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

3.2 The Penzin Model

Penzin et al. tried to link the microscopic mechanisms suggested by the Hess model with the degradation characteristics of the device [22]. Within this model, the Si/SiO (math image) interface is considered as a capacitor. Thus, Si-H bond dissociation leaving behind a depassivated bond and a charged hydrogen atom leads to an increased electric field in the capacitor. Due to this, the potential barrier between bonded and transport states increases. The bond-dissociation process is described by the kinetic equation:

$(3.5) \begin{equation} \frac {\mathrm {d}n}{\mathrm {d}t}=-kn+\gamma (N_{0}-n), \end{equation}$

with $n$ being the concentration of passivated Si-H bonds, $k$ the bond breakage rate, $\gamma$ the bond passivation rate, and $N_{0}$ the total concentration of Si-H bonds in the fresh device. The bond breakage rate $k$ is given by $k=k_{0}\exp \left (-E_{\mathrm {a}}/k_{\mathrm {B}}T_{\mathrm {L}}\right )k_{\mathrm {HC}}$ where $k_{0}$ is the attempt frequency. The hot-carrier acceleration factor $k_{\mathrm {HC}}$ is defined as $k_{\mathrm {HC}}=1+\delta _{\mathrm {HC}}\left |I_{\mathrm {HC}}\right |^{\rho _{\mathrm {HC}}}$ where $\rho _{\mathrm {HC}}$ and $\delta _{\mathrm {HC}}$ are fitting parameters, and $I_{\mathrm {HC}}$ is the local hot-carrier current [22]. With such a formalism for the defect generation kinetics, the Penzin model could be used for TCAD device simulations.

As per the Penzin model, the activation energy increases with depassivation of the interfacial Si-H bonds and is given as:

$(3.6) \{begin}{align} E_{\mathrm {a}=} & E_{\mathrm {a}}^{0}+\delta \left |E\right |^{\rho }+\beta k_{\mathrm {B}}T_{\mathrm {L}}\ln \frac {N_{0}-n}{N_{0}-n^{0}}\label {eq:Penzin-activation-energy}\\ \beta = & 1+\beta _{\perp }E_{\perp }\nonumber \{end}{align}$

where $E_{\mathrm {a}}^{0}$ is the activation energy when there are no mobile hydrogen atoms, and $n^{0}$ the concentration of pre-existing mobile hydrogen atoms. The increase in the capacitor electric field normal to the interface, due to bond dissociation is denoted as $E_{\perp }$ . The activation energy can also change due to stretching or squeezing the bond by an external electric field. This is represented by the term $\delta \left |E\right |^{\rho }$ in the expression for the activation energy, Equation 3.6.

The $N_{\mathrm {it}}$ concentrations simulated using the Penzin approach are shown in Figure 3.3. The simulated results compare quite well with the experimental ones. Similar to the Hess approach, the Penzin model considers the activation energy distribution which results in the sublinear slope seen in Figure 3.3. However, within this model, the carrier transport was not properly addressed and the $N_{\mathrm {it}}(x)$ profiles were not determined, except for some cumulative $N_{\mathrm {it}}$ shown in Figure 3.3. Also, the characteristics of the degraded device cannot be simulated with this approach.