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

3.5 The Bravaix Model

The group of Bravaix combined the Hess model and the Rauch and La Rosa approach [10, 134, 140]. The Bravaix model uses the concept of single- and multiple-carrier processes of defect generation within the energy-driven paradigm and evaluates the DF empirically. Three main modes of bond breakage are considered in the model, i.e. a single-carrier process, a multiple-carrier process, and electron-electron scattering which are then used to calculate the corresponding lifetimes. The single-carrier process is attributed to high energy carriers, while the drain current is low [169]. In this case, the lucky electron model can be used and the device lifetime can be estimated as $1/\tau _{\mathrm {SP}}\sim \left (I_{\mathrm {d}}/W\right )\left (I_{\mathrm {s}}/I_{\mathrm {d}}\right )^{m}$ . Here $I_{\mathrm {d}}$ , $I_{\mathrm {s}}$ are the drain and substrate currents, respectively, $W$ the device width, and $m$ the ratio of the powers in the impact ionization and interface state creation cross sections. The case with low carrier energies but high carrier density corresponds to the multiple-carrier process. For the MC-process, a truncated harmonic oscillator model is used to represent the breaking of the Si-H bond, as described in the Hess approach. The kinetic equation set which determines the bond passivation/depassivation rates is given by:

$(3.8) \{begin}{align} \frac {\mathrm {d}n_{0}}{\mathrm {d}t}= & P_{\mathrm {d}}n_{1}-P_{\mathrm {u}}n_{0}\nonumber \\ \frac {\mathrm {d}n_{i}}{\mathrm {d}t}= & P_{\mathrm {d}}(n_{\mathrm {i+1}}-n_{\mathrm {i}})-P_{\mathrm {u}}(n_{\mathrm {i}}-n_{\mathrm {i-1}})\\ \frac {\mathrm {d}n_{\mathrm {N_{l}}}}{\mathrm {d}t}= & P_{\mathrm {u}}n_{\mathrm {N_{l}-1}}-\lambda _{\mathrm {emi}}N_{\mathrm {it}}\left [H^{*}\right ],\nonumber \{end}{align}$

with $n_{\mathrm {i}}$ being the occupancy of the $i^{\mathrm {th}}$ oscillator level, $\left [H^{*}\right ]$ the concentration of mobile hydrogen atoms, while $N_{\mathrm {l}}$ labels the last bonded state. The rate of hydrogen release, $\lambda _{\mathrm {emi}}$ , is calculated as $\lambda _{\mathrm {emi}}=\nu _{\mathrm {emi}}\exp \left [-E_{\mathrm {emi}}/k_{\mathrm {B}}T_{\mathrm {L}}\right ]$ , where $\nu _{\mathrm {emi}}$ is the attempt frequency, and $E_{\mathrm {emi}}$ is the energy barrier between the last bonded state and the transport mode, see Figure 3.1. The bond excitation and decay rates ( $P_{\mathrm {u}}$ , $P_{\mathrm {d}}$ ) are written as:

$(3.9) \{begin}{align} P_{\mathrm {u}}= & \int I_{\mathrm {d}}\sigma \mathrm {d}\varepsilon _{\mathrm {e}}+\omega _{\mathrm {e}}\exp \left [-\hbar \omega /k_{\mathrm {B}}T_{\mathrm {L}}\right ]\label {eq:Bravaix-MPrate}\\ P_{\mathrm {d}}= & \int I_{\mathrm {d}}\sigma \mathrm {d}\varepsilon _{\mathrm {e}}+\omega _{\mathrm {e}},\nonumber \{end}{align}$

where $I_{\mathrm {d}}$ is the drain current. Since, the DF is calculated empirically, the acceleration integral (first term in Equation 3.9) is substituted by the term $S_{\mathrm {MP}}\left (I_{\mathrm {e}}/e\right )$ , where $S_{\mathrm {MP}}$ is a pre-factor [139]. This was an attempt to link the defect generation rate with the drain current. With a reasonable choice of the model parameters like $E_{\mathrm {emi}}$ , and $\hbar \omega$ the bond dissociation rates were accurately represented [169]. The device lifetime for the MC-process is given as $1/R_{\mathrm {MP}}$ , where $R_{\mathrm {MP}}$ is interface state generation rate due to low energy carriers given as:

$(3.10) \begin{equation} R_{\mathrm {MP}}=N_{0}\left [\frac {S_{\mathrm {MP}}\left (I_{\mathrm {d}}/e\right )+\omega _{\mathrm {e}}\exp \left [-\hbar \omega /k_{\mathrm {B}}T_{\mathrm {L}}\right ]}{S_{\mathrm {MP}}\left (I_{\mathrm {d}}/e\right )+\omega _{\mathrm {e}}}\right ]^{E_{\mathrm {B}}/\hbar \omega }\exp \left [-E_{\mathrm {emi}}/k_{\mathrm {B}}T_{\mathrm {L}}\right ]. \end{equation}$

The case with moderate carrier density is attributed to electron-electron scattering with lifetime $1/\tau _{\mathrm {EES}}\sim \left (I_{\mathrm {d}}/W\right )^{2}\left (I_{\mathrm {s}}/I_{\mathrm {d}}\right )^{m}$ . The quadratic relation is due to the electron-hole pairs, generated by impact ionization, not having enough energies to create interface states but are accelerated by electron-electron scattering to high energy levels where they can trigger bond dissociation. The major contribution of the Bravaix model was the prediction of the device lifetime by combining all the modes of interface state creation, see Figure 3.6.

$(3.11) \begin{equation} 1/\tau _{\mathrm {d}}=K_{\mathrm {SC}}/\tau _{\mathrm {SC}}+K_{\mathrm {EES}}/\tau _{\mathrm {EES}}+K_{\mathrm {MC}}/\tau _{\mathrm {MC}}, \end{equation}$

where $K_{\mathrm {SC}}$ , $K_{\mathrm {MC}}$ , $K_{\mathrm {EES}}$ are fitting parameters which represent contributions of each process and $\tau _{\mathrm {SC}}$ , $\tau _{\mathrm {MC}}$ , $\tau _{\mathrm {EES}}$ are the device lifetimes in the SC, MC, and EES driven regimes, respectively. However, the carrier DF is calculated empirically which leaves the entire physical picture behind HCD unclear. In order to understand the physics behind the HCD mosaic, ideally accurate carrier transport needs to be addressed via the solution of the Boltzmann transport equation.

Such an approach has been developed by Tyaginov et al. [25, 141], which is also used in this work (Section 4.4), which tries to link the entire physical mechanism consisting of carrier transport, defect generation, and degradation of the device characteristics. The Tyaginov method has been the most accurate in modeling HCD and its effects on different devices [44, 14, 147, 88]. The authors obtain the carrier DF by solving the Boltzmann transport equation using both Monte-Carlo and spherical harmonics expansion methods [171, 141]. The carrier acceleration integral is then calculated from the carrier DF which determines the rate of interface state generation. Once the bond breakage rates are determined, the trap density can be calculated. Since this model provides information about $N_{\mathrm {it}}$ profiles and the strong localization of HCD, it can predict device lifetimes and degradation characteristics.