## 2 Compact Model and Physics  
### Reaction-Diffusion Framework  
Similar to trap generation caused by NBTI, the mechanism of interfacial trap generation due to hot carrier (HC) stress can also be explained by reaction-diffusion (R-D) theory. This theory suggests that hot carriers acquire sufficient kinetic energy from the high transverse electric field, leading to the breaking of Si-H bonds and the formation of dangling bonds. If the energy is high enough, the carriers may also be trapped in the oxide layer, resulting in oxide defects. The reaction process can be represented as Figure 1 [18], while \(k_{f}\) is the forward reaction rate (the rate of bond dissociation), and \(k_{r}\) is the reverse rate.  
The R-D model consists of two stages: (1) the trap generation process, which involves the formation of interfacial traps at the gate oxide layer interface under external electrical stress, and (2) the diffusion process, which describes the movement of the reaction products from the interface toward the gate electrode due to the density gradient. These two processes are represented by the following equations:  
\[\frac{dN_{\Pi}}{dt}=k_{F}(N_{0}-N_{\Pi})-k_{R}N_{H}N_{\Pi} \tag{1}\]  
\[\frac{dN_{H}}{dt}=D_{H}\
abla^{2}N_{H} \tag{2}\]  
\(N_{0}\) is the maximum trap concentration. \(N_{\Pi}\) is the interfacial trap concentration, and \(N_{H}\) is the hydrogen atom concentration. \(D_{H}\) is the diffusion coefficient, which indicates the rate of hydrogen diffusion and varies across different dielectric layers. The H diffusion process is shown in Figure 2.  
To solve Equations (1) and (2), there are two assumptions: (1) The trap generation rate is assumed to be negligible at the beginning of device stress, so \(\frac{dN_{\Pi}}{dt}=0\); (2) the dissociation and diffusion of each hydrogen atom are assumed to create a trap vacancy at the interface. The trap concentration can be derived from the hydrogen atom concentration, \(N_{H}\), within the oxide layer across the entire diffusion region. The integral of \(N_{H}\) is then computed to obtain the desired result.  
\[N_{\Pi}(t)=\frac{1}{WL}\int N_{H}(r,t)\,d^{3}r \tag{3}\]  
Figure 1: The reaction process of Si-H bond dissociation.  
Figure 2: A schematic of the diffusion of H generated by the bond solution at the interface.  
For the solution of (2), it is usually in the form of a Gaussian solution, that is,  
\[N_{H}(t,r)=\frac{N_{H0}}{(4\pi D_{H}t)^{3/2}}\exp\left(-\frac{|r|^{2}}{4D_{H}t}\right) \tag{4}\]  
Bringing (4) into the reaction-diffusion equation (Equation (1)) gives  
\[\frac{dN_{\text{IT}}}{dt}=k_{F}(N_{0}-N_{\text{IT}})-k_{R}N_{\text{IT}}\frac{N_ {H0}}{(4\pi D_{H}t)^{3/2}}\exp\left(-\frac{|r|^{2}}{4D_{H}t}\right) \tag{5}\]  
Substituting \(dN_{\text{IT}}/dt=0\) in the assumption into (5), and also assuming that the process of Si-H bond dissociation occurs only where the Si-SiO\({}_{2}\) interface is, i.e., the vector r = 0, the final form of the solution obtained through (3) and (5) is  
\[N_{\text{IT}}\approx N_{0}[1-\exp(-AR\ t^{m})] \tag{6}\]  
AR represents the aging rate, which indicates how fast the device ages under hot carrier stress. The threshold voltage degradation caused by HCI is proportional to the trap concentration, written as  
\[\Delta V_{\text{TH}}=\frac{q}{C_{\text{ox}}}N_{\text{IT}} \tag{7}\]  
So, electron mobility degradation under HC stress is written as (8). \(\mu_{eff}\) is the effective mobility. The empirical parameters \(\alpha\) and \(m\) come from [19].  
\[\mu=\frac{\mu_{\text{eff}}}{(1+\alpha N_{\text{IT}})^{m}} \tag{8}\]