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5.2 Individual Defects

In addition to the large-area devices nano-scale devices are used to study the behavior of individual defects in greater detail. The capture and emission processes of single defects which contribute to recovery after NBTI stress, were analyzed. For this, 20 single defects in ten nano-scale pMOSFETs were measured and nine of them in four devices were fully characterized. A list of the extracted defects can be seen in Table 5.1. These particular nine defects were selected in order to best represent the supposed uniform lateral distribution. They were also selected according to their distribution in the spectral map and the ability to characterize properties like $\tau _{\mathrm {e}}$ , $\tau _{\mathrm {c}}$ , occupancy and step height over a wide range of stress and recovery voltages.

$ X_T/L $ — Table 5.1: **Relative lateral defect position and classification due to capture behavior:** By exploiting the recovery drain bias dependence of the step heights for constant gate recovery voltage $\VGrelax$ , the lateral position $X_\mathrm {T}$ $/$ $L$ (0 at source, 1 at drain) was extracted [105]. The uncertainty of $X_\mathrm {T}$ $/$ $L$ is about 20 %. Defects A3, B3, B4 and C2 showed a very complex behav- ior (e.g., due to an overlap with other defects in the spectral map at certain bias conditions) and were not characterized fully. The defects are assigned to three types according to their capture behavior during mixed NBTI/HC stress which is explained based on Figure 5.9.

$ \VGrelax $ — Table 5.1: **Relative lateral defect position and classification due to capture behavior:** By exploiting the recovery drain bias dependence of the step heights for constant gate recovery voltage $\VGrelax$ , the lateral position $X_\mathrm {T}$ $/$ $L$ (0 at source, 1 at drain) was extracted [105]. The uncertainty of $X_\mathrm {T}$ $/$ $L$ is about 20 %. Defects A3, B3, B4 and C2 showed a very complex behav- ior (e.g., due to an overlap with other defects in the spectral map at certain bias conditions) and were not characterized fully. The defects are assigned to three types according to their capture behavior during mixed NBTI/HC stress which is explained based on Figure 5.9.

For the characterization of the single defects, the TDDS framework described in Section 3.7 was used and following phases were applied:

1. Measure: $I_\mathrm {D}$ - $V_\mathrm {G}$ characteristics.
2. Stress: Application of a $($ $V_{\mathrm {G}}^\mathrm {str}$ $,$ $V_{\mathrm {D}}^\mathrm {str}$ $)$ combination within the region of stress conditions shown in Figure 3.1 for a certain $t_\mathrm {str}$ .
3. Measure: $\Delta V_{\mathrm {th}}$ at $V_{\mathrm {G}}^\mathrm {rec}$ and $V_{\mathrm {D}}^\mathrm {rec}$ for $t_\mathrm {rec}$ .
4. Repeat the second and third phase $100$ times in order to capture the statistics.

The recovery traces contain the typical steps due to charge exchange events between the channel and the oxide caused by single oxide defects. Each defect causes exponentially distributed steps with a particular step height at a particular mean value of $\tau _{\mathrm {e}}$ . By assigning the unique steps to a defect, the following parameters were extracted for each defect:

• $\tau _{\mathrm {e}}$ $($ $V_{\mathrm {G}}^\mathrm {rec}$ $,$ $V_{\mathrm {D}}^\mathrm {rec}$ $)$
• $\tau _{\mathrm {c}}$ $($ $V_{\mathrm {G}}^\mathrm {rec}$ $,$ $V_{\mathrm {D}}^\mathrm {rec}$ $)$
• Occupancy $($ $V_{\mathrm {G}}^\mathrm {rec}$ $,$ $V_{\mathrm {D}}^\mathrm {rec}$ $)$
• The lateral position $X_\mathrm {T}$

An analysis at the single defect level gives insight into the detailed behavior of individual defects already measured on average in large-area devices (Figure 5.1). The most surprising finding is that some of the source-side defects do not contribute to $R$ after mixed NBTI/HC stress although they do so after homogeneous NBTI stress with the same $V_{\mathrm {G}}^\mathrm {str}$ . As discussed above, this behavior cannot be explained by a simple electrostatic model only. In order to assign the behavior of a defect to its position, the relative lateral defect position $X_\mathrm {T}$ $/$ $L$ was extracted according to Equation 2.57 – the background is explained in Subsection 2.3.2 – by exploiting the readout drain bias dependence of the $\Delta V_{\mathrm {th}}$ step heights caused by the defects (see Figure 5.6). In the present case $P_\mathrm {0max}=$  10 mV because these were the largest step heights observed corresponding to defects in the middle of the channel. The results for the relative lateral positions are listed in Table 5.1 and shown in Figure 5.7 as a schematic sketch.

Figure 5.6: **Extraction of the lateral position:** The lateral position $X_\mathrm {T}$ $/$ $L$ (0 at source, 1 at drain) was extracted by exploiting the recovery drain bias dependence of the step heights for constant $V_{\mathrm {G}}^\mathrm {rec}$ [104]. The subfigures show the separation of the defects into three types according to their capture behavior during mixed NBTI/HC stress: blue group, green group and magenta group. Measurement data and linear fits are labled with the defect name and the extracted relative lateral position. Figure source: [105].

Figure 5.7: **Lateral defect distribution:** Schematic sketch of the positions of the nine characterized defects within the oxide. Figure source: [105].

In homogeneous NBTI measurements the defects show a typical behavior as discussed in Subsection 2.1.3. The emission behavior in dependence of $V_{\mathrm {G}}^\mathrm {str}$ is shown in Subfigure 5.8a for the device B. The emission times of the defects B1 and B2 at the readout conditions are within the experimental window. Therefore, as soon as B1 and B2 capture charge carriers during stress, the emission events are visible in the recovery trace as single steps. At $V_{\mathrm {G}}^\mathrm {str}$   $=$  −1.6 V only defect B1 captures a charge carrier during stress because $\tau _{\mathrm {c}}$   $<$   $t_\mathrm {str}$   $=$  10 s. Thus, emission events of B1 can be measured during recovery. At $V_{\mathrm {G}}^\mathrm {str}$   $=$  −2.2 V, $\tau _{\mathrm {c}}$ of B2 is low enough that it also captures charge carriers within $t_\mathrm {str}$   $=$  10 s. Thus, in the recovery traces steps caused by B1 and B2 are measured. In other words, by increasing $|$ $V_{\mathrm {G}}^\mathrm {str}$ $|$ , the occupancy grows while $\tau _{\mathrm {c}}$ decreases. This behavior is also shown in Subfigure 5.9a for all characterized defects.

The measurements at mixed NBTI/HC stress conditions illustrate a more complicated behavior. For a better understanding, it should be recalled what is discussed in Subsection 2.3.1. For mixed NBTI/HC, it is expected that the occupancy of defects near the drain will be reduced compared to homogeneous NBTI measurements due to the reduced $E_\mathrm {OX}$ and thus increased $\tau _{\mathrm {c}}$ . At the same time it is expected that source-side defects remain almost unaffected at mixed stress conditions compared to NBTI conditions. However, Subfigure 5.8b reveals that this assumption is not true. The defects A2 and A5, which are in the vicinity of the source, capture charge carriers at homogenous NBTI stress with $V_{\mathrm {G}}^\mathrm {str}$   $=$  −1.8 V and emit them during recovery. Contrary to expectations, at mixed NBTI/HC stress with $V_{\mathrm {G}}^\mathrm {str}$   $=$  −1.8 V and $V_{\mathrm {D}}^\mathrm {str}$   $=$  −2.8 V they do not emit charge carriers during recovery. This means that their behavior is affected by $V_\mathrm {D}$ .

(a) NBTI stress measured on device B: The traces contain discrete steps caused by B1 and B2. **Top:** B2 does not capture a charge carrier at $V_{\mathrm {G}}^\mathrm {str}$   $=$  −1.6 V and thus does not emit during the recovery measurement. Only one emission event of B1 can be observed here. **Bottom:** At $V_{\mathrm {G}}^\mathrm {str}$   $=$  −2.2 V, B1 and B2 capture a charge carrier in 60 % and 50 % of the stress phase, respectively, and emit them during the recovery measurement.

The behavior of all defects at mixed NBTI/HC stress conditions is shown in Subfigure 5.9b. At a fixed $V_{\mathrm {G}}^\mathrm {str}$ (around −2 V) and increasing $|$ $V_{\mathrm {D}}^\mathrm {str}$ $|$ the defects can be separated into three groups: Either the occupancy is constant for the whole $V_{\mathrm {D}}^\mathrm {str}$ range (defects A1, D1 and A4 – blue group) or it decreases continuously for $V_{\mathrm {D}}^\mathrm {str}$   $<$  0 V (C1 and B2 – green group) or it shows a local minimum at $V_{\mathrm {D}}^\mathrm {str}$   $\approx$  −0.8 V, a local maximum at $V_{\mathrm {D}}^\mathrm {str}$   $\approx$  −1.5 V and decreases to zero for $V_{\mathrm {D}}^\mathrm {str}$   $<$  −1.5 V (C3, B1, A5 and A2 – magenta group). The extracted $\tau _{\mathrm {c}}$ with respect to the drain bias shows a slightly increasing trend only for the green group. For the magenta and blue groups $\tau _{\mathrm {c}}$ is either constant or decreases.

The green and blue groups behave as expected and discussed previously. Drain-side defects (green group) show a decreasing occupancy and increasing $\tau _{\mathrm {c}}$ for mixed NBTI/HC stress due to the significantly reduced $E_\mathrm {OX}$ . Source-side to mid-channel defects (blue group) show a constant occupancy over the whole $V_{\mathrm {D}}^\mathrm {str}$ range. However, the defects in the magenta group, where also the two interesting defects A2 and A5 are assigned to, show an unexpected behavior. This can be visualized by a parameterization in terms of $V_{\mathrm {D}}^\mathrm {str}$ and $V_{\mathrm {G}}^\mathrm {str}$ in Figure 5.9. This parametrization is illustrated in Figure 5.10 and shows that the traces for increasing $|$ $E_\mathrm {OX}$ $|$ during NBTI stress and increasing $|$ $V_{\mathrm {D}}^\mathrm {str}$ $|$ at a fixed $V_{\mathrm {G}}^\mathrm {str}$ during mixed NBTI/HC stress follow reverse trends for the green group. In other words, the occupancy increases and $\tau _{\mathrm {c}}$ decreases for increasing $|$ $E_\mathrm {OX}$ $|$ while the occupancy decreases and $\tau _{\mathrm {c}}$ increases for increasing $|$ $V_{\mathrm {D}}^\mathrm {str}$ $|$ . In stark contrast, the magenta group shows a different behavior for increasing $|$ $V_{\mathrm {D}}^\mathrm {str}$ $|$ . For these defects, increasing $|$ $V_{\mathrm {D}}^\mathrm {str}$ $|$ causes a decrease in both, occupancy and $\tau _{\mathrm {c}}$ .

(a) NBTI stress: The defects show a typical behavior. By increasing $|$ $V_{\mathrm {G}}^\mathrm {str}$ $|$ the occupancy of the defects increases and $\tau _{\mathrm {c}}$ decreases.

The fact that A2 and A5 emit charge carriers after homogeneous NBTI but do not after mixed NBTI/HC stress does not mean that they are volatile as soon as mixed NBTI/HC stress is applied. The volatility of all defects was checked regularly by intermittently applying homogeneous NBTI conditions. A volatile defect would have remained neutral after stress independently from the stress conditions. To the contrary, all defects which remained neutral after mixed NBTI/HC stress with high $|$ $V_{\mathrm {D}}^\mathrm {str}$ $|$ were found to be charged after these intermittent homogeneous NBTI stress checks. None of the characterized defects showed a temporary electrical inactivity during the discussed measurements. The neutrality after mixed stress conditions must be attributed to microscopic changes in the charge transfer process with increasing $|$ $V_{\mathrm {D}}^\mathrm {str}$ $|$ . In this regard, Figure 5.11 shows that not only $\tau _{\mathrm {c}}$ can change but also $\tau _{\mathrm {e}}$ can change for different drain biases. Consequently, the ratio $\tau _{\mathrm {e}}$ / $\tau _{\mathrm {c}}$ changes for some defects, which affects the occupancy, illustrated in Figure 5.12. This means that as long as $\tau _{\mathrm {e}}$   $\gg$   $\tau _{\mathrm {c}}$ and $\tau _{\mathrm {c}}$   $<$   $t_\mathrm {str}$ at stress condition, the defect captures a charge carrier during stress and emits it during recovery (top panels of Figure 5.13). However, if the relation is reversed $\tau _{\mathrm {e}}$   $\ll$   $\tau _{\mathrm {c}}$ , the situation is more complicated. Then, it is more likely that a defect emits a captured charge carrier immediately after the capture event while the stress bias is still applied (central panels of Figure 5.13). Although the capture and emission events can repeat several times, it is very likely that no emission event can be measured at recovery conditions, which explains the considerable reduction in occupancy. By contrast, volatile defects do not capture or emit charge carriers at all (bottom panels of Figure 5.13).

Figure 5.10: **Occupancy versus capture time:** A parameterization of $V_{\mathrm {G}}^\mathrm {str}$ and $V_{\mathrm {D}}^\mathrm {str}$ demonstrates the differ- ence between green and magenta type shown for three defects. **Dashed lines:** For NBTI stress the occupancy increases and $\tau _{\mathrm {c}}$ decreases for increasing $|$ $E_\mathrm {OX}$ $|$ (corresponds to an increasing $|$ $V_{\mathrm {G}}^\mathrm {str}$ $|$ ). **Solid lines:** As soon as $V_{\mathrm {G}}^\mathrm {str}$ is held at a constant value and $V_{\mathrm {D}}^\mathrm {str}$   $<$  0 V, the occupancy of the green defects shows a reversed trend compared to NBTI. The occupancy decreases and $\tau _{\mathrm {c}}$ increases. This can be explained by the reduction of $E_\mathrm {OX}$ near the drain for $V_{\mathrm {D}}^\mathrm {str}$   $<$  0 V. By contrast, the occupancy of the magenta defects shows a completely different trend, namely towards decreasing $\tau _{\mathrm {c}}$ for a decreasing occupancy. This is an indication for a different process. Figure source: [105].

$ V_{\mathrm {G}}^\mathrm {str} $ — Figure 5.10: **Occupancy versus capture time:** A parameterization of $V_{\mathrm {G}}^\mathrm {str}$ and $V_{\mathrm {D}}^\mathrm {str}$ demonstrates the differ- ence between green and magenta type shown for three defects. **Dashed lines:** For NBTI stress the occupancy increases and $\tau _{\mathrm {c}}$ decreases for increasing $|$ $E_\mathrm {OX}$ $|$ (corresponds to an increasing $|$ $V_{\mathrm {G}}^\mathrm {str}$ $|$ ). **Solid lines:** As soon as $V_{\mathrm {G}}^\mathrm {str}$ is held at a constant value and $V_{\mathrm {D}}^\mathrm {str}$   $<$  0 V, the occupancy of the green defects shows a reversed trend compared to NBTI. The occupancy decreases and $\tau _{\mathrm {c}}$ increases. This can be explained by the reduction of $E_\mathrm {OX}$ near the drain for $V_{\mathrm {D}}^\mathrm {str}$   $<$  0 V. By contrast, the occupancy of the magenta defects shows a completely different trend, namely towards decreasing $\tau _{\mathrm {c}}$ for a decreasing occupancy. This is an indication for a different process. Figure source: [105].

Figure 5.11: **Emission time characteristics:** The emission time decreases with $|$ $V_\mathrm {D}$ $|$ . As a consequence, if $\tau _{\mathrm {e}}$ $\ll$ $\tau _{\mathrm {c}}$ at stress conditions, the defect cap- tures a charge carrier but immediately emits it before switching to recovery conditions. This holds true for all defects of the green and magenta group.

$ | $ — Figure 5.11: **Emission time characteristics:** The emission time decreases with $|$ $V_\mathrm {D}$ $|$ . As a consequence, if $\tau _{\mathrm {e}}$ $\ll$ $\tau _{\mathrm {c}}$ at stress conditions, the defect cap- tures a charge carrier but immediately emits it before switching to recovery conditions. This holds true for all defects of the green and magenta group.

It can be concluded that depending on their detailed configuration, defects at all lateral positions can remain neutral after mixed NBTI/HC stress and thus do not contribute to $R$ . This is the primary reason for the discrepancy between the experimental data and simulation at high $|$ $V_{\mathrm {D}}^\mathrm {str}$ $|$ as shown in Figure 5.1. So far, such a behavior has not been considered in the current models because oxide defects have been studied only under homogeneous NBTI conditions. In order to explain such a complex behavior like the distortion of the characteristics of source-side defects, also non-equilibrium carrier transport processes induced by the high $|$ $V_{\mathrm {D}}^\mathrm {str}$ $|$ have to be taken into account in addition to an inhomogeneous $E_\mathrm {OX}$ .

Figure 5.12: **Change of occupancy in respect of the ratio of emission to capture time:** Shifts of $\tau _{\mathrm {e}}$ and $\tau _{\mathrm {c}}$ by a few orders of magnitude af- fect the occupancy. **Top:** Schematic visualization of the shift from $\tau _{\mathrm {e}}$ $\ll$ $\tau _{\mathrm {c}}$ to $\tau _{\mathrm {e}}$ $\gg$ $\tau _{\mathrm {c}}$ at stress conditions. **Bottom:** Occupancy in respect to the ratio $\tau _{\mathrm {e}}$ / $\tau _{\mathrm {c}}$ is zero if $\tau _{\mathrm {e}}$ $\ll$ $\tau _{\mathrm {c}}$ and at its maximum if $\tau _{\mathrm {e}}$ $\gg$ $\tau _{\mathrm {c}}$ . Figure source: [124].

$ \tau _{\mathrm {e}} $ — Figure 5.12: **Change of occupancy in respect of the ratio of emission to capture time:** Shifts of $\tau _{\mathrm {e}}$ and $\tau _{\mathrm {c}}$ by a few orders of magnitude af- fect the occupancy. **Top:** Schematic visualization of the shift from $\tau _{\mathrm {e}}$ $\ll$ $\tau _{\mathrm {c}}$ to $\tau _{\mathrm {e}}$ $\gg$ $\tau _{\mathrm {c}}$ at stress conditions. **Bottom:** Occupancy in respect to the ratio $\tau _{\mathrm {e}}$ / $\tau _{\mathrm {c}}$ is zero if $\tau _{\mathrm {e}}$ $\ll$ $\tau _{\mathrm {c}}$ and at its maximum if $\tau _{\mathrm {e}}$ $\gg$ $\tau _{\mathrm {c}}$ . Figure source: [124].

Figure 5.13: **Schematic illustration of capture and emission events:** The charge state of the defect 0 if it is neutral and 1 if it is charged. **Top:** $\tau _{\mathrm {e}}$ $\gg$ $\tau _{\mathrm {c}}$ at stress condition. The defect captures a hole during stress and emits it during recovery. **Center:** $\tau _{\mathrm {e}}$ $\ll$ $\tau _{\mathrm {c}}$ at stress condition. The defect captures a charge carrier and emits it immediately afterwards at stress conditions. As a consequence, no emission event can be measured at recovery conditions. **Bottom:** Volatile defects are not electrically active.

In this context, the considerable change of $\tau _{\mathrm {e}}$ and $\tau _{\mathrm {c}}$ can be explained by a change of the transition rates $k_\mathrm {1,2’}$ and $k_\mathrm {2,1’}$ between the states $1$ and $2^\prime$ and the states $2$ and $1^\prime$ , respectively, in the four-state NMP model (Figure 2.18) [125]. The calculation of the transition rates includes among other factors the energy distribution function of the charge carriers (discussed in Subsection 2.1.5 and shown in Equations 2.22 and 2.24) illustrated in Figure 5.14. This figure shows clearly that under homogeneous NBTI conditions the carriers in the channel near the source are in equilibrium and thus properly described by the Fermi-Dirac distribution. As soon as a drain bias is applied this approximation is no longer valid. Carriers can gain energy by the channel field, exchange energy by various mechanisms, and can be severely out of equilibrium.

Furthermore, if the device is operated near or beyond pinch-off conditions, carriers with sufficient kinetic energy can trigger II and consequently generate secondary carriers. As a consequence, additionally to the minority charge carriers in the channel also majority charge carriers are available and may interact with the oxide defects. In Figure 5.14, this is shown as a change of the distribution function of the electrons. Thus a thorough carrier transport treatment by means of a solution of the BTE for each $($ $V_{\mathrm {G}}^\mathrm {str}$ $,$ $V_{\mathrm {D}}^\mathrm {str}$ $)$ combination and each lateral position is needed for such situations.

(a) Distribution function of holes: For different $V_\mathrm {D}$ and the same $V_\mathrm {G}$ .

$ V_\mathrm {D} $ — (a) Distribution function of holes: For different $V_\mathrm {D}$ and the same $V_\mathrm {G}$ .

For this purpose, the quasi-equilibrium model, termed NMP $_\mathrm {eq.}$ , which approximates the carrier energy distribution function by a Fermi-Dirac distribution independently from $V_{\mathrm {D}}^\mathrm {str}$ is expanded to the NMP $_\mathrm {neq.}$ model, which includes the distribution functions for holes and electrons evaluated with the higher-order spherical harmonics expansion simulator SPRING [125]. Thereby the bipolar BTE was solved self-consistently including phonon and impurity scattering mechanisms as well as impact ionization with secondary carrier generation.

The NMP $_\mathrm {eq.}$ model implies that oxide defects mainly interact with carriers in the valence band. Moreover, as it is discussed in Subsection 2.3.1, defects at the source-side are unaffected by $V_{\mathrm {D}}^\mathrm {str}$ . In contrast, the NMP $_\mathrm {neq.}$ model considers the interaction of high energetic carrier in the valence band as well as the interplay of defects with the secondary generated electrons in the conduction band. With this model, the observed defect behavior of $V_{\mathrm {D}}^\mathrm {str}$ -dependent transition rates even for defects located in the vicinity of the source contact can be captured quite well.

(a) Electric field: Lateral dependence of $E_\mathrm {OX}$ obtained from simulations with the de- vice simulator MINIMOS-NT.

$ E_\mathrm {OX} $ — (a) Electric field: Lateral dependence of $E_\mathrm {OX}$ obtained from simulations with the de- vice simulator MINIMOS-NT.

By coupling the NMP model with the device simulator MINIMOS-NT [126] and by considering the real distribution functions of the holes and electrons, the accurate transition rates between the states $1$ and $2^\prime$ and the states $2$ and $1^\prime$ for different $V_\mathrm {D}$ can be calculated. Thus the behavior of $\tau _{\mathrm {e}}$ and $\tau _{\mathrm {c}}$ under different stress conditions can be simulated. The electric field and the carrier concentration obtained in simulations using MINIMOS-NT are shown in Figure 5.15. After obtaining the NMP parameters of a defect based on the gate bias dependence of $\tau _{\mathrm {e}}$ and $\tau _{\mathrm {c}}$ (Figure 5.16 for the defect B1), the NMP parameters under consideration of the correct distribution function of the charge carriers in the channel the $\tau _{\mathrm {e}}$ and $\tau _{\mathrm {c}}$ behavior for different $V_\mathrm {D}$ can be calculated without introducing any new parameters.

Figure 5.16: **Gate bias dependence of the characteristic times of switching defect B1 modeled with the four-state NMP model:** The left panel shows the measurement data (circles) and the simulation results (solid lines). The right panels show the shift of the defect due to an increased $E_\mathrm {OX}$ (top) and the different capture and emis- sion pathways which cause the switching behavior. The switching point describes the change from the preferred path for emission $2 \rightarrow 2^\prime \rightarrow 1$ to $2 \rightarrow 1^\prime \rightarrow 1$ . Charging the defects always proceeds over the path $1 \rightarrow 2^\prime \rightarrow 2$ . Figure source: [125].

Figure 5.17 shows the difference between modeling the characteristic quantities of defect B1 using the NMP $_\mathrm {eq.}$ model and the NMP $_\mathrm {neq.}$ model. The characteristic times $\tau _{\mathrm {c}}$ and $\tau _{\mathrm {e}}$ modeled using the NMP $_\mathrm {eq.}$ model show an increasing trend simply due to the change of $E_\mathrm {OX}$ at this lateral position and at $V_\mathrm {G}$   $=$  −1.5 V. This does not correspond to the experimental data, which shows a slightly decreasing trend. Furthermore, although the simulated occupancy captures the general decreasing trend for increasing $|$ $V_\mathrm {D}$ $|$ , it does not reflect the complex experimental behavior. Only if the non-equilibrium conditions are correctly considered an agreement with experimental data is obtained. The NMP $_\mathrm {neq.}$ model is able to capture the rather complex experimental trends, like the decrease of the occupancy to zero at high drain voltages, and properly describes all characteristic quantities of B1.

(a) NMP model assuming equilibrium channel carriers: Neither $\tau _{\mathrm {c}}$ and $\tau _{\mathrm {e}}$ nor the occupancy can be modeled properly. The simulation data of $\tau _{\mathrm {c}}$ and $\tau _{\mathrm {e}}$ show an increasing trend due to the change of $E_\mathrm {OX}$ , which does not correspond to the ex- peremental trend.

$ \tau _{\mathrm {c}} $ — (a) NMP model assuming equilibrium channel carriers: Neither $\tau _{\mathrm {c}}$ and $\tau _{\mathrm {e}}$ nor the occupancy can be modeled properly. The simulation data of $\tau _{\mathrm {c}}$ and $\tau _{\mathrm {e}}$ show an increasing trend due to the change of $E_\mathrm {OX}$ , which does not correspond to the ex- peremental trend.

The behavior shown in Figure 5.17 depends strongly on the configuration of the defect and its lateral position. The impact of the different distribution functions on the defect’s behavior cannot be formulated generally. From the experimental results, one can observe that defects assigned to the magenta group are more affected by changes in the distribution functions of the holes and electrons than defects assigned to the green or blue groups. However, the lateral position of a defect is not a meaningful measure for the classification of the three color groups. For example, the defects A5, D1 and A2 are located near the source and quite close to each other but only A5 and A2 show the typical behavior of the defects in the magenta group.

Using the NMP $_\mathrm {neq.}$ model, not only the behavior of individual defects can be simulated with a excellent agreement with the experimental data, also the recoverable component of $\Delta V_{\mathrm {th}}$ of large-area devices can be modeled. This can be done by assuming a large number of defects with different NMP parameters and by considering the lateral and bias dependent distribution functions of the charge carriers in the channel. The recovery $R$ of the measurements discussed in Section 5.1 can be modeled for different bias stress conditions. Figure 5.18 shows that as long as the carriers in the channel are assumed to be in equilibrium independently of the drain bias, $R$ is reduced only due to the change of $E_\mathrm {OX}$ and no agreement between the experimental data and the simulation can be obtained, quite similar to the discrepancy shown in Figure 5.1 using a simplified electrostatic model. Quite to the contrary, using the non-equilibrium distribution function, the modeled $R$ captures the experimental observations for different stress bias combinations very well.

Figure 5.18: **Recovery in large-area pMOSFETs:** The open circles show the experimental data and the solid and dashed lines illustrate the simulated threshold voltage shifts. **Top:** Comparison of simulation and experimental data for homogeneous BTI conditions. This data set was used to calibrate the NMP model to extract a unique parameter set for all simulations. **Bottom:** $R$ after mixed stress conditions. The NMP $_\mathrm {neq}$ model (solid lines) captures the experi- mental trend, while the equilibrium NMP model (dashed lines) fails to predict the recovery behavior. Figure source: [125].

$ R $ — Figure 5.18: **Recovery in large-area pMOSFETs:** The open circles show the experimental data and the solid and dashed lines illustrate the simulated threshold voltage shifts. **Top:** Comparison of simulation and experimental data for homogeneous BTI conditions. This data set was used to calibrate the NMP model to extract a unique parameter set for all simulations. **Bottom:** $R$ after mixed stress conditions. The NMP $_\mathrm {neq}$ model (solid lines) captures the experi- mental trend, while the equilibrium NMP model (dashed lines) fails to predict the recovery behavior. Figure source: [125].

Finally, Figure 5.19 highlights the main difference between the NMP $_\mathrm {eq.}$ model and the NMP $_\mathrm {neq.}$ model based on the lateral distribution of charged oxide defects directly after stress. Similar to the discussion in Subsection 2.3.1 and the illustration in Figure 2.32, without taking into account non-equilibrium effects defects located near the source remain unaffected. Their behavior does not depend on the drain bias. By contrast, defects located near the source or in the middle of the channel may be uncharged after mixed NBTI/HCD stress due to the reduces oxide field. The NMP $_\mathrm {neq.}$ model, which takes non-equilibrium effects as discussed in the current chapter into account, predicts a faster reduction of charged defects with increasing drain bias. Remarkably, not only defects located near the drain but also near the source may remain uncharged, which corresponds to the experimental observations.

Figure 5.19: **Comparison of the distribution of charged oxide defects directly after stress in large-area pMOSFETs:** Using the NMP $_\mathrm {eq.}$ model defects in the source region are unaffected by an increased $V_\mathrm {D}$ . Defects located near the drain as well as in the middle of the channel may remain uncharged due to the reduced oxide field. The NMP $_\mathrm {neq.}$ model predicts a faster reduction of charged defects (highlighted areas) with increasing drain bias. Remarkably, defects located near the source may remain uncharged as well, which corresponds to the experimental observations. Figure source: [125].

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Transistor	Def. Nr.	Def. Name	$X_T/L$	Type
A	1	A1	0.40	blue
	2	A2	0.21	magenta
	3	A3	–	–
	4	A4	0.32	blue
	5	A5	0.17	magenta
B	1	B1	0.71	magenta
	2	B2	0.82	green
	3	B3	–	–
	4	B4	–	–
C	1	C1	0.81	green
	2	C2	–	–
	3	C3	0.86	magenta
D	1	D1	0.20	blue