The fundamentals of bias temperature instability and charge trapping in the oxide based on nonradiative multiphonon theory is already explained in Chapter 2.2. However, capturing the charging and discharging kinetics of oxide defects under non–equilibrium conditions is beyond the
applicability of the current 4–state framework. Nevertheless, the NMP\(_\mathrm {eq.}\) model already provides a physically complete description as well as a native extension towards the integration of such processes. In the
following the implications of a heated carrier ensemble in the valence and conduction band of a MOSFET will be discussed and two model extensions will be derived.
4.2.1 Extended 4–State Model
The described NMP\(_\mathrm {eq.}\) model does not account for non–equilibrium carrier dynamics in full {VG , VD } bias space. Only an inhomogeneous shift of the effective trap level along the Si/SiO\(_2\)
interface is taken into account using the relation to the oxide field Fox \((x)\) given in (2.15). However, this is a purely
electrostatic effect. Emerging changes of the energy distribution functions (EDFs) due to the application of a drain bias \(\Vd \), such as the formation of a heated, non–equilibrium carrier ensemble and the generation of secondary
carriers, are not included in the aforementioned approach. On the other hand, a thorough and consistent description requires a full solution of the coupled BTE for holes and electrons for each bias point which is computationally
very costly. Moreover, using the EDFs in conjunction with the (classical) lineshape function (LSF) for a large ensemble of defects is again a very computation intensive task. In order to capture the charging dynamics of oxide traps
also at higher \(\Vd \) stress conditions in a more practical TCAD model, a simplified approach is highly desired.
Figure 4.8: The extended 4–state model includes the effect of impact ionization (II) using semi–empirical approaches within the drift–diffusion (DD) simulations. Thus, interactions
with electrons in the conduction band yield a non–vanishing contribution to the NMP transitions (Left). In particular, the hole emission processes (which are equivalent to capturing an electron), i.e. \(2^\prime \Rightarrow
1\) and \(2\Rightarrow 1^\prime \), are expected to be affected by an increasing concentration of carriers in the conduction band. On the other hand, the reverse hole capture processes are still dominated by the interactions with
the VB. The NMP\(_\mathrm {eq.+II}\) model assumes the carriers to be in thermal equilibrium (Right).
A possible approximation is given by including the effect of II within the DD simulations using semi–empirical II models. This simplified picture neglects the interplay of defects with a heated carrier ensemble, but includes
the interaction with secondary created carriers in the conduction band (CB) of a pMOS, see Fig. 4.8. Particularly
for the charge transitions \(1\Leftrightarrow 2^\prime \) and \(1^\prime \Leftrightarrow 2\) this can have a large impact. While charging is associated with the capture event of a hole (or the emission of an electron),
discharging is linked to the defect emitting a hole into the VB (or to capture an electron from the CB). Therefore, the emission processes \(2^\prime \Rightarrow 1\) and \(2\Rightarrow 1^\prime \) are expected to be affected
by an increasing carrier concentration \(n\) in the conduction band, see Fig. 4.8. This variant of the 4–state
model is termed extended equilibrium model, NMP\(_\mathrm {\mathbf {eq.+II.}}\). Due to the (crude) approximation that carriers remain in thermal equilibrium, thus properly described by a FD
distribution, the computationally effective band edge approximation is still applicable.
4.2.2 Non–Equilibrium 4–State Model
Despite the appealing model introduced above, it certainly lacks a detailed description of the interaction between oxide defects and energetic carriers described by a non–equilibrium energy distribution function (EDF). Their
interaction, however, is given by the lineshape function (LSF), see (2.11) (or the classical limit) which is
already a substantial part of the NMP\(_\mathrm {eq.}\) model variant. The LSF is the sum of a multitude of contributions due to the various vibrational modes involved, see Fig. 2.4 in Chap. 2, for a specific configuration of the
two parabolas. Furthermore, Fig. 2.5 and the set of equations, (2.13) in Sec. 2.2, show
that the LSF additionally depends on the energy \(E\) of the reservoir state due to the interaction with a continuous band, which is represented by a shifted (charged) parabola. This shifted alignment translates into a changing
intersection point of the parabolas and, thus, an energy dependent LSF. For a decreasing (increasing) barrier \(\varepsilon _{i,j}\) with the reservoir energy \(E\) the LSF is therefore (usually) an increasing (decreasing)
function over \(E\). As already mentioned in Sec. 2.2, the rate \(k_{i,j}\) is determined by the product
\(f_p(E)f_{i,j}^\mathrm {LSF}(E,E_\mathrm {T})\).
Figure 4.9: The impact of non–equilibrium energy distribution functions (EDFs) onto the NMP rates \(k_{i,j}\). Heated carrier ensembles in the valence (Left) as well as conduction bands (Right) potentially
increase the corresponding transition rates and therefore alter the defects’ charging and discharging dynamics.
Fig. 4.9 shows that in the case of non–equilibrium EDFs, e.g. hole EDFs at the drain end of the channel or
electron EDFs at the source end, the integral can be substantially different compared to a FD distribution. This not only implies considerably modified transition rates \(k_{i,j}\) with increasing \(\Vd \) conditions, but also that
the (energetically) accessible defects change within {VG , VD } bias space. The full consideration of this effect is termed non–equilibrium 4–state model, NMP\(_\mathrm {\mathbf {neq.}}\), and is
based on a self consistent calculation of the coupled BTE for holes and electrons. Consequently, heated carrier ensembles for holes and electrons – which are created at the drain side and accelerated by the electric field towards the
source – are taken into account. This variant of the 4–state model yields the most accurate description and reveals interesting phenomena as will be shown below.
