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The central foundations of the following Section is the current state–of–the–art understanding of charge trapping at pre–existing defect sites, accurately described within the framework of the 4–state NMP model by Grasser *et. al* [6, 7, MJJ7]. Its development is heavily based on the pioneering work of Kirton and Uren [5], who investigated noise due to single defects in the context of NMP theory and the subsequent application of Tewksbury to model the threshold voltage drift in MOSFETs [126]. Furthermore, the original formulation was inspired by *irradiation
damage* in SiO\(_2\), which has been explained by taking the neutral oxygen vacancy and its two (meta–) stable charged configurations into account, known as \(E^\prime _\gamma \) and \(E^\prime _\delta \) [127, 128].

The dynamics of charge exchange between a defect in the gate stack and the carrier reservoir of a MOSFET is determined by a combination of field – via an applied **bias** – accelerated nonradiative multiphonon transitions
together with **temperature** activated structural reconfigurations. Its universal formulation is independent of the actual defect structure and relies on the description using one–dimensional **potential energy
curves** (PECs). Their knowledge reveals the respective barriers which determine the transition rates and hence the dynamics of charge trapping. As already outline in the Introduction of this Chapter, transitions can be split
into thermally activated structural relaxations *within* a PEC and charge transfer reactions *across* different PECs, see Fig. 2.3.

**Thermal transitions** can be accounted for using classical transition state theory (TST) [64, 129–131] which assumes the initial state to be in thermal equilibrium with a distribution given
by the Boltzmann form. The transitions state (TS) between the initial and final state represents a bottleneck for the reaction to occur with a barrier high enough that the time between escape events into the
final configuration is longer than the (vibrational) relaxation time. Furthermore, an essential factor in TST is a negligible back flow once the TS has been crossed which implies that the system spends an extended time in the final
state compared to being in the vicinity of the transition point. According to TST the rate is given by the probability that the system reaches the TS multiplied with the flux through the TS, i.e. the rate crossing from the initial
state I to the final state F

\begin{equation} k^\mathrm {TST}=\frac {\int _\mathrm {TS}\mathrm {e}^{-V(\mathbf {q})/\kb T}\mathrm {d}\mathbf {q}}{\int _\mathrm {I}\mathrm {e}^{-V(\mathbf {q})/k_\mathrm {B}T}\mathrm {d}\mathbf {q}}v_\perp . \end{equation}

The average velocity \(v_\perp \) in the direction perpendicular to the TS surface can be calculated from the Maxwell distribution

\(\seteqnumber{0}{2.}{2}\)\begin{equation} v_\perp =\sqrt {\frac {\kb T}{2\pi \mu _\perp }}, \end{equation}

where \(\mu _\perp \) is the effective mass for the motion across the TS due to the movement of the involved atoms. The general rate for a multidimensional TST is given as

\(\seteqnumber{0}{2.}{3}\)\begin{equation} k^\mathrm {TST}=\sqrt {\frac {\kb T}{2\pi \mu _\perp }}\frac {\mathcal {Z}_\mathrm {TS}}{\mathcal {Z}_\mathrm {I}}, \end{equation}

with \(\mathcal {Z}\) being the integrals of the Boltzmann factors over the specified regions TS and I in configuration space, often referred to as partition functions or configuration integrals, respectively. Due to the exponential dependence on the barrier height \(E_\mathrm {B}\), the escape rates are determined by the path with the lowest possible barrier, also known as the minimum energy path (MEP). Choosing the MEP for the reaction coordinate (RC) where the transition state as a first order saddle point on the energy surface allows one to use harmonic transition state theory [64, 129]. The total energy of the system can be expanded up to second order around the initial minimum \(\mathbf {q}_\mathrm {I}\) and the saddle point \(\mathbf {q}_\mathrm {TS}\) which allows for an analytic evaluation of \(\mathcal {Z}_\mathrm {I}\) and \(\mathcal {Z}_\mathrm {TS}\). The harmonic TST yields [132, 133]

\(\seteqnumber{0}{2.}{4}\)\begin{equation} k^\mathrm {TST}=\frac {1}{2\pi \sqrt {\mu _\perp }}\sqrt {\frac {\prod _{i=1}^{3N}k_{\mathrm {I},i}}{\prod _{i=1}^{3N-1}k_{\mathrm {TS},i}}}\mathrm {e}^{-E_\mathrm {B}/\kb T}, \label {eq:chap2_tst_k} \end{equation}

with \(k_{\mathrm {I},i}\) and \(k_{\mathrm {TS},i}\) are the eigenvalues of the Hessian matrix. Since only vibrational contributions to the partition functions \(\mathcal {Z}\) are to be considered, (2.5) can be written in terms of the angular frequencies \(\omega =\sqrt {k/\mu }\)

\(\seteqnumber{0}{2.}{5}\)\begin{equation} k^\mathrm {TST}=\frac {1}{2\pi }\frac {\prod _{i=1}^{3N}\omega _{\mathrm {I},i}}{\prod _{i=1}^{3N-1}\omega _{\mathrm {TS},i}}\mathrm {e}^{-E_\mathrm {B}/\kb T}. \label {eq:chap2_tst_w} \end{equation}

In the denominator the imaginary frequency corresponding to the unstable vibrational mode at the saddle point is left out from the product. Finally, (2.6) can be interpreted as a prefactor, the so–called *attempt frequency*, which is related to the number of attempts per unit
time to overcome the barrier, and an activation term associated with the barrier height. The multi–dimensional transition rate can be simplified by assuming a single reaction coordinate \(q\) and a one–dimensional PEC \(V(q)\).
In this case (2.6) reduces to the well known empirical Arrhenius law

\begin{equation} k^\mathrm {TST}=\frac {\omega _0}{2\pi }\mathrm {e}^{-E_\mathrm {B}/\kb T}. \label {eq:chap2_arrhenius} \end{equation}

In the context of charge trapping at oxide defects (2.7) is used to describe thermally activated structural reconfigurations within a single PEC.

Contrary to thermal transitions, **nonradiative multiphonon transitions** involve the capture or emission event of a charge which is characterized by a change of the adiabatic potential energy curve (assuming again a
one–dimensional reaction coordinate) [MJJ7, 134–138]. Such phonon assisted reactions are associated with a deformation and subsequent relaxation of the defect site [63]. The charge exchange process can be assumed as instantaneous and preferably occurs *at* or *near* the intersection point of the involved
PECs [10, 11, MJJ7, 63].

Using a first order perturbation theory approach the rate \(k_{i\alpha ,j\beta }\) for a NMP transition from the inital state \(\ket {\Phi _i\otimes \eta _{i\alpha }}\) to the final state \(\ket {\Phi _j\otimes \eta _{j\beta }}\) is given by FGR [139]

\(\seteqnumber{0}{2.}{7}\)\begin{equation} \begin{aligned} k_{i\alpha ,j\beta }&=\frac {2\pi }{\hbar }\big |M_{i\alpha ,j\beta }\big |^2\delta (E_{i\alpha }-E_{j\beta }),\\ M_{i\alpha ,j\beta }&=\matrixel *{\eta _{i\alpha }}{\!\mel *{\Phi _i}{\hat {H}^\prime }{\Phi _j}}{\eta _{j\beta }}. \end {aligned} \label {eq:chap2_nmp_fgr} \end{equation}

Here, \(\hat {H}^\prime \) is the perturbation operator, \(i\) and \(j\) denote the electronic states and \(\alpha \) and \(\beta \) are the vibrational states. The delta function ensures the conservation of energy for the transition, where \(E_{i\alpha }\) and \(E_{j\beta }\) include the phonon as well as electronic energies. Invoking the Born Oppenheimer (more precisely the Born Huang) or adiabatic approximation [58, 59] together with the Franck–Condon principle (FCP) [140–142] the perturbation Hamiltonian can be split into an electronic \(\hat {H}^\prime _\mathrm {el}\) and a phonon \(\hat {H}^\prime _\mathrm {phon}\) part which allows to rewrite the matrix element \(M_{i\alpha ,j\beta }\) as

\(\seteqnumber{0}{2.}{8}\)\begin{equation} M_{i\alpha ,j\beta }=\mel {\Phi _i}{\hat {H}^\prime _\mathrm {el}}{\Phi _j}\braket {\eta _{i\alpha }}{\eta _{j\beta }}. \label {eq:chap2_nmp_mel} \end{equation}

The BOA decouples the electronic and nuclear wave functions, while the Franck–Condon principle (FCP) splits vibronic transitions, the simultaneous occurrence of electronic and vibrational transitions, into
individual products due to their different timescales. Therefore, (2.9) can be interpreted as the product
\(A_{i,j}f_{i\alpha ,j\beta }^\mathrm {LSF}\), where \(A_{i,j}\) describes the instantaneous electronic excitation and is the electronic matrix element and \(f_{i\alpha ,j\beta }^\mathrm {LSF}\), the so–called
*lineshape function* (LSF), considers the vibrational interactions occurring during the lattice reconfigurations. In order to capture the effect of charge trapping at oxide defects under elevated temperatures with NMP theory,
all vibrational modes of a potential representing one electronic state have to be taken into account [11, MJJ7, 143]. The total transition rate \(k_{i,j}\) is the thermal average of all
individual rates \(k_{i\alpha ,j\beta }\) across the canonical ensemble. The partial rates must be averaged over all initial states \(\alpha \) weighted with a Boltzmann factor and all final states have to be summed over

\begin{equation} k_{i,j} = \underset {\alpha }{\mathrm {average}}\big (\sum _\beta k_{i\alpha ,j\beta }\big ). \end{equation}

Finally, the total NMP transition rate which describes a charge transfer process can be written as:

\(\seteqnumber{0}{2.}{10}\)\begin{equation} \begin{aligned} f_{i,j}^\mathrm {LSF}\widehat {=}&\underset {\alpha }{\mathrm {ave}}\big (\sum _\beta \big |\braket {\eta _{i\alpha }}{\eta _{j\beta }}\big |^2\delta (E_{i\alpha }-E_{j\beta })\big ),\\ k_{i,j}=& A_{i,j}f_{i,j}^\mathrm {LSF},\\ A_{i,j}\widehat {=}& \frac {2\pi }{\hbar }\big |\mel {\Phi _i}{\hat {H}^\prime _\mathrm {el}}{\Phi _j}\big |^2. \end {aligned} \label {eq:chap2_nmp_rate_contributions} \end{equation}

Regarding physical aspects, the electronic matrix element \(A_{i,j}\) defines the coupling strength between the involved electronic wavefunctions and the LSF is the overlap integral of the initial and final vibrational eigenmodes, see Fig. 2.4. In the context of charge trapping in MOSFETs, \(A_{i,j}\) determines the interaction of a band state in the silicon conduction/valence band and the defect wavefunction in the gate stack. Due to the strong localization of the defect wavefunction, \(A_{i,j}\) can be reasonably well approximated using the Wentzel–Kramers–Brillouin (WKB) approximation [11, MJJ7]. Calculating the LSF requires the quantum mechanical vibrational eigenstates of the PECs. Analytical solutions are only available for simple potentials, such as the harmonic approximation, and even then their overlaps need to be calculated using recurrence schemes to obtain the necessary accuracy [144, 145]. For more generic energy profiles \(f_{i,j}^\mathrm {LSF}\) solely relies on numerical results which is unfeasible for TCAD simulations. Therefore, a harmonic PEC together with the classical limit of the LSF is mostly used in such applications. In the classical limit the LSF is a Dirac peak at the intersection point of the two PECs. However, also in the quantum mechanical picture using the overlap integrals of the vibrational wavefunctions, mainly contributions in the direct vicinity of the classical crossing point determine the total LSF, compare with Fig. 2.4, which justifies this seemingly crude approximation.

Oxide defects in MOSFETs and the associated charge trapping involves more than just two isolated states. Charge capture and emission events can occur from (to) a **whole band of states**, namely the valence and
conduction bands of the silicon substrate (as well as of the gate material) [8, MJJ7, 146]. The present work investigates hole traps in a pMOSFET, i.e. a defect
which can be in its neutral or positively charged (meta–) state with the transitions \(0\Leftrightarrow +\). Such defects mainly interact with carriers in the valence band (VB), assuming pMOSFETs and
negative bias conditions, and thus the invoked transitions are governed by trapping a hole from the VB (or emitting an electron, which is equivalent). Therefore, in the following the energy minimum of the positively charged state is
chosen as the reference energy, \(V_\mathrm {+,min}\,\widehat {=}\,\SI {0}{eV}\). It is convenient to introduce the electrostatic trap level \(E_\mathrm {T}\) of the neutral configuration with respect to the VB edge in the
absence of an electric field as \(E_\mathrm {T}\,\widehat {=}\,V_{0,\mathrm {min}}-E_\mathrm {V}\)^{5}. Interacting with a continuous spectrum of carriers in the valence or conduction band, the energy \(E\) of
the charge in the initial state is not uniquely defined, which is represented by a shifted charged (positive) PEC, see Fig. 2.5. This is also reflected in the NMP transition rate, which now depends on the reservoir energy \(E\) and the trap level
\(E_\mathrm {T}\)

\begin{equation} k_{i,j}=A_{i,j}(E,E_\mathrm {T})f_{i,j}^\mathrm {LSF}(E,E_\mathrm {T}). \end{equation}

Integration of all band states yields the final set of transition rates:

\(\seteqnumber{0}{2.}{12}\)\begin{equation} \begin{aligned} k_{0,+}^\mathrm {VB}&=\int _{-\infty }^{E_\mathrm {V}}g_p(E)f_p(E)A_\mathrm {0,+}(E,E_\mathrm {T})f_\mathrm {0,+}^\mathrm {LSF}(E,E_\mathrm {T})\mathrm {d}E\\ k_{+,0}^\mathrm {VB}&=\int _{-\infty }^{E_\mathrm {V}}g_p(E)(1-f_p(E))A_\mathrm {+,0}(E,E_\mathrm {T})f_\mathrm {+,0}^\mathrm {LSF}(E,E_\mathrm {T})\mathrm {d}E\\ k_{0,+}^\mathrm {CB}&=\int _{E_\mathrm {C}}^{\infty }g_n(E)(1-f_n(E))A_\mathrm {0,+}(E,E_\mathrm {T})f_\mathrm {0,+}^\mathrm {LSF}(E,E_\mathrm {T})\mathrm {d}E\\ k_{+,0}^\mathrm {CB}&=\int _{E_\mathrm {C}}^{\infty }g_n(E)f_n(E)A_\mathrm {+,0}(E,E_\mathrm {T})f_\mathrm {+,0}^\mathrm {LSF}(E,E_\mathrm {T})\mathrm {d}E \end {aligned} \label {eq:chap2_nmp_rates} \end{equation}

Here, \(g_p\) (\(g_n\)) is the density of states of the valence band (VB) and conduction band (CB), respectively, and \(f_p\) (\(f_n\)) denotes the EDF, how carriers are distributed over energy. For the reaction \(0\Rightarrow +\) a filled (hole) state in the VB, or an empty (electron) state in the CB, must be taken into account. The opposite applies for the reverse transition \(+\Rightarrow 0\).

The last component is the **interaction of a defect with the electric field** present in the oxide. Applying a gate bias imposes an electric potential inside the oxide which changes the defect’s energy and hence its trap
level \(E_\mathrm {T}\) [61, 144]. A simple model, neglecting all
trapped charges inside the oxide, predicts a linear relation between the oxide field Fox and the defect’s depth

\begin{equation} \Delta _\mathrm {S}=qx_\mathrm {T}F_\mathrm {ox}. \end{equation}

This additional energy shifts the trap level accordingly

\(\seteqnumber{0}{2.}{14}\)\begin{equation} E_\mathrm {T,eff}=E_\mathrm {T}+\Delta _\mathrm {S}, \label {eq:chap2_nmp_field} \end{equation}

and its effect can be seen in Fig. 2.5.

For the sake of completeness, it is worth noting that the discussed framework exhibits a few **shortcomings** due to the approximations used owing to the broad distribution of defect parameters associated with the
amorphous nature of SiO\(_2\). The subsequently discussed advances are performed (mainly) in the context of crystalline materials where defect structures and their characteristics are well defined. Each of the discussed effects and
its extension of the NMP model described above would make the corresponding DFT calculations prohibitively expensive and even the TCAD simulations unfeasible.

First, the theory derived above relies on the calculated (and approximated) PECs and their crossing points. Therefore, the rate of a charge transfer process is determined by the classical energy barriers and the probability to reach a
certain value of the reaction coordinate characterized a vibrational mode. This classical approximation is usually referred to as *Marcus theory* [147]. However, recent progress of computational methods for atomistic calculations allows one to calculate the *electron–phonon coupling* which is beyond
the BOA [148–152], see (2.8) and (2.9).
Usually the additional terms are referred to as nonadiabatic terms and contain derivatives of the electronic wavefunction with respect to the nuclear coordinates. They describe the coupling of electronic states with respect to the
dynamics of the lattice atoms and are called electron–phonon coupling.

Second, quite recently it has been demonstrated that the electronic matrix element, the first term in (2.9) and
\(A_{i,j}\) in (2.11), respectively, can be calculated explicitly. Using an atomistic Si/SiO\(_2\)
model, however, it is not possible to directly calculate a positively charged defect below the Fermi level (or a negatively charged defect above \(E_\mathrm {F}\)), and hence the involved wavefunctions. Nevertheless, the authors
in [153, 154] showed that by applying an external perturbation, e.g. an electric field,
the respective band and *neutral* defect wavefunctions can be shifted with respect to each other until an avoided crossing is observed. The minimum energy gap between both states is directly proportional to the electronic
coupling strength. However, this approach relies upon the assumption that the defect wavefunction does *not* substantially change upon the charge capture process^{6}. A more rigorous approach is provided by the
method of constrained density functional theory (CDFT) which allows one to directly construct the Hamiltonian in a charge localized *diabatic* basis, as was shown in [156–158]. However, the studies attempted to model the coupling of degenerated
defects within a bulk material and not across an interface system.

Furthermore, the effect of an applied electric field is accounted for only via a linear response upon the electric potential inside the oxide thereby shifting the energy position of the defect, see (2.15). On the other hand, the interaction with the defects’ dipole moment, which potentially alters the PECs and, hence, the energy barriers [159, 160], is neglected. Contrary to recent investigations on symmetric systems where the migration or dissociation pathway follows a certain direction, the situation is much more complicated for amorphous systems like SiO\(_2\) and which results in enormous computational efforts.

Last but not least, the utilized methods to calculate the MEP, within a PEC or across different PECs, obey classical mechanics for the nuclei. Over the past decade, however, methods have been developed which allowed one to take nuclear quantum effects into account which opened new insights into chemical processes, particularly for light atoms such as hydrogen [161]. Practical applications, such as quantum transition state theories and ring polymer molecular dynamics methods have shown that they yield accurate rates at low temperatures where the effects of tunneling dominate [162–165]. Even though defects in SiO\(_2\) are associated with hydrogen where tunneling effects are potentially important, the aforementioned methods rely on an extensive sampling of the free energy surface which is beyond computational possibilities due to the variety of defect structures encountered in amorphous SiO\(_2\).

Note that each of these advances and dedicated studies is a very challenging research field on its own and virtually impossible to be applied within the description of amorphous structures exhibiting a broad distribution of defect parameters using currently available computational resources.

^{5} Conversely, the conduction band (CB) edge is used for electron traps \(E_\mathrm {T}\,\widehat {=}\,E_\mathrm {C}-V_{0,\mathrm {min}}\).

^{6} The methodology would break down e.g. for intrinsic electron traps (self–trapped polarons) in SiO\(_2\) [155] which dramatically
change its wavefunction upon charge capture.

Striking evidence for the 4–state NMP formulation has been found using TDDS [3] and the characteristics of single defects. Based on the defect’s
emission behaviour, two different types of oxide traps can be identified: *fixed positive charge traps*, which feature a rather constant emission time τe over \(\Vg \), and *switching traps*, which show a rapid
decrease of τe towards low gate voltages [4, 7]. A consistent and unified
approach to explain both defect types is given by the 4–state NMP model [4, 7, 166], see Fig. 2.6. It accounts for two *stable* states representing the neutral and charged defect (\(1\) and \(2\)) as well
as two *metastable* configurations for both charge states (\(1^\prime \) and \(2^\prime \)). While charging is assumed to always proceed via the metastable (positive) state \(2^\prime \) (\(1\Rightarrow 2^\prime
\Rightarrow 2\)), the discharging dynamics are governed by two different pathways. Defect configurations with an unfavourable energy barrier via state \(1^\prime \) will possess a constant emission time determined by the (bias
independent) thermal barrier \(\varepsilon _{2,2^\prime }\). On the other hand, if the lowest energy path to emit a charge is given by \(2\Rightarrow 1^\prime \Rightarrow 1\), discharging is governed by the bias
dependent barrier \(\varepsilon _{2,1^\prime }\). Since the energetic alignment of the PECs, hence their crossing point, depends on the electric field Fox and the VB (CB) state energy \(E\), the favourable emission
pathway potentially changes from \(2\Rightarrow 2^\prime \Rightarrow 1\) to \(2\Rightarrow 1^\prime \Rightarrow 1\) and vice versa with applied recovery conditions.

The proposed four states also impose very stringent limits on the atomistic defect configuration. In order to be compatible with the 4–state model, the microscopic defect candidate needs to exhibit a stable and a metastable
configuration in each charge state which are connected via structural relaxations. The oxygen vacancy (OV) has been suggested to be the main cause of BTI for the last decades [167–170]; however, recent publications have shown that its charge transition levels are too far
below the silicon VB which rules the OV out as a BTI active trapping site [62]. On the other hand, an inevitable connection between hydrogen and BTI
has been demonstrated [17, 18, 171, 172], suggesting hydrogen based defects such as the hydrogen bridge (HB) and (or) the hydroxyl–E\(^\prime \) center (HE\(^\prime \)) as potential defect structures. Recent studies employing *ab initio* calculations have
indicated that the respective energy levels of the HB and the HE\(^\prime \) are much closer to the Si VB which renders them compatible with experimental BTI investigations [62, 173–175].
Nevertheless, searching for a suitable atomistic defect structure in SiO\(_2\) (SiON) as well as in high–\(\kappa \) materials such as HfO\(_2\) is still an active research field. The four states proposed above can be translated into
a one dimensional reaction coordinate (RC) using the parabolic approximation for the PECs which is shown in Fig. 2.6. The shape of the respective quantum harmonic oscillators together with their intersection points is uniquely
defined by the following relations

\begin{equation} \begin{aligned} \Delta E_{i,j}&=E_{j,\mathrm {min}}-E_{i,\mathrm {min}},\\ S_{i,j}&=\mathcal {S}_{i,j}\hbar \omega =c_i(q_j-q_i)^2,\\ R_{i,j}^2&=\frac {c_i}{c_j}, \end {aligned} \label {eq:chap2_nmp_parameters} \end{equation}

where \(S_{i,j}\) is the relaxation energy, \(R_{i,j}\) describes the ratio of the curvatures and \(\Delta E_{i,j}\) is the energy difference. The crossing points of the parabolas determine the classical energy barriers \(\varepsilon _{i,j}\) which can be calculated as

\(\seteqnumber{0}{2.}{16}\)\begin{equation} \begin{aligned} \varepsilon \idx &=\frac {S\idx }{R\idx ^2-1}\big (1-R\idx \sqrt {\frac {S\idx +\Delta E_{j,i}(R\idx ^2-1)}{S\idx }}\big )\quad &\mathrm {for}\quad &R\idx \neq 1,\\ \varepsilon \idx &=\frac {(S\idx +\Delta E_{j,i})^2}{4S\idx }\quad &\mathrm {for}\quad &R\idx = 1. \end {aligned} \label {eq:chap2_nmp_barriers} \end{equation}

Both sets of equations, (2.16) and (2.17), in conjunction with the parameters highlighted in Fig. 2.6, unambiguously define the 4–state model with all its barriers \(\varepsilon \idx \). The NMP transition rates \(1\Leftrightarrow 2^\prime \) and \(1^\prime \Leftrightarrow 2\) as well as the thermal transitions \(2^\prime \Leftrightarrow 2\) and \(1\Leftrightarrow 1^\prime \) can be calculated using (2.13) (NMP) and (2.7) (thermal). The resulting rates \(k\idx \) determine the occupation probability \(P_i\) of the system to be in state \(i\) which is given by the master equation

\(\seteqnumber{0}{2.}{17}\)\begin{equation} \frac {\dd P_i(t)}{\dd t}=k_{j,i}P_j(t)+k_{k,i}P_k(t)-(k_{i,j}+k_{i,k})P_i(t), \end{equation}

with the initial condition \(P_{i=1}(0)=1\) and \(P_{i\neq 1}(0)=0\). The time to reach the stable configuration \(2\), describing a hole capture process, for the first time is called *first passage time* (FPT). The
transitions between the states \(1\) and \(2\), however, proceed via the metastable states \(1^\prime \) or \(2^\prime \). The FPTs in such an effective 3–state system which describes the transitions \(1\Leftrightarrow 2\)
via the states \(1^\prime \) or \(2^\prime \) is given by the following approximation^{7}

\begin{equation} \tau _\mathrm {c,FPT}^i=\frac {k_{1,i}+k_{i,1}+k_{i,2}}{k_{1,i}+k_{i,2}}\quad \quad \quad \tau _\mathrm {e,FPT}^i=\frac {k_{i,1}+k_{i,2}+k_{2,i}}{k_{i,1}+k_{2,i}}, \label {eq:chap2_fpt} \end{equation}

with \(i\) being the intermediate state. The final capture and emission times τc and τe within the 4–state model which can be compared to experimental results are

\(\seteqnumber{0}{2.}{19}\)\begin{equation} \tau _\mathrm {c}=\big (\sum _i\frac {1}{\tau _\mathrm {c,FPT}^i}\big )^{-1}\quad \quad \quad \tau _\mathrm {e}=\big (\sum _i\frac {1}{\tau _\mathrm {e,FPT}^i}\big )^{-1}. \end{equation}

The 4–state NMP model is designed to accurately describe the degradation due to BTI as well as (anomalous) random telegraph noise. It takes the interaction of the defect with charge carriers in the substrate’s valence and
conduction band into account and assumes their occupation probability to be given by the FD distribution. During the worst–case conditions for BTI, namely \(\Vg \gg \Vdd \) and \(\Vd =\SI {0}{V}\), charge carriers are
indeed in thermal equilibrium, which validates this assumption. Within this approach, the total transition rates are mainly dominated by the interaction with the valence band due to the negligible (intrinsic) concentration of
electrons in the CB^{8}, see Fig. 2.7. This model variant will be termed **equilibrium 4–state
model**, **NMP\(_\mathrm {\mathbf {eq.}}\)**, in the following. Further simplifications can be made by considering the set of equations given in (2.13). The integrals are dominated by the product \(f_\mathrm {p}(E)f_{i,j}^\mathrm {LSF}(E,E_\mathrm {T})\) which
features the biggest contribution close the band edges as shown in Fig. 2.7. It can, therefore, be inferred that the
rates \(k\idx \) and the individual terms only depend on the respective band edges which can be factored out from the integral. Using the classical limit of the LSF together with the fact that the integral over the product
\(g_p(E)f_p(E)\) equals the carrier concentration \(p\), finally yields an analytical expression for the rate \(k\idx \) which exponentially depends on the barrier. This simplification is termed the *band edge approximation*.
However, the quality of this approximation strongly depends on the curvature and the corresponding intersection points of the involved parabolas. For example, if the neutral state does *not* intersect the parabola which
represent the valence band *edge* at a certain bias condition, the band edge approximation breaks down and fails to describe the corresponding time constants. Although excited carriers in the valence band can still interact
with the defect, this effect is not included in the original formulation of the band edge approximation. However, a recently presented extension overcomes this limitations, see [146, MJJ8, MJC14].

^{7} This approximation assumes that there is only one *active* capture or emission path for a given bias condition, which is typically fulfilled.

^{8} This, of course, is only true for hole traps in pMOSFETs, where a negative gate voltage is applied. For positive bias temperature instability (PBTI) in nMOSFETs electron traps
mainly interact with the Si CB.

The NMPeq. model described above is not intended or conceived to simulate the behaviour of oxide defects in full {\(V_\mathrm {G},V_\mathrm {D}\)} bias space. Only an inhomogeneous oxide field \(F_\mathrm {ox}(x)\) along
the Si/SiO\(_2\) interface which, according to (2.15), determines the *effective* trap level is taken into account.
Important non–equilibrium effects related to the carrier ensemble and the energy distribution functions are not considered in the aforementioned approach. However, due to its physically sound description, the interaction of oxide
defects with a whole band of charge carriers is already included via the term \(f_{n/p}(E)f^\mathrm {LSF}_{i,j}(E,E_\mathrm {T})\), where \(f\) is the EDF and \(f^\mathrm {LSF}\) is the lineshape function, see (2.13). Therefore, the NMP\(_\mathrm {eq.}\) model can naturally be extended towards additional non–equilibrium effects,
such as the formation of a heated, non–equilibrium carrier ensemble and the generation of secondary carriers, by using a full solution of the Boltzmann transport equation. This model variant will be referred to as full non–equilibrium
NMP\(_\mathrm {neq.}\) model. On the other hand, this procedure is computationally very costly and not practical for modern TCAD applications. Instead, semi–empirical approximations will be introduced to consider the
interactions with secondary generated carriers. Within this approach, which will be referred to as extended–equilibrium NMP\(_\mathrm {eq.+II}\) model, the EDFs will be still assumed to obey a Fermi–Dirac distribution.