1.4.3 Reaction-Diffusion Framework

Another approach, which focused on the physical picture behind hot-carrier degradation was developed by the group of Alam [26,149]. The assumption was that NBTI and HCD are related to the breakage of silicon-hydrogen bonds, differing only in the driving force triggering this dissociation. Therefore, both phenomena are to be coupled within the same modeling framework. The authors claimed that since NBTI is just the breakage of Si-H bonds followed by hydrogen release and diffusion, NBTI and HCD are to be united within the reaction-diffusion concept.

Experimental observations demonstrated that time signatures of NBTI and HCD have different power-law slopes, i.e. the former one can be approximated by a t^1/4 law while the latter one better obeys a t^1/2 dependence, see Figure 1.6 and [26,149]. The reaction-diffusion framework includes the following stages (are depicted in Figure 1.7 [149,150]):

Creation of interface states via breaking Si-H bonds. This stage is reaction-limited and described by a t¹ dependence.
Hydrogen diffusion begins to take over with no more interface states created: N_it~t⁰.
Diffusion-limited phase with t^1/4 behavior.
Hydrogen diffuses away with unlimited diffusion velocity resulting in the 1/2 degradation time slope, i.e. N_it~t^1/2.
Finally, saturation occurs when all the "virgin" Si-H bonds are depassivated: N_it~t⁰.

**Figure 1.6:** Different time slopes of hot-carrier induced degradation and NBTI. The data are borrowed from [26].

Therefore, it was assumed that NBTI is diffusion-limited, which explains its t^1/4 behavior while HCD is controlled by the 4^th phase. However, this scenario presumes that in the case of HCD a transition from t^1/4 to t^1/2 is to be observed but the authors of [149] claim that in practice no experimental evidence of such a transition is known. Instead, they suggest that the difference in time slopes is related to the circumstance that NBTI is a 1D problem while HCD is a 2D phenomenon due to the non-uniform N_it distribution over the lateral coordinate. Since the Si-H bond-breakage event generates one mobile hydrogen and one interface trap one writes N_it=∫N_H(r,t)d³r, (N_H is the coordinate-dependent hydrogen concentration). The diffusion front moves as (D_Ht)^1/2 and thus NBTI- and HCD-related N_it are

(1.10)

where D_H is the hydrogen diffusivity, A_d the area of the degraded spot and $N_\mathrm{H}^\mathrm{(0)}$ is the H density at the interface. Assuming that $N_\mathrm{it}N_\mathrm{H}^\mathrm{(0)}\sim \mathrm{constant}$ , one obtains that $N_\mathrm{it}^\mathrm{(NBTI)}\sim (D_\mathrm{H}t)^{1/4}$ and $N_\mathrm{it}^\mathrm{(HCD)}\sim (D_\mathrm{H}t)^{1/2}$ .

Despite its ability to explain the different time slopes of NBTI and HCD, this reaction-diffusion model suffers from serious shortcomings. First, within this framework it is assumed that both phenomena are diffusion limited. This implies, however, that once the stress is removed recovery should occur rather quickly. Recent NBTI data suggest, however, that interface state creation is reaction rather than diffusion limited [151,136,138]. Concerning HCD, the recovery is in general rather weak if there is any recovery at all. Second, the model does not rely on carrier transport, that is, it does not consider the driving force behind the trap generation. As a consequence, the N_it distribution and the localized nature of the damage are not addressed. To conclude, the attempt to describe only one essential aspect of HCD - the mechanisms of defect creation - not capturing others has been undertaken.

**Figure 1.7:** The main phases of the reaction-diffusion model applied to NBTI with different time slopes being marked. Data from [150].

I. Starkov: Comprehensive Physical Modeling of Hot-Carrier Induced Degradation