3.2 Extensions of the Reaction-Diffusion Model

The lack of a decent description of recovery based on the reaction-diffusion theory, cf. (3.2), soon urged the development of modified and extended RD models [63, 33, 64, 41, 17, 59, 65, 66, 67, 55]. In the following variants thereof are summarized.

1. As modern oxide layers are only a few nanometers thick, the explanation based on diffusing hydrogen inside the oxide was questioned. Therefore the diffusion process of was suggested to continue inside the polysilicon as well [63]. By assuming two different diffusion coefficients, namely fast diffusion in the oxide and slow diffusion inside the polysilicon, the so-called two-region RD model was expected to be able to explain the much larger observed recovery range. Actually this range only increased at little [55].
2. The two-interface RD model [33], presented by Krishnan et al., focuses on the quick diffusion of atomic hydrogen inside the oxide. Once having reached the polysilicon interface, a second chemical reaction takes place creating molecular hydrogen ( ). Like in the two-region model, the molecular hydrogen diffuses further into the polysilicon. Since the diffusivity in the oxide is regarded to be very high compared to the diffusivity in the polysilicon, the stored in the oxide is indeed able to cause a fast initial recovery. For large stress times, on the other hand, it is this higher oxide diffusivity that locks the hydrogen in the polysilicon for a long time. This means that the short recovery effect vanishes.
3. In contrast to the two-region RD model, where instant dimerization at the interface is assumed, the RD model with explicit dimerization is based on a continuous dimerization process inside the oxide, what allows both hydrogen species to coexist while diffusing into the oxide [64]. Whereas the initial stress phase is thereby altered to , the recovery characteristic remains the same compared to the standard RD model.
4. Since the experimentally observed recovery revealed a log-like characteristics (cf. Section 4.1), Islam et al. questioned the interface states to be fast enough to follow the gate voltage switches. They suggested an RD model assuming slow interface states. Unfortunately, such a model is in stark contradiction to the Shockley-Read-Hall theory (SRH) used to describe the trapping dynamics at the interface with transients due to electron capture being within the nano-second regime. Under the assumption of exessively small capture cross sections some sort of fast relaxation in the microsecond-regime within one or two decades in time is indeed obtained. However, this form of recovery is not observed in any experimental data [55].
5. Extended reaction-dispersive-diffusion (RDD) models using a broad distribution of energy levels were discussed in [17, 59, 65, 66,  55]. They describe the hydrogen transport occuring via the highest energetic states only (transport level). Hydrogen being located in a deeper energy level needs to be thermally activated prior to be able to diffuse further into or out of the oxide, i.e. without any activation this hydrogen is trapped. Further, in these models only hydrogen sitting at the interface is allowed to re-passivate which slows down the reverse rate as most of the hydrogen is trapped.

   In contrast, a simplified version of the RDD model does not differentiate between trapped and untrapped hydrogen, i.e. all hydrogen is allowed to interact with the interface [67]. This implies a faster initial recovery, compared to the non-simplified RDD model, cf. simulations performed in [55].

   Although with increasing dispersion of the bond breaking at the interface the recovery can be slowed down, none of the RDD variants is finally able to describe the actual experiment.

The following conclusion can be drawn for RD theory in general. While during recovery solely passivation occurs, the stress is modeled using depassivation and passivation simultaneously [31]. At present, no extension of the RD-model is able to describe recovery after stress in a reasonable form. Whether such a model is then able to describe the much more complex stress-relaxation patterns during the operation of a MOSFET is very questionable. The premises are simply not correct. This leads to the conclusion that hydrogen diffusion is very unlikely to be a main player when dealing with NBTI degradation. For this reason completely new approaches are inevitable [11, 6, 40, 61, 68, 69, 18].