Discussion

7.6 Discussion

As demonstrated in great detail, the eNMP model successfully reproduces the features of the time constants extracted from TDDS measurements. These experiments reveal the behavior of single defects by monitoring the response of single defects to different stressing conditions. This has made it possible to identify the underlying physical process involved in charge trapping. Reisinger et al. [54] established the link between single defects and NBTI by showing that the NBTI degradation is primarily caused by a large number of individual trapping events. The capture and emission times of these events are characterized by a wide distribution. In the eNMP this can be explained by large variations in the adiabatic potentials of defects, as usual for defects in an amorphous host materials [20, 19]. Fig. 7.12 illustrates how single hole capture events sum up to smooth degradation curves obtained in NBTI measurements. It becomes clear that the increase in the number of traps leads to an averaging of the $ΔVth(t)$ curves and the steps of single events cannot be resolved anymore. The distribution of hole capture times determines the form of the stress curves while the corresponding emission times yields the recovery curves. Consequently, the field acceleration and temperature activation of the time constants may explain the field and temperature dependences of the NBTI degradation curves. But one should keep in mind that the permanent component of NBTI is not captured by TDDS and thus have not been accounted for in the eNMP model. As a result, a significant contribution to the overall NBTI degradation is not described by the hole capture and emission process.

Figure 7.12: Normalized degradation curves for a varying number of defects. With an increasing number of traps, the stepped curve becomes smoothed out and approaches the expectation value of $ΔVth(t)$ . The barriers for hole capture are assumed to be homogeneously distributed, resulting in a logarithmic time behavior. The step height due to one hole capture follow approximately an exponential dependence in agreement with [25].

In the TSM the hole capture is modeled by an MPE process, which relies on a concept similar to the charge transfer reactions in the eNMP model. In both cases, the defect has to overcome an energy barrier resulting from the intersection of two adiabatic potentials in the configuration coordinate diagram. While the height of this barrier is a random variable in the TSM, it is determined from the shape of the adiabatic potentials in the eNMP model. In the latter, the intersection point varies with the relative position of the adiabatic potentials and consequently with the oxide field. Together with the intermediate state $2′$ , which is involved in the hole capture process, the eNMP model can reproduce the time constants seen in TDDS. By contrast, a field enhancement factor in the TSM had to be phenomenologically introduced in order to capture the field dependence observed in the experimental data. For the aforementioned reasons, the eNMP model is viewed as an improvement in the description of the hole capture process.

Apart from these physical details, the higher-level picture of the hole capture process remains the same for both models. In the precursor configuration (state $1$ in both models), the defect features a trap level located far below the substrate valence band. Note that this level is referred to as $Et,1$ in the TSM or $Et$ in the eNMP model. If $Et,1$ / $Et$ is shifted upwards by the oxide field, the defect can capture a substrate hole. This is accompanied by a structural relaxation of the defect configuration and leads to a new trap level $E ′t$ located within or at least close to the substrate bandgap.

The TSM as well as the eNMP model can describe defects which show a field dependence in the emission times and the recovery, respectively. This behavior is linked to a hole emission process, which neutralizes the defect via the transition $T2→3$ in the TSM or $T2→1′$ in the eNMP. As mentioned before, the corresponding trap level is $Et,2$ in the TSM and $E′t$ in the eNMP model and lies within or close to the substrate bandgap. As a result, the occupation of $Et,2$ or $E′t$ is strongly dependent of small variations of the substrate Fermi level, also known as the ‘switching trap’ behavior of defects. Only from the neutral charge state (the state $3$ in the TSM and the state $1′$ in the eNMP model), the defect is allowed to return to its initial state $1$ by structural relaxation. In the state diagrams, the last step corresponds to the transition $T3→1$ in the TSM or $T1′→1$ in the eNMP model. It is important to note here that the hole emission times in both models are eventually controlled by the position of $Ef$ relative to $Et,2$ or $E′t$ . This effect is reflected in the field dependence of the ‘anomalous’ defects, on the one hand, and NBTI recovery, on the other hand. Consider that the transition $T2↔3$ in the TSM and $T2↔1′$ in the eNMP model are actually based on a different description of the hole capture and emission process. However, the corresponding transition barriers are assumed to be small in both models. Therefore, $T2↔3$ in the TSM and $T2↔1′$ in the eNMP model occur fast so that the occupancies of the involved states reach their equilibrium values, which are unaffected by the barrier heights. Insofar the different field dependences of both models do not enter the occupancies of the states and thus do not impact the model behavior.

Besides the ‘anomalous’ defects, the eNMP model also gives an explanation for ‘normal’ defects. They are characterized by the fact that the alternative pathway from the state $2$ back to $1$ is taken over the metastable state $2′$ . Thus the hole emission process is determined by the thermal transition $T2→2′$ resulting in field independent emission times as required for this kind of defects. But note that the TSM has no analog for the ‘normal’ defect behavior and thus must be viewed as an insufficient description of charge trapping in NBTI and TDDS.

In summary, it has been pointed that the physical picture behind of hole trapping in NBTI is the same for the TSM as well as for the eNMP model. Nevertheless, the eNMP model should be regarded as an improvement for the two reasons: First, it is extended by the metastable state $2′$ , which allows for the curvature in $τcap$ and the field independence of $τem$ . Second, the NMP formalism is expected to be a better description of the investigated charge transfer reactions than its simplified MPE variant used in the TSM. In contrast to the TSM, the eNMP model has been rigorously derived from one configuration coordinate diagram, which is regarded as the most complete description of a defect with respect to energy.


Used Models	TSM	eNMP


Stress	$T 1→2$	$T ′ 1→2→2$	NMP Transfer Reaction + Intermediate State
‘Normal Defects’	✗	$T2→2′→1$
‘Anomalous Defects’	$T2→3→1$	$T2→1′→1$	Field-Dependent Recovery

Table 7.2: Comparison between the TSM and the eNMP model.

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