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

4.4 Defect Generation

Once the DFs are determined from the solution of the BTE, the defect generation rates are calculated. The HCD model considers the single- and multiple-carrier bond breakage mechanisms as two pathways of the same reaction, self-consistently, which converts electrically passive Si-H bonds to active interface traps. As already explained, the bond breakage event induced by a solitary hot carrier in a single interaction with the Si-H bond corresponds to the single-carrier process, see Figure 4.3. Since the hydrogen nucleus is much heavier than an electron, the most plausible explanation of such a mechanism is the energy exchange between the incoming high-energy electron and one of the bonding electrons. The excited binding electron is excited into an antibonding state which exerts a repulsive force acting on the H atom which leads to hydrogen release. Thus, the SC-process is often referred to as anti-bonding (AB) process. The multiple-carrier process, on the other hand, is triggered by a series of colder carriers which subsequently bombard the Si-H bond, excite and eventually break it as depicted in Figure 4.3. Previously, the single- and multiple-carrier processes of bond dissociation were assumed to be independent [25]. The interface state density was calculated as a superposition of SC- and MC-induced contributions weighted with some probability coefficients [27, 25, 10]. However, in an updated version of the model, the Si-H bond depassivation process has been modeled considering all possible superpositions of the SC- and MC-mechanisms with the corresponding rates determined by the carrier DF, see Figure 4.5.

The carrier DFs are used to compute the carrier acceleration integral which determines the rates of both single- and multiple-carrier processes [15, 13, 14]:

$(4.29) \begin{equation} I_{\mathrm {SC/MC}}^{\mathrm {e/h}}=\int f(\varepsilon )g(\varepsilon )v(\varepsilon )\sigma _{\mathrm {0,SC/MC}}(\varepsilon -\varepsilon _{\mathrm {t\mathrm {h}}})^{p_{\mathrm {SC/MC}}}\mathrm {d}\varepsilon ,\label {eq:AI} \end{equation}$

where $f(\varepsilon )$ is the DF, $g(\varepsilon )$ the density-of-states, $v(\varepsilon )$ the carrier group velocity, and $\sigma _{0}(\varepsilon -\varepsilon _{\mathrm {th}})^{p}$ the reaction cross section. The integrand in Equation 4.29 can be interpreted as multiplication of the carrier flux having an energy in the range $\left [E+dE\right ]$ by the probability of bond-excitation by these carriers. The acceleration integrals for the SC- and MC-processes differ in the parameters $\varepsilon _{\mathrm {th}}$ (threshold energy) and $p$ (11 for the SC-process and 1 for the MC-process). In the case of the single-carrier bond breakage mechanism the reaction cross section is Keldysh-like [187, 171], with the threshold energy equal to the bond breakage activation energy. For the multiple-carrier mechanism, the bond is modeled as a truncated harmonic oscillator [188, 139] comprising of an energy ladder with $N_{\mathrm {l}}$ vibrational energy levels, as shown in Figure 4.5. Due to the multi-vibrational model, the multiple-carrier pathway is also called multi-vibrational excitation process. The threshold energy for the MC-process corresponds to the distance between the bond vibrational states ( $\hbar \omega$ ). The bond can gain/loose energy which leads to bond excitation/deexcitation. $E_{\mathrm {B}}$ is the energy required to reach the highest bonded energy level, while $E_{\mathrm {emi}}$ is the energy required for H emission from a Si-H bond that is in the highest bonded energy state. Thus, $E_{\mathrm {B}}+E_{\mathrm {emi}}$ forms the activation energy $E_{\mathrm {a}}$ for Si-H bond breakage. Conversely, the energy barrier for the passivation process, i.e. hydrogen forming the bond again, is $E_{p}$ .

Two vibrational modes of the Si-H bond have been identified, i.e. stretching (with $\hbar \omega =0.25\,$ eV and $E_{\mathrm {a}}=2.56\,$ eV) and bending ( $\hbar \omega =0.075\,$ eV and $E_{\mathrm {a}}=1.5\,$ eV) [140, 189]. In one of the most successful HCD models developed by the Bravaix group it is assumed that bond dissociation occurs via the bending mode with the corresponding values of $\hbar \omega$ and $E_{\mathrm {a}}$ [140]. The vibrational lifetime ( $\tau _{\mathrm {e}}$ ) chosen in that model is $10\,$ ps while its temperature dependence is not considered [190, 140]. Experimental investigations, however, suggest that H desorption from the $\mathrm {Si}\slash \mathrm {SiO_{2}}$ interface occurs with $E_{\mathrm {a}}=2.56\,$ eV [114]. This value is close to that typical for the stretching mode with the corresponding vibrational lifetimes at $T=25$ and $75\,^\circ$ C of $1.5$ and $1.3\,$ ns, respectively, see [191]. These parameter values are used in this work.

The rates of the bond excitation/deexcitation processes triggered by the multiple-carrier mechanism are:

$(4.30–4.31) \{begin}{align} P_{\mathrm {u}} & =I_{\mathrm {MC}}+\omega _{\mathrm {e}}\exp [-\hbar \omega /k_{\mathrm {B}}T_{\mathrm {L}}]\label {eq:bond-excitation-rate}\\ P_{\mathrm {d}} & =I_{\mathrm {MC}}+\omega _{\mathrm {e}},\label {eq:bond-deexcitation-rate} \{end}{align}$

where $\omega _{\mathrm {e}}$ represents the reciprocal phonon lifetime. The bond activation/passivation rates from the $i^{\mathrm {th}}$ level are:

$(4.32) \begin{equation} R_{\mathrm {\mathrm {a}/p},i}=\omega _{\mathrm {th}}\exp [-(E_{\mathrm {a}}-E_{i})/k_{\mathrm {B}}T_{\mathrm {L}}]+I_{\mathrm {SC},i},\label {eq:Ra,i} \end{equation}$

with $\omega _{\mathrm {th}}$ being the attempt frequency. The first term in Equation 4.32 corresponds to thermal activation of the H atom, while the second one incorporates the effect of hot carriers. Each bond level, $i$ , can contribute to the bond breakage mechanism as per Equation 4.32. A self-consistent consideration of the SC- and MC-processes means that first the bond can be excited by a series of cold carriers to a certain intermediate level, see Figure 4.5, and then dissociated by a solitary hot carrier which induces hydrogen release to the transport mode [136, 157, 188, 192, 13]. Note that in the case of a preheated bond the potential barrier which separates the bonded and transport states is reduced and the probability of finding a carrier which brings a bond breakage portion of energy is higher [15, 13, 14]. The system of rate equations for bond passivation/depassivation for all the energy levels in the potential well can be written as:

$(4.33) \{begin}{align} \frac {\mathrm {d}n_{\mathrm {0}}}{\mathrm {d}t}= & P_{\mathrm {d}}n_{1}-P_{\mathrm {u}}n_{\mathrm {0}}-R_{\mathrm {a,0}}n_{0}+R_{\mathrm {p,0}}N_{\mathrm {it}}^{2}\nonumber \\ \frac {\mathrm {d}n_{i}}{\mathrm {d}t}= & P_{\mathrm {d}}(n_{i+1}-n_{i})-P_{\mathrm {u}}(n_{i}-n_{i-1})-R_{\mathrm {a},i}n_{i}+R_{\mathrm {p},i}N_{\mathrm {it}}^{2}\label {eq:rate-equations}\\ \frac {\mathrm {d}n_{\mathrm {N_{\mathrm {l}}}}}{\mathrm {d}t}= & P_{\mathrm {u}}n_{\mathrm {N_{\mathrm {l-1}}}}-P_{\mathrm {d}}n_{\mathrm {N_{l}}}-R_{\mathrm {a,N_{l}}}n_{\mathrm {N}_{\mathrm {l}}}+R_{\mathrm {p,N_{l}}}N_{\mathrm {it}}^{2}\nonumber \{end}{align}$

where $n_{i}$ represents the occupation number of the $i^{\mathrm {th}}$ energy level. Due to the large difference between phonon and Si-H bond lifetimes, the time constants for oscillator transitions are much shorter than those typical for passivation/depassivation processes. Thus, the time scale hierarchy can be used to solve the system of equations 4.33 which reduces to a single equation [13]:

$(4.34) \begin{equation} \frac {\mathrm {d}N_{\mathrm {it}}}{\mathrm {d}t}=(N_{0}-N_{\mathrm {it}}\Re _{\mathrm {a}}-N_{\mathrm {it}}^{2}\Re _{\mathrm {p}}). \end{equation}$

Here $N_{0}$ is the density of pristine Si-H bonds present in the fresh device, while $\Re _{\mathrm {a/p}}$ are the cumulative bond activation/passivation rates. $\Re _{\mathrm {a}}$ can be obtained by summation of the bond breakage rates from each level weighed with its population number [13, 14, 15]:

$(4.35) \begin{equation} \Re _{\mathrm {a}}=\frac {1}{k}\sum _{r=0}R_{a,i}\left (\frac {P_{\mathrm {u}}}{P_{\mathrm {d}}}\right )^{i},\label {eq:Ra} \end{equation}$

while $\Re _{\mathrm {p}}$ is attributed to thermal activation and is represented by an Arrhenius term over a single energy level as:

$(4.36) \begin{equation} \Re _{\mathrm {p}}=\nu _{\mathrm {p}}\exp (-E_{\mathrm {p}}/k_{\mathrm {B}}T_{\mathrm {L}})\label {eq:Rp} \end{equation}$

Finally, the following analytic expression for the interface state density $N_{\mathrm {it}}$ is obtained:

$(4.37) \{begin}{align} N_{\mathrm {it}} & =\cfrac {1}{2\tau \Re _{p}}\cfrac {1-f(t)}{1+f(t)}-\cfrac {\Re _{\mathrm {a}}}{2\Re _{\mathrm {p}}},\label {eq:Nit}\\ f(t) & =\cfrac {1-\tau \Re _{\mathrm {a}}}{1+\tau \Re _{\mathrm {a}}}\exp \left (-t/\tau \right ),\nonumber \\ \cfrac {1}{\tau } & =2\sqrt {\Re _{a}^{2}/4+N_{0}\Re _{\mathrm {a}}\Re _{\mathrm {p}}},\nonumber \{end}{align}$

The activation energy for this reaction can be reduced by the interaction of the bond dipole moment $d$ with the oxide electric field $E_{\mathrm {ox}}$ [13, 15, 14]. The acceleration integral in Equation 4.29 then becomes:

$(4.38) \begin{equation} I_{\mathrm {SC/MC}}^{\mathrm {e/h}}=\int f(\varepsilon )g(\varepsilon )v(\varepsilon )\sigma _{\mathrm {0,SC/MC}}(\varepsilon -\varepsilon _{\mathrm {t\mathrm {h}}}+dE_{\mathrm {ox}})^{p_{\mathrm {SC/MC}}}\mathrm {d}\varepsilon \end{equation}$

The Si-H bond activation energy also varies due to the structural disorder at the $\mathrm {Si}\slash \mathrm {SiO_{2}}$ interface which can lead to substantial changes in the device degradation characteristics simulated with the model. The energy $E_{\mathrm {a}}$ is assumed to obey a normal distribution with a mean value $\left \langle E_{\mathrm {a}}\right \rangle$ of $2.56$ $\,$ eV and standard deviation $\sigma _{\mathrm {E_{\mathrm {a}}}}$ of 0.15 $\,$ eV [193, 194, 13].

Equation 4.37 allows evaluation of the interface state density for each position at the $\mathrm {Si}\slash \mathrm {SiO_{2}}$ interface and for each stress time step. $N_{\mathrm {it}}(x,t)$ profiles calculated from Equation 4.37 are then used as input for the device simulator to model the characteristics of the degraded device for each stress time step. The effect of charged interface traps is twofold: they perturb the local band-bending of the device and degrade the carrier mobility due to scattering of the charge carriers. The former effect is considered while solving the coupled Poisson and Boltzmann equations, while for the latter one an empirical expression for mobility degradation is used [195, 196]:

$(4.39) \begin{equation} \mu _{\mathrm {degr}}=\cfrac {\mu _{0}}{1+\alpha N_{\mathrm {it}}\mathrm {exp}(-r/r_{\mathrm {ref}})},\label {eq:mobility} \end{equation}$

where $\mu _{0}$ is the mobility in the virgin device, $r$ is the shortest distance from this local point to the interface, $r_{\mathrm {ref}}$ = 10 $\,$ nm defines the maximum range within which a carrier is still influenced by the field of the trapped charge, while the parameter $\alpha =10^{-13}\,$ cm $^{2}$ determines the magnitude of the effect.

Parameter	Value	Comments
$\tau _{\mathrm {n/p}}$	$\tau _{\mathrm {n}}=0.35$ $\,$ ps, $\tau _{\mathrm {p}}=0.4$ $\,$ ps	Energy relaxation time
$\sigma _{0}$	$\sigma _{0,\mathrm {SC}}=$ $5\times 10^{\text {\textminus }18}\,$ cm $^{2}$ , $\sigma _{0,\mathrm {MC}}=$ $5\times 10^{\text {\textminus }19}\,$ cm $^{2}$	Pre-factor in reaction cross section
$p$	$p_{\mathrm {SC}}=11$ , $p_{\mathrm {MC}}=1$	Exponent in reaction cross section
$\left \langle E_{\mathrm {a}}\right \rangle$	$2.56$ $\,$ eV	Si-H bond activation energy
$\sigma _{\mathrm {Ea}}$	$0.15$ $\,$ eV	Standard deviation of activation energy
$\hbar \omega$	$0.25$ $\,$ eV	Distance between bond vibrational states
$\omega _{\mathrm {e}}$	$\omega _{\mathrm {e}}=1/\tau$ , $\tau =1.5$ $\,$ ns	Reciprocal phonon lifetime
$\omega _{\mathrm {th}}$	$5\times 10^{6}$ $\,$ s $^{-1}$	Attempt frequency for thermal activation
$\nu _{\mathrm {p}}$	$10^{8}$ $\,$ s $^{-1}$	Passivation attempt frequency
$E_{\mathrm {p}}$	$1.5$ $\,$ eV	Passivation energy
$N_{0}$	$1\times 10^{11}$ $\,$ cm $^{-2}$	Density of precursors