The concept of NMP has been used in a slightly modified variant termed multiphonon field-assisted tunneling (MPFAT) [116, 115, 116, 117, 125, 55], which was proposed for ionization of deep impurity centers. The underlying theory accounts for the fact that the emission of charge carriers out of bulk traps is accelerated in the presence of an electric field. This effect is eventually related to the shortened tunneling distance through a triangular barrier when considering thermal excitation of the charge carriers. According to theoretical calculations of Ganichev et al. [117], it yields a field enhancement factor , which is suspected to have a strong impact on hole capture processes in NBTI and has therefore been phenomenologically introduced in the two-stage model (TSM) [61].

The TSM relies on the Harry-Diamond-Laboratories (HDL) [15] model but is
extended by a second stage accounting for the permanent component of NBTI (cf.
Fig. 6.3). The defect precursor, an oxygen vacancy according to the HDL model, is
capable of capturing substrate holes via the aforementioned MPFAT mechanism. The
trap level of the precursor is located below the substrate valence band and
subject to a wide distribution due to the amorphousness of . Upon hole
capture, the defect undergoes a transformation to an center, which is visible in
ESR measurements [43]. In this new configuration, it features a dangling bond
associated with a defect level within or close to within the substrate bandgap
in accordance to [162]. The level shift from to arises from the change to
a new ‘stable’ defect configuration, namely the dangling bond. In the
center configuration, the defect can be repeatedly charged and discharged by
electrons tunneling in or out of its dangling bond. The associated switching
behavior^{1}
is in agreement with the experimental observations made in electrical
measurements [15, 16]. Only in the neutral state , in which the dangling bond
is doubly occupied by an electron, the center can be annealed, thereby becoming
an oxygen vacancy again.

The second stage involves an amphoteric trap, most probably a center, which has been found to interact with the switching trap as observed in irradiation experiments [43]. That is, a hydrogen is detached from an interfacial - bond and leaves behind a center. In a subsequent reaction, it saturates the dangling bond of the center. This stage fixes the positive charge at the oxide defect and creates a new interface state, whose charge state is controlled by the substrate Fermi level. Since the hydrogen transition is assumed to last much longer than the hole capture or emission process, this stage corresponds to the permanent or slowly recoverable component of NBTI.

Mathematically, the dynamics of this complex mechanism are described by the set of the following rate equations:

The subscript of stands for the state according to the numbering in Fig. 6.3. The transition rates are denoted as , with and as the initial and the final states, respectively. The rate is derived from the SRH equations (2.69), in which the empirical enhancement factor for the MPFAT transitionThe following simulations are based on the same numerical scheme as has been presented in Section 3.2 and are used for the ETM and the LSM in Chapter 4 and 5. Each representative trap in this scheme is characterized by its individual set of defect levels and barriers. The generated random numbers are homogeneously distributed for , , , and while they follow a Fermi-derivative (Gaussian-like) distribution [90] for and . The remaining quantities including , , , , and are assumed to be single-valued.

In contrast to previous models, oxide charges (state 2) as well as interface traps (state 4) are incorporated into the TSM so that two states must be considered for the calculation of . It is important to note that only a part of the overall degradation during stress is observed within the experimental time window. As demonstrated in Fig. 6.5, a large fraction already occurs before the beginning of the OTF measurement () and only a part of the degradation can be monitored by this technique (cf. Section 1.3.2). As a consequence, the measured threshold voltage shift must be calculated as

The TSM [61] has been compared to a large set of measurement data, including various combinations of stress voltages and temperatures. For illustration, a fit to the eMSM (cf. Section 1.4) data at is depicted in Fig. 6.6. The findings of this model are evaluated in the following:

- Hole trapping during stress sets in before the first OTF measurement point () is determined and no sign of saturation appears until . Detrapping during relaxation extends over the whole experimental time window ranging from to .
- follows a logarithmic time behavior during the whole stress phase.
- During the initial phase of recovery, a logarithmic time dependence is clearly recognizable. However, this behavior is obscured for relaxation times larger than where the curve starts to level off. This is due to the fact that the TSM also accounts for the permanent component of NBTI.
- The logarithmic slopes of during the stress and the relaxation phase
exhibit a ratio of in agreement with [61]. This asymmetry is
demonstrated by the temporal change of the trap occupancy in Fig. 6.6 (right).
After the level shift from to , when the defect transforms from an
oxygen vacancy to an center, the trap level lies closer to the
substrate valence bandedge than . This results in higher trapping rates
for hole capture () and emission () so that the subsystem
of states and is only weakly affected by during recovery.
Therefore, this reduced system is close to equilibrium, meaning that
- The simulations fit the experimental stress and relaxation curves for a wide range of gate biases meaning that the TSM correctly reproduces the field dependence seen in measurements.
- An equal quality of fits (not shown here) has been achieved for other temperatures with the same set of parameters, meaning that the TSM is able to describe the temperature dependence observed experimentally.

The TSM is found to satisfy all criteria of Table 6.2 and therefore seems to properly describe NBTI degradation. Besides that, it also agrees well with the observation of a field-dependent recovery, which is demonstrated by the measurements shown in Fig. 6.7. Due to the occupancy effect, the substrate Fermi level controls the portion of neutral centers (state 3) which can return to state by structural relaxation and contribute to the NBTI recovery. Interestingly, this field dependence is compatible with the finding that the emission times of ‘anomalous defects’ are field-sensitive.

The distributions of trap levels obtained from the model calibration are depicted in Fig. 6.8. The trap levels of the precursors (state ) are uniformly distributed between and in qualitative agreement with the values in [39]. The defects located the highest have also the highest substrate hole capture rates and therefore have already been transformed centers (states and ) after a stress time of . In this new configuration, they feature a trap level in the range between and in qualitative agreement with the values published in [39]. According to equation (6.21), the occupancy of the levels is determined by the substrate Fermi energy and thus the number of neutralized defects in the center configuration (state ) increases with a lower energies. Since only defects in this state transformed to a precursor (state ) again, the number of traps in state diminished towards the substrate Fermi energy (cf. Fig. 6.8). The donor levels of the interface states have been assumed to be uniformly distributed and are located within the lower part of the substrate bandgap consistent with [64].

So far, classical calculations of the band diagram have been performed to obtain the interface quantities, such as the position of the bandedges (, ), the Fermi level (), and the electric field () within the dielectric. These quantities enter the expressions of the rates and will significantly alter them due to their exponential dependences. However, the SRH rates (6.9) used in Section 6.3.1 are valid for a three-dimensional electron gas [55] but this assumption breaks down for an inversion layer of MOS structures. In the one-dimensional triangular potential well in the channel, quasi-bound states build up and form subbands, which correspond to the new initial or final energy levels for the charge carriers undergoing an NMP transitions. The quantum mechanical transition rates are obtained following the derivation in Section 2.5.2 but using the DOS for one-dimensionally confined holes. Then the rate equation (2.59) modifies to

It is noted here that the exact shape of does not enter this derivation and consequently the DOS can be expressed asThese rates have been used to incorporate the aforementioned quantum effects into the TSM, which has been evaluated against the same set of experimental data. For a proper comparison with the classical variant of TSM, only the NMP parameters (, , , and ) have been optimized while all other parameters have been held fixed. The simulated degradation curves show a good agreement with experimental data (see Fig. 6.10) so that the quantum mechanically refined variant of the TSM still fulfills all criteria listed in Table 6.2. It is noted here that these simulations yield an the uppermost trap levels , which have been shifted downwards by about the same energy as the separation of and (ranging between and ). This can be explained when considering that, first, is replaced in the rate equations (6.27) and (6.28) and, second, the NMP barrier for hole capture is reduced by this energy difference. From this it follows that also the trap levels must be shifted down by approximately the same energy in order to obtain hole capture rates of an equal magnitude. In summary, it has been assured that also the quantum mechanically refined variant of the TSM can explain the NBTI data and must therefore be considered as a reasonable NBTI model.

The criteria in Table 4.1 have been successfully satisfied by the TSM. With this respect, the TSM should be regarded as model qualified to describe NBTI. However, these criteria only evaluate the degradation produced by an ensemble of defects but do not consider whether the behavior of a single defect is correctly reproduced. For this reason, the TSM will be investigated using the time constant plots in the following. Since the TDDS measurements cannot capture the permanent component of NBTI, the transition state diagram must be reduced to stage one. This means that the equation (6.8) and the rates and in equation (6.6) must be omitted. Since the trap level is assumed to lie closer to the valence band edge than , the accociated rate and are much larger than . Thus the fast switching between state and produces noise, which is undesired for the analysis of in the time constant plots. Therefore, a compact rate expression for the transition is sought. One can calculate the corresponding emission time as the mean first passage time in continuous time Markov chain theory [131] (discussed in Section 3.2).

- The simulated in the time constant plot of Fig. 6.11 has a wrong curvature. This can be traced back to the fact that is proportional to according to equation (6.31).
- The NMP barriers allow for a marked temperature activation of .
- The TSM gives field-insensitve as demonstrated in Fig. 6.11. However, this is only true for defects with a trap level above . In this case, approximates to and is given by .
- The TSM also allows for field-sensitive if is located slightly below at the relaxation voltage. Then the field dependence is caused by the term in equation (6.32).
- In both defects, is thermally-activated.

As demonstrated in the previous section, the TSM is indeed an important improvement of the NBTI model. Regarding the time constant plots (cf. Table 6.3), the introduction of the state gives an explanation for ‘normal’ as well as the ‘anomalous’ defect behavior. However, the TSM predicts a wrong curvature of and thus cannot be reconciled with the TDDS data. As a consequence, it can be concluded that the TSM performs well for stress and relaxation curves but fails to describe the behavior of single defects.