So far, only qualitative statements about the defect behavior could be made based on the switching levels. In order to make quantitative predictions, the ETM must be generalized in a way to account for the levels shift. Recall that the conventional concept of the ETM is based on the assumption that the energy levels for tunneling ‘into’ and ‘out of’ a defect coincide. This is only the case for unrealistic defects which do not deform after a tunneling event. But as proven in the previous Section 5.1, defects do undergo structural relaxation and therefore feature two switching levels, say for hole capture and for hole emission for instance, which can even be separated by some electron Volts (see Fig. 5.9). To be precise, the switching levels and actually originate from the one defect orbital and thus must be correctly interpreted as one trap level, which shifts after each charging or discharging event. For the trapping kinetics, this means that only one of these levels can be present in the band diagram at a time. For instance, when the defect in Fig. 5.10 is in its neutral charge state, it has a trap level for hole capture while its corresponding trap level remains inactive for the time being. If a hole capture process takes place, the level vanishes and thus the level appears. Based on the considerations above, the ETM must be regarded as a special case of the level shift model (LSM) but with a negligible defect relaxation. Consequently, the formulation of the ETM must be modified in order to account for the level shift. Thus equation (4.6) is rewritten as
where the rates are defined as
and denote the two charge states involved in the tunneling process and the trap levels and corresponds to the switching traps introduced in Section 2.3. The above rate equation is reminiscent of the ETM presented in the previous Section 2.5.2. The peculiarity of the LSM is that the particular terms on the right-hand side of equation (5.1) must be evaluated for different energies, namely or , depending on the charge state of the defect before the tunneling transition occurs. For instance, the positively charged defect of Fig. 5.9 has a trap level , which must be applied for calculation of the electron capture rate (see Fig. 5.10). By contrast, the neutralized defect features a trap level used for the hole capture rate . The calculation of the corresponding time constants is illustrated in Fig. 5.10. It is important to note here that the expressions for , , , and remain the ‘same’ as in the ETM and only change in the energy they are evaluated for. This is due to the fact that the tunneling mechanism itself is not affected by the structural relaxation. Thus, analogously to the ETM, the tunneling process can be described by the tunneling rates (2.45) and (2.46) of the ETM and reasonably approximated by (5.2) and (5.3).
In the following, a new quantity, referred to as the demarcation energy1, will be introduced. It determines equilibrium occupancy of the defects and is defined by the condition
While the equilibrium occupation of the defects is given by the demarcation energy, the trapping dynamics directly follow from the electron and hole capture time constants, whose dependence on the Fermi level and the trap depth will be discussed in the following. The ‘interesting’ instance is when the Fermi level is situated inbetween the levels and (cf. Fig. 5.10). Using Boltzmann statistics (5.7) and (5.8) and approximative WKB factor (5.9), the capture time constants can be estimated by
In this section, the LSM will be employed to investigate the impact of the level shift on the trapping dynamics in NBTI experiments. Based on the NBTI checklist established in Section 1.4, it will be tested whether this model is capable of reproducing the NBTI degradation seen in experiments. The following simulations are carried out on a pMOSFET () with a strongly doped p-poly gate (). The thickness of the oxide layer has been chosen to be for demonstration purposes. Thereby, the traps can be homogeneously spread within the dielectric but are still sufficiently separated () from the poly interface in order to be able to neglect trapping from the gate. Furthermore, this wide range of trap depths ensures a large distribution of capture and emission times over 14 decades in time (cf. Fig. 5.11). The energy levels of the traps have been assumed to be uniformly distributed with the and the levels being uncorrelated and thus independently calculated using a random number generator. The operation temperature is set to and thus lies in the middle of the range relevant for NBTI. It is noted that only charge injection from the substrate is accounted for in the following simulations for simplicity.
In the following, the basic properties of the LSM will be discussed on the basis of a simple showcase. Therefore, this model is evaluated for a type of defect whose trap level has a wide distribution below while the counterpart is sharply peaked slightly above (see Fig. 5.12). At the beginning of the stress phase nearly all defects are neutral and thus occupied by one electron. In this state, the traps are characterized by the hole capture levels located below the substrate valence band. The corresponding electron capture levels lie above the substrate conduction band but are inactive for the time being. During the stress phase, the substrate holes must be thermally excited to the defect level in order that a tunneling process can occur. In equation (5.6) their energy-dependent concentration is linked to the factor , which decays exponentially with decreasing energies assuming Boltzmann statistics. This decay would lead to a temporal filling of traps from energetically higher towards lower traps in the band diagram. Furthermore, the tunneling of electrons and holes has an additional trap depth dependence, which is reflected in the WKB factor of equation (5.6). Analogously to the ETM, this would cause a temporal filling of traps starting from close to the interface and continuing deep into the oxide. As demonstrated in Fig. 5.13, the superposition of both effects results in a tunneling hole front which proceeds from high defect levels close to the interface towards lower ones deep in the oxide. The resulting time evolution of the trap occupancies is visualized in Fig. 5.12.
The temporal filling is also reflected in the occupancies of the demarcation energies, shown in Fig. 5.14. As already mentioned before, only defects located above can participate in hole capture. As a consequence, the temporal filling of traps does not proceed below , which thus marks a border to the tunneling hole front.
After the stress phase, a large part of the hole capture levels has disappeared and is replaced by their corresponding electron capture levels . The latter are assumed to be concentrated in a small trap band slightly above the substrate conduction band. During the recovery phase, electrons in the substrate conduction band must be thermally excited up to the level where the trap depth-dependent tunneling process can take place. According to Fig. 5.12, the defects are found to be filled according to their trap depth, visible as a horizontally moving tunnel front. The small separation of the levels on the energy scale results in a narrow distribution of electron capture times. From this it follows that two particular charging events at the upper and the lower edge of the trap band can be hardly resolved in time. As a consequence, no vertical component in the motion of the tunnel front is observed during the recovery phase of Fig. 5.12. Analogously to the stress phase, the tunneling hole front also appears in the occupancies of the demarcation levels displayed in Fig. 5.14. During the relaxation phase, the levels are shifted below where they can be neutralized if equilibrium has been reached. However, Fig. 5.14 reveals that the discharging of traps has not been completed even until an unrealistic long relaxation time of . It is important to note here that during stress and relaxation determines the active area in which hole capture is possible. Defects above this area are already unoccupied before stress and thus cannot capture a further hole, while the ones below will remain neutral due to the high hole emission rate. As a result, only defects within this area can change their charge state and thus contribute to the net amount of captured holes and in further consequence to NBTI.
The LSM has been employed to simulate NBTI degradation in a pMOSFET for a wide range of different stress conditions. The calculated stress/relaxation curves for the aforementioned showcase are presented in Fig. 5.15. In contrast to the ETM, they exhibit a marked temperature dependence in addition to the obvious field acceleration. While the former one mainly stems from the temperature dependent Fermi-Dirac distribution in (cf. equation (5.6)), the latter one cannot be simply interpreted by the lowering of the tunneling barrier at higher . The field acceleration originates form the larger shift of the trap levels at higher , as visualized in Fig. 5.16. As such, the field acceleration strongly depends on the distribution of the trap levels in space and energy but is not inherent to the LSM itself. For instance, a defect with and during stress will not be able to capture a hole at all (see Fig. 5.17).
In order to obtain more realistic results, tunneling from interface states  has been incorporated into LSM. The obtained degradation curves for defects with and are depicted in Fig. 5.18. Based on these results, it will be evaluated whether the LSM can satisfactorily reproduce the basic features seen in NBTI experiments. In the NBTI checklist of Table 5.2, these features are formulated as necessary criteria, where each of them will be judged in the following.
The above list provides strong evidence that the LSM cannot be reconciled with the experimental NBTI data. As a consequence, pure tunneling must be discarded as a possible cause for hole trapping in NBTI.