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The Physics of Non–Equilibrium Reliability Phenomena

2.2 Non–Equilibrium Bias Temperature Instability

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].

2.2.1 Nonradiative Multiphonon Framework

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.


Figure 2.3: Schematic representations of the potential energy curves (PECs) involved for charge transitions (Left) and thermal transitions (Right). Thermally activated structural relaxation proceed within a PEC, while (nonradiative) charge transfer provokes a change of the PEC from the neutral to the charged state.

Thermal transitions can be accounted for using classical transition state theory (TST) [64, 129131] 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

\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

\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]

\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 \(\omega =\sqrt {k/\mu }\)

\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, 134138]. 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]

\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) [140142] 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

\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}


Figure 2.4: The lineshape function (LSF) of NMP transitions. Left: Potential energy profiles with the corresponding vibrational eigenmodes of two differently charged defect configurations schematically representing a typical NMP process. According to (2.11), the LSF is proportional to the overlap integral of the wavefunctions \(\ket {\eta _{i\alpha }}\) and \(\ket {\eta _{j\beta }}\) within the two PECs. The delta function guarantees the conservation of energy, thus only pairs of energy where \(E_{i\alpha }=E_{j\beta }\) is fulfilled contribute to \(f_{i,j}^\mathrm {LSF}\). For all practical applications the delta peaks are broadened depending on the temperature. Right: Individual contributions to the total LSF. While the overlap of the vibrational wavefunctions rapidly increases towards the classical crossing point and approximately stays constant above, the (temperature dependent) occupation probability of the eigenmodes decays. The interplay of both terms yields the lineshape function (LSF) indicated as the blue area.

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:

\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:

\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

\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.


Figure 2.5: Interactions with an oxide field as well as a whole band of states. Left: Applying an electric field shifts the defects’ trap level accordingly (gray parabolas). Additionally, due to the interaction with the carrier reservoir in the channel, the charged PEC is shifted with respect to the carrier energy \(E\). The defect can interact with holes in the valence band (red parabolas) and electrons in the conduction band (blue parabolas). The combination of these two effects leads to a distribution of configurations and hence crossing points which is reflected in the LSF \(f_{i,j}^\mathrm {LSF}(E,E_\mathrm {T})\). Right: The valence band lineshape function for the neutral \(\Rightarrow \) charged transition. With increasing carrier energy \(E\) the transition barrier is reduced, thus the LSF increases. Furthermore, also the shift \(\Delta s\) induced by the electric field Fox  alters the energetic alignment of the two parabolas and modifies the LSF.

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 [148152], 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 process6. 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 [156158]. 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 [162165]. 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.

2.2.2 Equilibrium 4–State Model

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.


Figure 2.6: Schematic reaction coordinate (RC) diagram together with a simple state diagram. Left: An example of a 4–state NMP defect which is uniquely defined given the defect energies \(\Delta E_{i,j}\), the relaxation energies \(S_{i,j}\) and the energy barriers \(\varepsilon _{i,j}\). Right: A state diagram visualizing the NMP transitions between the state \(1\Leftrightarrow 2^\prime \) and \(1^\prime \Leftrightarrow 2\) as well as the thermally activated relaxations \(1\Leftrightarrow 1^\prime \) and \(2\Leftrightarrow 2^\prime \).

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 [167170]; 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, 173175]. 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

\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

\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 approximation7

\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

\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}


Figure 2.7: The PECs as the cornerstone of the NMPeq.  model. Due to the negligible carrier concentration in the conduction band, assuming normal operating conditions of a pMOSFET, the transition rates \(k_{i,j}\) are determined by the interaction with holes in the valence band (Left). The biggest contribution to the total rate using the LSF together with the Fermi–Dirac statistics for the carriers is close to the VB edge and, therefore, validates the so–called band edge approximation (Right).

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 CB8, 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.

2.2.3 Key Concepts

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.