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Predictive and Efficient Modeling of Hot Carrier Degradation with Drift-Diffusion Based Carrier Transport Models

3.3 The Reaction-Diffusion Model

Another attempt to model the physical process behind HCD was the application of the reaction-diffusion model developed for NBTI to HCD by Alam et al. [162, 163]. They assumed both NBTI and HCD to be involved in Si-H bond breakage with different driving forces, and tried to model the two phenomena using the same framework. However, the experimental \( N_{\mathrm {it}} \) obtained from NBTI and HCD were found to have different times slopes as shown in Figure 3.4.

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Figure 3.4: The experimentally obtained power-laws for NBTI and HCD curves plotted versus stress times from data in [162].

The different stages in the reaction-diffusion model are summarized in Tab. 3.1. The degradation process starts with the reaction limited phase of Si-H bond breakage. In this phase, \( N_{\mathrm {it}} \) increases linearly with time. During the second stage, no more interface states are generated and hydrogen diffusion begins. The third step comprises of a diffusion limited phase with \( N_{\mathrm {it}} \) generation time slope of \( 1/4 \). In the next stage, hydrogen diffuses away with unlimited velocity and \( N_{\mathrm {it}} \) depending on time as \( t^{1/2} \). In the final phase, all the Si-H bonds are depassivated thus leading to saturation of interface states.

From Tab. 3.1 and Figure 3.4, it was inferred that NBTI is diffusion-limited while HCD is dominated by stage \( 4 \). The different slopes discussed above were explained by the RD model by considering the interface state density modeled as \( N_{\mathrm {it}}=\int N_{\mathrm {H}}(r,t)\mathrm {d}^{3}r \) with \( N_{\mathrm {H}}(r,t) \) being the coordinate-dependent density of H-atoms. Using the hydrogen diffusivity \( D_{\mathrm {H}} \), \( A_{\mathrm {d}} \) the degraded area of the device, and \( N_{\mathrm {H}}^{0} \) the hydrogen density at the interface, the \( N_{\mathrm {it}} \) distribution for HCD and NBTI are calculated as:

(3.7–3.7) \begin{eqnarray} N_{\mathrm {it}}^{\mathrm {NBTI}} & = & 1/A_{\mathrm {d}}\int _{0}^{(D_{\mathrm {H}}t)^{1/2}}N_{\mathrm {H}}^{0}\left [1-r/\left (D_{\mathrm {H}}t\right )^{1/2}\right ]A_{\mathrm {d}}\mathrm {d}r=(1/2)N_{\mathrm {H}}^{0}\left (D_{\mathrm {H}}t\right )^{1/2},\label {eq:RD-Nit}\\ N_{\mathrm {it}}^{\mathrm {HCD}} & = & \pi /2A_{\mathrm {d}}\int _{0}^{(D_{\mathrm {H}}t)^{1/2}}N_{\mathrm {H}}^{0}\left [1-r/\left (D_{\mathrm {H}}t\right )^{1/2}\right ]r\mathrm {d}r=(\pi /12A_{\mathrm {d}})N_{\mathrm {H}}^{0}\left (D_{\mathrm {H}}t\right ),\nonumber \end{eqnarray}

Equation 3.7 gives \( N_{\mathrm {it}}^{\mathrm {NBTI}}\sim \left (D_{\mathrm {H}}t\right )^{1/4} \) and \( N_{\mathrm {it}}^{\mathrm {HCD}}\sim \left (D_{\mathrm {H}}t\right )^{1/2} \).

Stage Phase time dependence of \( N_{\mathrm {it}} \)
1 Reaction limited (Si-H bond breakage) \( t^{1} \)
2 Hydrogen diffusion (no more \( N_{\mathrm {it}} \) generation) \( t^{0} \)
3 Diffusion-limited \( t^{1/4} \)
4 Hydrogen diffusion with infinite diffusion velocity \( t^{1/2} \)
5 Saturation (all Si-H bonds depassivated) \( t^{0} \)

Table 3.1: Different stages of the reaction-diffusion model [163, 66]

Although the different time slopes could be explained, the reaction-diffusion model has several failings. The framework assumes that both processes are diffusion limited implying that a quick recovery should be observed on removal of the stress. This, as suggested by recent data, is not true for both NBTI and HCD [164, 165, 166]. Interface state generation has been shown to be reaction limited and HCD, in most cases, does not show any recovery. Moreover, since the model does not rely on carrier transport, the spatial \( N_{\mathrm {it}} \) distribution cannot be obtained.