As mentioned in Chap. 1, the reaction diffusion model has been mainly used for the interpretation of BTI experiments for almost four decades. Its apparent inability to explain NBTI recovery has led to several modifications of the concept, including trapping of the diffusing species and different types of diffusors [11, 9].

Quite recently it was claimed that the misprediction of recovery is due to the one-dimensional description of the diffusing species in the macroscopic model (1.5)–(1.6). As illustrated in Fig. 3.1, it was suggested that this formulation makes it too easy for the hydrogen atom to find a dangling bond to passivate because the one-dimensional diffusion considers only two options of motion: forward and backward jumping. In a higher-dimensional description the diffusion and reaction kinetics are much richer:

- The atoms can move in all three dimensions equally likely, leading to a distribution of arrival times at the interface during recovery.
- H
2 -molecules dissociate at a dangling bond, creating a passivated dangling bond and a free hydrogen atom that does not immediately find another dangling bond to passivate. - Hydrogen atoms arriving later have to hover along the interface to find an unoccupied dangling bond.

A simple estimate of the recovery in the hypothetical three-dimensional model is given in [16]. This estimate mimics the different repassivation kinetics arising in the atomic description within the framework of the usual macroscopic RD model. To account for the longer “effective” recovery paths, the diffusion coefficients in the macroscopic model are reduced by different factors during recovery and the resulting recovery traces are averaged. Although this approach gives a recovery that proceeds over more time scales, as shown in Fig. 3.2, no derivation for the quasi-three-dimensional description is given and its physical validity is at least questionable. One of our targets is to test the claims of [16] within a firm theoretical framework.

Another motivation for the study of the microscopic limit of the reaction-diffusion model for NBTI
is based on general issues with the rate-equation-based description. As a literature study reveals,
reaction-diffusion systems have been studied in numerous scientific communities from both the
theoretical and the experimental side for more than a century [145, 146, 147, 148, 149, 150, 151].
Although the mathematical framework of the RD model (1.4)–(1.6) seems physically sound
and the description using densities and rate equations is commonly considered adequate,
it is a well known and experimentally confirmed result of theoretical chemistry that the
partial differential equation based description of chemical kinetics breaks down for low
concentrations [145]. Additionally, in reaction-diffusion systems bimolecular reactions, such as the
passivation and the dimerization reaction, require a certain proximity of the reactant species,
termed

It is easy to show that diffusion must play a dominant role in the bimolecular reactions in the RD
model for NBTI. Fig. 3.3 schematically shows a uniform random distribution of dangling bonds on
a silicon (100) surface that corresponds to a density of 5 ^{12}cm^{-2}, which is a usual
assumption for ^{-1∕2}

Once established by the atomic viewpoint above, the diffusive influence on the bimolecular reactions
leads to contradictions in the RD model and its physical interpretation. The predicted degradation of
the RD model that is compatible with experimental data is only obtained if the hydrogen atoms
that are liberated during stress compete for the available dangling bonds and dimerize
at a certain rate. Both requirements involve diffusion over distances much larger than
the nearest neighbor distance, which takes about ^{-13}cm^{2}

| (3.1) |

While a reasonable reaction radius is in the regime of the average radius of the oxide interstitials,
which is about 4Å [152], the application of (3.1) to published dimerization rates gives values ranging
from 70

For the present study of the microscopic properties of the RD mechanism we have developed our own atomistic reaction-diffusion simulator based on the considerations of the previous chapter. The microscopic reaction-diffusion picture developed for this purpose also serves as a framework to assess whether a description based on rate-equations is applicable to the physical mechanism behind the RD model.

3.1 The Microscopic RD Model

3.2 Results and Discussion

3.2.1 General Behavior of the Microscopic RD Model

3.2.2 Recovery

3.2.3 Approximations in the Macroscopic Model

3.2.4 A Real-World Example

3.2.5 Increased Interface Diffusion

3.3 Related Work

3.2 Results and Discussion

3.2.1 General Behavior of the Microscopic RD Model

3.2.2 Recovery

3.2.3 Approximations in the Macroscopic Model

3.2.4 A Real-World Example

3.2.5 Increased Interface Diffusion

3.3 Related Work

Our microscopic RD model attempts to mimic the proposed mechanisms of the reaction-diffusion
model at a microscopic level. The basic actors are H-atoms, H

Several algorithms have been used in the chemical literature for the stochastic simulation of reaction-diffusion systems [148, 149]. The most difficult task in the modeling of these systems is the diffusion of the reactants and several different approaches are available which can roughly be categorized as grid-based methods or grid-less methods [149], see Fig. 3.5.

Grid-less methods propagate the coordinates of the diffusing species quite similarly to molecular dynamics methods. Instead of explicitly treating all atoms of the solvent and their effect on the trajectory of the diffusors, the motion of the diffusing particles is perturbed by an empirical random force to generate a Brownian motion. Bimolecular reactions happen at a certain rate as soon as two reaction partners approach closer than a given radius. Although this technique suffers from its sensitivity to the time-step and the specific choice of the random force, it is a popular choice for the simulation of reaction-diffusion processes in liquid solutions where real molecular-dynamics simulations are not feasible [148, 149]. In grid-based methods the simulated volume is divided into small domains and each diffusing particle is assigned to a specific domain. The motion of the diffusors proceeds as hopping between the grid-points. In these models the bimolecular reactions happen at a certain rate as soon as two reactants occupy the same sub-volume. The advantage of this approach is that it can be formulated on top of the chemical master equation (2.20). The master equation can then be solved using the stochastic simulation algorithm (see Sec. 2.11), which does not depend on artificial time-stepping. A problem of the grid-based method that is repeatedly discussed in chemical literature is the choice of the spacial grid as it induces a more or less unphysical motion in liquid solutions. Additionally, the probability to find two particles in the same grid point and in consequence the rate of bimolecular reactions are quite sensitive to the volume of the sub-domains [149].

In the reaction-diffusion model for NBTI, the diffusion of the particles proceeds inside a solid-state solvent. Contrary to diffusion in gases or liquids, the motion of an impurity in a solid-state host material proceeds via jumps between metastable states as illustrated in Fig. 3.6.

This hopping diffusion is understood as a barrier-hopping process as explained in Sec. 2.8. In the
case of H or H

We conclude that the most appropriate description of the physics considered in the present work is
obtained from the reaction-diffusion master equation approach [151, 150, 148, 149]. Within the
natural lattice of interstitial positions the actors of our RD system exist in well-defined and discrete
states. In accord with the considerations of Sec. 2.6, it is now possible to define a state vector that
contains the interstitial positions and bonding states of all actors as well as a set of reaction channels
which cause transitions between the states of this vector. The RD system then becomes a
time-dependent stochastic process (_{ω}
and whose evolution over time can then be described by the chemical master equation
(2.20). As explained in Sec. 2.6, the physics behind the reaction channels are contained in
the propensity functions _{γ} and the state-change vectors _{γ}, which are given in Fig. 3.7.
The reactions employed in our simulations are the hopping transport between interstitial
sites, the passivation/depassivation reaction and the dimerization/atomization reaction.

Reaction | Macroscopic | Microscopic | Illustration | |

| Si^{*} + H | | | |

| SiH ^{*} + H | | | |

| H: _{1} _{2} | | | |

| H_{1} _{2} | _{2} | | |

| 2H | ^{2} | 2 | |

| H | | |

…Dangling bond …H …H

The stochastic chemical model is solved using the stochastic simulation algorithm (SSA) explained in Sec. 2.11.

In the microscopic RD model employed in this work the interstitial sites form a regular and
orthogonal three-dimensional grid and the hopping rates for the diffusors are assumed to be constant
in accord with the isotropic and non-dispersive diffusion underlying the conventional macroscopic RD
model [9]. In a real SiO

As mentioned above, the choice of the grid size requires special attention as it determines the probability of the bimolecular reactions. The interstitial size of amorphous silica has been calculated for molecular-dynamics generated atomic structures and is about 4Å [152]. We take this value as the physically most reasonable grid size.

Once the microscopic model is defined, the relation to the macroscopic RD model (1.4)–(1.6) has to
be established. Using the number of dangling bonds in the simulation box

= | (3.2) | |

= | (3.3) | |

H(_{i}) | = | (3.4) |

H_{i}) | = | (3.5) |

| (3.6) |

in accordance with the assumptions of the macroscopic RD model.

Two different systems have been studied in detail: a model system and a “real-world-example”. The model system is used to study the general features of the microscopic reaction-diffusion process. It is parametrized in order to clearly show all relevant features at a moderate computational effort. The parametrization of the real-world system is based on a published parametrization of the modified reaction-diffusion model. This system is used to relate our microscopic model to published data.

| |

Depassivation | 0^{-1} |

Passivation | 4 ^{4}^{-1} |

Dimerization | 2 ^{5}^{-1} |

Atomization | 5^{-1} |

H-hopping | 100^{-1} |

H | 100^{-1} |

The parametrization that is used to study the general behavior is given in Tab. 3.1. The number of diffusing
particles is a trade-off between accuracy and computational speed. Due to the large computational
demand^{1} ,
different regimes of the degradation curve have to be calculated with different numbers of diffusing
particles.

The earliest degradation times are dominated by the depassivation of the silicon dangling bonds
leading to a linear increase of the degradation, which is equivalent to the initial “reaction limited”
degradation of the macroscopic RD model [8]. However, the degradation predicted by the
microscopic RD model quickly saturates as an equilibrium forms between depassivation
and repassivation

( | = | (3.7) |

= | (3.8) |

| (3.9) |

The main difficulty in the calculation of the early degradation times in the microscopic RD model is the very low degradation level in this regime, which requires a high accuracy, i.e. a large number of particles to obtain smooth results. Fortunately, as reactions between the hydrogen atoms or between hydrogen atoms and neighboring dangling bonds are not happening in this regime, a good parallelization can be obtained by averaging over separate simulation runs, see Fig. 3.9.

A comparison of the microscopic RD model and (3.9) is shown in Fig. 3.10.

The initial behavior of the microscopic RD model stands in stark contrast to the degradation in the macroscopic model where the linear regime continues until a global equilibrium has formed at the interface.

As this initial behavior takes a central position in our further discussion, it requires a deeper analysis. According to Sec. 2.12 the microscopic single-particle regime can be accurately described using rate equations, as it does not contain any second-order reactions. The required equations are basically those of the RD model, but as every hydrogen atom can be assumed to act independently, the expressions for the hydrogen bonding as well as the competition for dangling bonds are neglected. As the kinetic behavior in this regime is strongly determined by the first diffusive steps of the hydrogen atoms, the diffusion part of this approximation must have the same interstitial topology as the microscopic model. As all hydrogen atoms act independently, only one atom and one dangling bond need to be considered. The interface reaction and the diffusion of the hydrogen atom is thus described as

= | (3.10) | |

= _{j∈(i)}( | (3.11) |

After the atoms have traveled sufficiently long distances, the interaction between the particles becomes relevant and the single-particle approximation becomes invalid. In Fig. 3.11 this is visible as a transition away from the single-particle behavior towards the macroscopic solution between 1s and 1ks. Due to the relatively large level of degradation, the long-term simulations do not require as much accuracy as the short-term simulations. Consequently the number of particles can be reduced for longer simulation times, which makes the prediction of long-term degradation possible.

Finally, Fig. 3.12 compares the microscopic RD model to the macroscopic version over the
course of one complete stress cycle, where the microscopic curve was obtained by combining
calculations of different accuracy, as explained above. Instead of the three regions which arise
from the macroscopic RD model — reaction-limited, equilibration and diffusion-limited —
the H-H

- The earliest degradation times (
t < 20μ s in this case) are dominated by the depassivation of dangling bonds. In this regime, the microscopic and the macroscopic model give identical degradation behavior. - After the passivation and depassivation have reached an equilibrium between
k f andk r separately for each Si-H bond, the fraction of depassivated dangling bonds remains constant until the diffusion of the hydrogen atoms becomes dominant. This regime only shows when the individual hydrogen atoms are considered and consequently is not obtained from any model based on rate equations. - As more and more hydrogen atoms leave their initial position, the degradation is determined
by the buildup of a diffusion front along the Si-SiO
2 interface and the equilibration between the dangling bonds. This regime has a very large power-law exponent of almost one^{2}that is not experimentally observed. The stress time range in which this regime is observed depends on the average distance between two dangling bonds, the diffusion coefficient and the interstitial size. - As the bimolecular reactions become relevant, the macroscopic diffusion-limited regime
begins to emerge. For some parametrizations we have observed a time window in which a
sufficient amount of H
2 has not formed yet, and thus the initial diffusion-limited regime has the typicalt ^{1∕4}-form that arises from the classical RD model without H2 [25].

The initial single-particle phase of the degradation is a remarkable feature of the microscopic model. As it is incompatible with experimental data and very sensitive to the parametrization, its relevance for real-world reliability projections has to be investigated. For this purpose we have run calculations based on a published parametrization of the reaction-diffusion model for NBTI, see Sec. 3.2.4.

In agreement with our investigations on two-dimensional systems [25, 24], the three-dimensional
stochastic motion of the hydrogen atoms ^{1∕6} degradation regime requires
an equilibration and thus a quasi-one-dimensional behavior. The recovery proceeds on a
timescale that is at least two orders of magnitude longer than the stress time. The lateral
search of hydrogen atoms for unoccupied dangling bonds was suggested to dominate at
the end of the recovery. However, due to the logarithmic time scale on which recovery is
monitored, the equilibration along the interface has negligible impact at the end of the
recovery trace if this equilibration proceeds about two orders of magnitude faster. Thus, the
hovering of hydrogen atoms along the interface does not influence the shape of the recovery
transient.

After the microscopic RD theory Fig. 3.7 has been established and its general behavior has been investigated, one can use this framework to analyze the assumptions that are implicit to the macroscopic RD model (1.4)–(1.6), which is still widely considered to be an adequate approximation.

The most obvious approximation in the macroscopic RD model is the one-dimensional description of
diffusion. While this may seem to be appropriate as boundary effects in the diffusion of both H and
H

All the liberated hydrogen atoms at the interface (H it in (1.1) and (1.4)) compete instantaneously withall the other free interfacial hydrogen atoms forall the available dangling bonds.All the pairs of hydrogen atoms at a certain distance from the Si-SiO2 interface are equally likely to undergo dimerization and form H2 , independently of their spatial separation.

As was shown above, a hydrogen atom liberated during stress initially stays in the vicinity of its original dangling bond and thus the lateral homogeneity has to be considered a long-term approximation. It is accurate when the diffusion of hydrogen has led to enough intermixing so that there is no significant variability in the concentration of free hydrogen along the interface. Following [115] and the discussion in Sec. 2.12, this condition can be called “lateral well-stirredness” of the system.

The second and more delicate approximation in the macroscopic RD model is the mathematical
description using rate- and diffusion-equations. In the microscopic RD model, the rate at which an
atom at the interface passivates a dangling bond depends not only on the rate

Similar to the passivation rate, the rate at which H^{2}. As thoroughly explained in [145], this approximation is
only valid for large numbers of particles, as the number of pairs of hydrogen atoms in an
interstitial goes as ^{2} if

All in all, the macroscopic RD model can only be considered a valid approximation of the microscopic RD model for very long stress times and a sufficient amount of liberated hydrogen atoms. The time it takes for the macroscopic approximation to become valid, however, may exceed the time range in which it is usually applied, depending on the parametrization.

| 3 | s |

| 6 | cm |

| 5 | cm |

| 95 | s |

| 10^{-13} | cm |

_{2} | 1 | cm |

| 5 | cm |

To study the behavior of the atomistic model for a real-world example, we compare to the
measurements of Reisinger et al. [12] using the parametrization of Islam et al. [20] in a modified
form, see Tab. 3.2. Fig. 3.14 shows the results of our calculations for several interstitial sizes. While
the macroscopic one-dimensional RD model fits the data very well, the kinetic Monte Carlo data
shows a completely different behavior. Again, the single-particle regime is clearly present. However,
due to the low density of dangling bonds at the interface, the single-particle regime dominates the
degradation for a large part of the stress time. For a realistic interstitial size of 4Å [152, 155], the
onset of the ^{1∕6} regime lies far beyond the experimental window of 10^{5}s. When the interstitial size is
increased, the onset of the ^{1∕6} regime moves to earlier times, which is due to the increase of the
reaction radius for the bimolecular reactions as explained above. For the given parameter set, an
interstitial size of ^{1∕6} regime touch the experimental window.

A shift of the onset of the experimentally observed regime to earlier times at a realistic
interstitial size requires a dramatic increase of either the hydrogen diffusion coefficient or the
availability of free hydrogen near the interface. An increase of the hydrogen diffusion coefficient,
however, breaks the dominance of H^{1∕6} towards ^{1∕4}. Increasing
the availability of hydrogen at the interface by adjusting the ratio

This indicates that in the given microscopic model it is impossible to obtain the experimentally
observed ^{1∕6} degradation within the experimental window at a reasonable interstitial size.

The behavior predicted by the microscopic model is completely incompatible with any experimental
data, while the description is much closer to the physical reality than the macroscopic RD model.
Only two interpretations are possible to resolve this dilemma. Either the ability of the macroscopic
RD model to fit degradation measurements has to be regarded as a mathematical artifact without
physical meaning, or the structure of the Si/SiO_{2} interface somehow accelerates the lateral
equilibration considerably. We investigated the second option more closely by considering
first-principles calculations that have shown a lowering of diffusion barriers for hydrogen (molecules)
along the Si

As can be seen in Fig. 3.15, the increase of the interface diffusion coefficient accelerates the
degradation during the initial phase as it increases the transport of hydrogen atoms away from
their dangling bonds. Interestingly, even if the interface diffusion coefficient is increased by
four orders of magnitude there is no ^{1∕6} behavior visible, but instead the degradation
takes on the typical ^{1∕4} behavior of a hydrogen-only reaction-diffusion model. While the
competition for dangling bonds sets in earlier for increased interface diffusion coefficients, the
formation of H

As a side note we remark that in a real wafer, a nearly infinite diffusion coefficient along the
Si

Three other scientific groups have put forward microscopic RD models recently [157, 23, 158] and interestingly those investigations find a reasonable agreement between their microscopic description and the macroscopic RD model. In the work of Islam et al. [23] the atomic description is basically equivalent to the work presented here but is built upon a one-dimensional foundation which carries the same implicit assumptions as the macroscopic model. Clearly this model cannot capture the effects discussed in this chapter as those are solely due to higher-dimensional effects. From a physical point of view, however, the one-dimensional approximation lacks justification considering the results presented above.

The work of Choi et al. considers the three-dimensional diffusion of the particles based on a
grid-less stochastic formulation [158]. Although the degradation in that work seems to
match the macroscopic RD model quite well at first sight, also strong discrepancies arise
between the two for longer stress times. Interestingly, for situations where the approach
presented above predicts a degradation far below the prediction of the macroscopic model, the
degradation predicted by Choi et al. overshoots the macroscopic model considerably. Only for an
enormous density of dangling bonds or a very large reaction radius the macroscopic behavior is
obtained, in accord with our results. The degradation behavior in [158] initially follows

The work of Panagopoulos et al. uses a grid-based stochastic RD model that seems to be compatible with our description. The surprisingly good agreement between their results and the macroscopic RD model may be an artifact of the employed method which is based on an adaptive time-stepping [157]. Also, the paper states that the passivation reaction occurs if a hydrogen atom is “close” to a dangling bond. This indicates an artificial capture radius, but this is not explicitly stated. Also, the grid spacing is not given in the paper and its physical relevance is not discussed. However, as shown by our calculations, an unphysically large grid spacing strongly promotes bimolecular reactions and thus induces a degradation behavior that is (falsely) compatible with the macroscopic RD model.