causes resonant tunneling (red lines). While this dependence is less clear when normalized to (math image) 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 \( \eta \) has been reported before in [224, 242]. In these studies a uniform potential fluctuation model was used for describing the dependence of \( \eta \) on (math image)

(7.17) \{begin}{align} \eta = \frac {\partial \id }{\partial \phi }\Delta \phi =\frac {\partial \id }{\partial \vg }\Delta \vg =\gm \Delta \vg . \{end}{align}

According to this simple model, RTN should vanish at the turning points where \( \gm =\SI {0}{\siemens } \), which is in contradiction to our measurements. Therefore, a more complex model is needed to describe the relation between source-drain current and \( \eta \). 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 \( \eta \) on (math image). Developing such a model, however, exceeds the scope of this thesis but might be investigated in future studies.