# The Physics of Non–Equilibrium Reliability Phenomena

#### 5.2 Non–Equilibrium Dynamics of Individual Oxide Defects

In a number of publications [295, MJC15, MJJ6] conducted on small–area SiON pMOSFETs, it has been shown that the charging and discharging characteristics of individual oxide defects can substantially change upon applying a drain bias. While in [MJC15, MJJ6] only defects with a decreasing occupancy, i.e. a reduced likelihood for the defect to trap a charge, have been found for increasing $$\Vd$$, the authors in [295] concluded that oxide defects could actually be affected in both ways, i.e. an increasing occupancy with increasing drain voltage was also measured. Furthermore, in a recent study [296] the authors claim that the fingerprints of single defects in the oxide, such as their step height and emission time $$\tau _\mathrm {e}$$, change with intermediate hot–carrier stress cycles, but their initial characteristics can be restored after an extended rest period.

The following Section presents simulations applying different variants of the 4–state NMP model to explain the behaviour of single oxide defects. Individual defects have been explicitly chosen for this benchmark study due to the direct measurement of fundamental parameters such as their capture and emission time. These quantities are not accessible when measuring large–area devices and, hence, the influence of a heated carrier ensemble would be blurred by a collective response of a large number of traps. A detailed analysis provides insight into the charging and discharging dynamics to finally unravel the experimentally recorded characteristics. The non–equilibrium EDFs have been obtained using the higher order spherical harmonics expansion simulator Spring [289, 297299], see Sec. 5.1 for details.

The measurement results used to validate the model approaches described here are performed on 2.2  ; nm plasma nitrided SiON pMOSFETs of a 130  ; nm commercial technology with an operating voltage of $$\Vdd =\SI {-1.5}{V}$$ [MJJ6, 300, MJJ4], the same pMOSFET as has been used for the HCD study above. Individual oxide defects have been studied on nano–scale devices ($$W=\SI {160}{nm}$$) applying the time–dependent defect spectroscopy (TDDS) technique [MJC15, MJJ6, MJJ4]. The devices have been subjected to a stress phase to induce the capture of holes by oxide traps. Subsequently $$\Vg$$ and $$\Vd$$ have been switched to recovery conditions to allow the trapped holes to be emitted again. This sequence has been repeated 100 times in order to collect statistics on active defects [MJC15, MJJ6].

##### 5.2.1 Defect Characteristics

Two defects have been chosen for studying the behaviour of single oxide traps [MJJ6, MJJ4]. Both defects and their respective capture and emission times have been characterized over a broad range of stress and recovery conditions rendering them ideal candidates for a simulation study. Furthermore, both traps possess a rather unexpected and intriguing trend with increasing drain bias $$\Vd$$ which is beyond explanations solely based on electrostatic considerations.

The basis for subsequent simulations provides their behaviour for pure BTI ($$\Vd \sim \SI {0}{V}$$) conditions. Applying the NMP$$_\mathrm {eq.}$$ model described in Sec. 2.2 is legitimate for such bias conditions and is able to represent the experimentally recorded characteristic capture and emission times for both defects, B1 and B2, see Fig. 5.9 (left panels). Defect B1 shows a switching trap behaviour: Its emission time below $$\Vg \sim \SI {-0.5}{V}$$ is strongly bias dependent and determined by the discharging path $$2\Rightarrow 1^\prime \Rightarrow 1$$ given by the first passage time (FPT) $$\tau _\mathrm {e,FPT}^{1^\prime ,1}$$, see (2.19). Increasing $$\Vg$$ shifts the trap level accordingly and the preferred emission path changes to $$2\Rightarrow 2^\prime \Rightarrow 1$$ at high $$\Vg$$. On the other hand, B2 is a fixed trap with an (almost) constant emission time over $$\Vg$$ which is dominated by $$2\Rightarrow 2^\prime \Rightarrow 1$$.

Interestingly, the behaviour of both defects changes in a complex way when the device is stressed at $$\Vg =\SI {-2.8}{V}$$ and an increasing drain bias $$\Vd =[\SI {0.0}{V},\SI {-2.8}{V}]$$ is applied, see Fig 5.9 (middle and right panels). Defect B1, which has been determined to be in the middle of the channel ($$0.6L_\mathrm {G}-0.7L_\mathrm {G}$$) [MJC15, MJJ6], reveals a rather constants capture time. However, at the same time the defects’ occupancy dramatically decreases. These observations have two implications. First, defects which are not located in the vicinity of the drain end of the channel can be affected by a drain bias. Second, a feasible explanation for the decreasing occupancy, despite the constant $$\tau _\mathrm {c}$$, would be an even faster reduction of the emission time $$\tau _\mathrm {e}$$. Thus, the defect would have already partially emitted its charge before the measurement switched to recovery conditions. Separate measurements at different $$\Vg$$ conditions actually support this idea, see [MJJ6]. The lateral defect position of B2, on the other hand, was extracted to be in the drain region of the MOSFET ($$0.8L_\mathrm {G}-0.9L_\mathrm {G}$$) [MJC15, MJJ6]. Although defects on the drain end of the channel should be heavily affected (due to the reduction of Fox ), its capture time is almost unperturbed and actually slightly decreasing. Even more remarkably, the occupancy of B2 is still $$0.4$$ at $$\Vg =\Vd =\SI {-2.8}{V}$$. This means that in $$40\%$$ of the TDDS traces the defect was active and captured a charge during stress. Both observations clearly reveal the conceptual limits of simple electrostatic considerations to explain the charging dynamics of oxide defects for $$|\Vd |\gg \SI {0.0}{V}$$. Naturally, a unique parameter set for each defect, extracted by representing their characteristics for pure BTI conditions, has been used in all subsequent simulations.

Applying the NMP$$_\mathrm {\mathbf {eq.}}$$ model yields the results shown in Fig. 5.9. While for B1 both characteristic trap parameters, $$\tau _\mathrm {c}$$ and $$\tau _\mathrm {e}$$, increase due to the changed electrostatics, the emission time of B2 changes its trend at $$\Vd \sim \SI {-0.5}{V}$$. This can be explained due to the formation of the pinch–off region and the associated reduction of the hole concentration. The extracted occupancies of the defects for a stress time of $$t_\mathrm {s}=\SI {2}{s}$$ show a decreasing trend for both traps, see Fig. 5.9 (right panels). However, the measurement data are not properly captured. While the experimental occupancy of B1 shows a plateau up to $$\Vd \sim \SI {-2.0}{V}$$ followed by a rapid decrease, the simulations predict a continuous reduction which saturates at $$\sim 0.4$$. In contrast, the simulations for B2 yield a prompt reduction starting from $$\Vd =\SI {-1.0}{V}$$ and an inactive defect at high $$\Vd$$, whereas the experimental data show an occupancy of $$40\%$$. Even worse, $$\tau _\mathrm {c}$$ actually shows an opposite trend compared to the measurements. As expected, only taking electrostatic considerations into account in the NMP$$_\mathrm {eq.}$$ model is not enough to explain the measured observations.

In order to improve the quality of the simulations, non–equilibrium effects can explicitly be included within the simulation framework NMP$$_\mathrm {\mathbf {neq.}}$$. A thorough description of the interaction with energetic carriers in the valence as well as the conduction band can dramatically increase the corresponding NMP transition rates $$k_{i,j}$$, see Sec. 4.2.2 and Fig. 4.9. As shown in Fig. 5.10, this refinement of the 4–state NMP framework yields an accurate description of the defect parameters. All relevant characteristics are well represented by the model. The capture times $$\tau _\mathrm {c}$$ for both defects stay approximately constant while the emission times decrease with $$\Vd$$. This translates into a very good agreement with the measured occupancies of B1 and B2. Specific features, such as the constant occupancy for B1 between $$\Vd =\SI {-0.5}{V}$$ and $$\SI {-2.0}{V}$$ as well as the remaining active defect B2 at $$\Vd ={-2.8}{V}$$, are properly captured by the model approach.

A detailed analysis of the respective capture and emission pathways reveals the nature of these interesting phenomena, see Fig 5.11. While the path for charging the defect stays the same over the whole $$\Vd$$ bias range for both defects ($$1\Rightarrow 2^\prime \Rightarrow 2$$), the discharging path changes for B1, similar to its switching behaviour for pure BTI conditions. At low $$\Vd$$, B1 emits its charge via $$2\Rightarrow 2^\prime \Rightarrow 1$$, while at elevated $$\Vd$$ the emission time $$\tau _\mathrm {e}$$ is determined by the faster pathway $$2\Rightarrow 1^\prime \Rightarrow 1$$. In order to understand the interactions with the valence and conduction band as well as the resulting characteristic times, all relevant NMP rates $$k_{i,j}$$ are given in Fig. 5.11. The relationship between the rates $$k_{i,j}$$ and the first passage times are given in (2.19). Two important implications can be seen in Fig. 5.11. First, the valence band rates are rather constant (or even slightly increasing) for all stress conditions. This is quite surprising, since the electric field Fox , and thus the effective trap level, typically determines the transition rates. An increasing drain bias, therefore, results in an inhomogeneous field which decreases towards the drain end of the channel. However, the non–equilibrium EDFs exhibit a pronounced high energy fraction of particles in the VB which compensate this effect, see Fig. 5.1. This phenomenon is particularly important for defects located in the vicinity of the drain end, e.g. B2. Second, it is clear that interactions with the conduction band potentially play a major role, specifically for B1. Starting from $$\Vd <\SI {-1.5}{V}$$ impact ionization (II) leads to an increasing concentration of (energetic) carriers in the CB. The interaction of the defect with electrons in the CB dominates the respective transition rate $$k_{2,1^\prime }$$ and thus determines the new and faster emission pathway via $$1^\prime \Rightarrow 1$$.

Although the measurement data have been rather puzzling, the non–equilibrium 4–state model NMP$$_\mathrm {neq.}$$, which takes the full spectrum of effects into account, is able to fully explain the peculiar measurement trends. The presented analysis helps to understand how (energetic) holes in the VB and secondary generated (energetic) electrons in the CB interact with defects in the gate stack and thereby validates the modeling approach.

However, accessing the EDFs and including them within the NMP model is typically computationally very demanding and, therefore, not practical for TCAD simulations. The extended variant NMP$$_\mathrm {eq.+II}$$ provides a considerably simplified implementation which does not require an additional solution of the Boltzmann transport equation (BTE) and where further approximations are applicable, such as the band edge approximations. To establish a full picture of the various model variants, Fig. 5.12 shows the simulation results using the NMP$$_\mathrm {eq.+II}$$ model for the two individual defects B1 and B2.

For defect B1 the simplified extended model approach is still able to capture the qualitative features of the full non–equilibrium model NMP$$_\mathrm {neq.}$$, in particular the rapidly decreasing emission time with increasing drain bias $$\Vd$$. This is a direct consequence of the interaction with carriers in the CB, as described in detail in Sec. 4.2. On the other hand, for the second defect, B2, the results are almost the same as for the NMP$$_\mathrm {eq.}$$ model variant. However, this was to be expected due to the driving force of the defects’ behaviour at increasing $$\Vd$$ bias being the interaction with hot holes in the VB. Such a peculiar feature can not be properly described by the NMP$$_\mathrm {eq.+II}$$ model.

Nevertheless, the accelerated decrease of the recoverable component $$R$$ of degradation with increasing $$\Vd$$ bias [MJJ6, 300303] is assumed to be well captured by the NMP$$_\mathrm {eq.+II}$$ model. On the contrary, a potential increase of the defects’ occupancy, see  [295], due to the interaction with energetic carriers, in contrast, can not be reproduced with this approach. Due to the rather large physical parameter set of a 4–state defect as well as their broad distribution, a general behaviour can hardly be deduced from two single oxide defects. Furthermore, note that the two defects, B1 and B2, have been particularly chosen for this study due to their abnormal behaviour. Nevertheless, out of the nine single traps characterized in [MJC15], six defects exhibited expected trends with respect to their lateral position.