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2 Degradation Mechanisms

The term of degradation is associated with unwanted shifts of certain MOSFET characteristics which can endanger the correct interaction with other circuit components in digital circuits. For example, drifts of $V_{\mathrm {th}}$ , also called threshold voltage shift ( $\Delta V_{\mathrm {th}}$ ), during operation seriously reduce the time-to-failure of integrated applications. Degradation phenomena like stress induced leakage current (SILC), time-dependenc dielectric breakdown (TDDB), bias temperature instability (BTI) and hot-carrier degradation (HCD) describe the origins of these uncontrolled drifts. Especially the last two are commonly listed as the most prominent challenges which have to be properly understood [4]. In order to meet these challenges, advanced simulations based on physical models are required for a robust design of circuits.

Throughout the process of understanding and modeling degradation mechanisms, different pieces of the puzzle have been put together in order to get the big picture. In this thesis, one important piece of the puzzle in the context of BTI and HCD and their interplay is contributed. Therefore, the theoretical background and state of the art modeling attempts are introduced in this chapter.

2.1 Bias Temperature Instability

The phenomenon BTI has been known for more than 50 years [16, 28–30] and describes basically the temperature and gate bias, or in other words the oxide electric field ( $E_\mathrm {OX}$ ), dependent shift of transistor parameters. For example, at BTI conditions the transconductance ( $g_\mathrm {m}$ ), the linear drain current ( $I_\mathrm {D,lin}$ ), the saturation drain current ( $I_\mathrm {D,sat}$ ) and the channel mobility ( $\mu _\mathrm {eff}$ ) decrease while the sub-threshold swing ( $SS$ ), the off-current and $V_{\mathrm {th}}$ increase as shown in Figure 2.1. Further, as devices have been downscaled, the vulnerability to BTI has increased. The oxide field at nominal operating conditions has increased due to the scaling of $t_\mathrm {OX}$ , the nominal operating temperature has increased due to the higher power dissipation, and charge exchange events of single defects have a detrimental impact on the MOSFET electrostatics. All these aspects together with the challenge of keeping pace with the downscaling of device dimensions have led to several modeling attempts, which are introduced in this section.

Figure 2.1: **Transfer characteristics and transconductance of a large-area pMOSFET after negative bias temperature instability (NBTI) stress:** The $I_\mathrm {D}$ - $V_\mathrm {G}$ (top) is recorded in the linear region for drain voltage ( $V_\mathrm {D}$ )  $=$  −0.1 V and the transconductance is extracted (bottom). After 1 ks of NBTI stress, the $V_{\mathrm {th}}$ is shifted and $I_\mathrm {D,lin}$ as well as maximum transconductance ( $g_\mathrm {m,max}$ ) are reduced.

$ I_\mathrm {D} $ — Figure 2.1: **Transfer characteristics and transconductance of a large-area pMOSFET after negative bias temperature instability (NBTI) stress:** The $I_\mathrm {D}$ - $V_\mathrm {G}$ (top) is recorded in the linear region for drain voltage ( $V_\mathrm {D}$ )  $=$  −0.1 V and the transconductance is extracted (bottom). After 1 ks of NBTI stress, the $V_{\mathrm {th}}$ is shifted and $I_\mathrm {D,lin}$ as well as maximum transconductance ( $g_\mathrm {m,max}$ ) are reduced.

2.1.1 Experimental BTI Characteristics

Although BTI affects all device parameters, as it is shown illustratively in Figure 2.1, it is commonly studied and expressed in terms of an equivalent $\Delta V_{\mathrm {th}}$ since BTI affects the threshold voltage significantly. Typically, BTI is characterized as a gate bias is applied to the gate contact while drain, source and well contacts are at ground. In experiments BTI is classified according to the sign of the gate bias, namely NBTI if a negative $V_\mathrm {G}$ is applied and positive bias temperature instability (PBTI) if a positive $V_\mathrm {G}$ is applied, commonly studied in pMOSFETs and nMOSFETs, respectively. By contrast, the study of NBTI in nMOSFETs and PBTI in pMOSFETs receives less attention due to the difficulty of the experimental characterization. This is because most of the transistors are protected against electrostatic discharge, which is realized by a diode. This allows only for the operation in inversion mode but not in accumulation. Thus, in most real devices of a commercial technology it is not possible to study NBTI in nMOSFETs and PBTI in pMOSFETs.

Both, NBTI and PBTI, are usually characterized at accelerated stress conditions, which allow for obtaining meaningful parameter shifts within feasible experimental time slots of minutes, days or weeks instead of years. Such accelerated stress is associated with $V_\mathrm {G}$ above nominal operating conditions and elevated temperatures, e.g., 80 °C to 150 °C. Most of the BTI studies in this regard are realized by the on-the-fly (OTF) method, the measure-stress-measure (MSM) or the extended measure-stress-measure (eMSM) method. Although these methods are discussed in the Sections 3.1, 3.4 and 3.5 in detail, they are introduced in this subsection briefly.

• MSM measurements are realized by short interruptions of the applied gate and drain voltages in order to characterize the degradation state of the device, e.g., by taking an $I_\mathrm {D}$ - $V_\mathrm {G}$ curve. From this, the monitored parameter is extracted and the degradation and/or recovery over time is obtained.
• The OTF method obtains $\Delta V_{\mathrm {th}}$ during operation without interruption of the applied voltages. By a periodic modulation of the gate voltage at stress conditions ( $V_{\mathrm {G}}^\mathrm {str}$ ) at a certain $V_\mathrm {D}$ and simultaneous monitoring of $I_\mathrm {D}$ , $\Delta V_{\mathrm {th}}$ can be estimated at a stress level.
• The eMSM method comprises of basically three phases. First, the unstressed device is characterized by taking an $I_\mathrm {D}$ - $V_\mathrm {G}$ . Subsequently, $V_{\mathrm {G}}^\mathrm {str}$ typically higher than nominal operating conditions is applied for a certain stress time ( $t_\mathrm {str}$ ). Finally, the recovery is obtained with the gate voltage set to recovery conditions ( $V_{\mathrm {G}}^\mathrm {rec}$ ) near $V_{\mathrm {th}}$ or below by monitoring the evolution of $\Delta V_{\mathrm {th}}$ over a certain recovery time ( $t_\mathrm {rec}$ ) without interruption of the applied voltages.

Figure 2.2: **Shift of the threshold voltage during stress and recovery:** $\Delta V_{\mathrm {th}}$ during stress and recovery of a multi-gate field-effect transistor (FinFET) is shown. During stress $\Delta V_{\mathrm {th}}$ increases and decreases again as soon as the stress bias is removed. Figure source: [31].

The impact of a stress and a recovery bias on $V_{\mathrm {th}}$ of a FinFET is illustrated in Figure 2.2. During stress, $V_{\mathrm {th}}$ drifts and $\Delta V_{\mathrm {th}}$ , being the difference between the current $V_{\mathrm {th}}$ and the threshold voltage before stress ( $V_{\mathrm {th,0}}$ ), increases while it decreases again as soon as the stress is removed. The decrease of $\Delta V_{\mathrm {th}}$ is also called recovery or relaxation. The sign of $\Delta V_{\mathrm {th}}$ illustrated on a linear scale, as in Figure 2.2, commonly corresponds to whether an n-channel or a p-channel device is probed and whether the absolute value of $V_{\mathrm {th}}$ increases or decreases. In the case that $|$ $V_{\mathrm {th}}$ $|>|$ $V_{\mathrm {th,0}}$ $|$ , $\Delta V_{\mathrm {th}}$ is negative for p-channel devices and positive for n-channel devices while it has the opposite sign for both in case that $|$ $V_{\mathrm {th}}$ $|<|$ $V_{\mathrm {th,0}}$ $|$ . If $|$ $V_{\mathrm {th}}$ $|$ drifts towards larger $|$ $V_\mathrm {G}$ $|$ , which is typically associated with the term of degradation, $\Delta V_{\mathrm {th}}$ drifts to more negative values for p-channel devices and more positive values for n-channel devices. By contrast, the dynamics show the opposite trend if $|$ $V_{\mathrm {th}}$ $|$ shifts towards smaller $|$ $V_\mathrm {G}$ $|$ , which is typically associated with a recovery of the degradation.

The particular behavior of $\Delta V_{\mathrm {th}}$ over time during stress and recovery strongly depends on whether NBTI or PBTI is applied and on almost every device characteristics and probe condition, namely $t_\mathrm {OX}$ , $E_\mathrm {OX}$ , temperature, $W$ and $L$ , transistor type and many more. Useful for the further discussion, the most important dependencies are introduced briefly, starting with the fact that NBTI and PBTI have considerably different impacts on $\Delta V_{\mathrm {th}}$ [32]. A characterization of these different impacts for an $\ch{SiO2}$ oxide can be seen in Figure 2.3. While NBTI ( $V_\mathrm {G}$ $<0$  V applied) has the greatest impact on pMOSFETs, PBTI ( $V_\mathrm {G}$ $>0$  V applied) nearly does not affect nMOSFETs. Based on this, it is not surprising that the most studied case is NBTI on pMOSFETs. In the following, the focus is mainly on NBTI on pMOSFETs.

Figure 2.3: **Shift of the threshold voltage under NBTI and PBTI stress:** Both BTI classifications for $\ch{SiO2}$ nMOSFET and pMOSFET. NBTI stress conditions with $V_\mathrm {G}$ $<0$  V have the greatest impact on pMOSFET. Figure source: [32].

Figure 2.3: **Shift of the threshold voltage under NBTI and PBTI stress:** Both BTI classifications for $\ch{SiO2}$ nMOSFET and pMOSFET. NBTI stress conditions with $V_\mathrm {G}$ $<0$  V have the greatest impact on pMOSFET. Figure source: [32].

Moreover, the evolution of $\Delta V_{\mathrm {th}}$ is affected by the gate bias applied during stress as well as during recovery as illustrated in Figure 2.4. Based on the impact of $V_{\mathrm {G}}^\mathrm {str}$ and the gate voltage at recovery conditions ( $V_{\mathrm {G}}^\mathrm {rec}$ ) on $\Delta V_{\mathrm {th}}$ measured for a large-area $\ch{SiON}$ pMOSFET it can be seen that while a higher $V_{\mathrm {G}}^\mathrm {str}$ accelerates the increase of $\Delta V_{\mathrm {th}}$ over time a higher $V_{\mathrm {G}}^\mathrm {rec}$ suppresses the recovery of $\Delta V_{\mathrm {th}}$ [33].

(a) The $\Delta V_{\mathrm {th}}$ behavior depends strongly on $V_{\mathrm {G}}^\mathrm {str}$ . With increasing $V_{\mathrm {G}}^\mathrm {str}$ the absolute value of $\Delta V_{\mathrm {th}}$ increases.

Similar to the acceleration of the $V_{\mathrm {th}}$ shift over time due to a higher gate bias at stress conditions, degradation is also accelerated because of an elevated temperature ( $T$ ). In this regard, Figure 2.5 shows that the degradation of $\Delta V_{\mathrm {th}}$ is higher and changes faster with elevated $T$ [33]. By contrast, the $\Delta V_{\mathrm {th}}$ recovery shows only a weak, almost negligible, temperature dependence for large-area devices, which is not shown here.

Figure 2.5: **Temperature dependence of degradation at the same stress bias:** $T$ accelerates the degradation of $\Delta V_{\mathrm {th}}$ , which is shown here for a FinFET at three different temperatures (open and closed symbols; two similarly processed wafers). The data is described by a power-law dependence. Figure source: [31].

$ T $ — Figure 2.5: **Temperature dependence of degradation at the same stress bias:** $T$ accelerates the degradation of $\Delta V_{\mathrm {th}}$ , which is shown here for a FinFET at three different temperatures (open and closed symbols; two similarly processed wafers). The data is described by a power-law dependence. Figure source: [31].

All shown dependencies on gate bias and temperature are different for different device dimensions and different architectures. While the latter are not discussed in detail here, the changes in device dimensions together with the limitations in time of the experimental window are quite important in regard of understanding the different approaches of modeling device degradation. For an explanation, one should take a look at the lower limit of the experimental window during stress ( $t_\mathrm {str,min}$ ) of the measurements in Figure 2.3 and Figure 2.5. This lower limit is 1 s and 10 s, respectively, which are common lower limits in literature until approximately 2004. The reason lies in the measurement sequences for the characterization of degradation. The interruptions of the applied voltages using the MSM method, although as short as possible, can take 50 ms or more and disturb the degradation or recovery state seriously. Due to such a distortion of the degradation or recovery state the MSM method often cannot capture short-term effects because the interruption of the applied voltages might reverse their impact on device characteristics. This makes reliable short-term measurements ( $t_\mathrm {str}$   $<$  1 s or $t_\mathrm {rec}$   $<$  1 s) impossible.

As a result, for a long period of time models were developed based on mainly long-term $\Delta V_{\mathrm {th}}$ measurements from 1 s to nearly 10 ks. The long-term characterization, at a first glance and in a very simplified way, shows that $\Delta V_{\mathrm {th}}$ in stress measurements of large-area devices has power-law-like behavior. Thus, simple empiric models based on a power-law-like life-time estimation as shown in Figure 2.5 have been popular [32, 34–36].

The power-law covers a stress bias dependent pre-factor, a temperature dependence following an Arrhenius’ law and the power-law in time. Based on this, the $\Delta V_{\mathrm {th}}$ can be extrapolated in order to estimate the parameter shift over time. However, such an empirical description is quite inaccurate and overestimates the $\Delta V_{\mathrm {th}}$ over time because it neither describes the short-term behavior for stress times below 1 s nor the very long-term behavior for stress times larger than 10 ks nor recovery effects due to interruptions of the stress properly as discussed in the following.

2.1.2 The Reaction-Diffusion Model

Beside the empirically found power-law as an attempt to estimate the life-time of a transistor, also physics-based models have been introduced in order to describe the processes leading to device degradation. A widely accepted model in this regard is the reaction-diffusion model. This model is capable of reproducing the time evolution of device degradation of large-area MOSFETs. It was introduced in 1977 [30] and continuously adapted [32, 37, 38]. The basic assumption of the reaction-diffusion model is that $\ch{Si}$ - $\ch{H}$ bonds at the interface between the substrate and the oxide can be broken by NBTI stress, schematically shown in Figure 2.6. Consequently, the remaining $\ch{Si}$ dangling bonds (interface-states or interface-traps) are positively charged, which can be expressed by a interface-charge density ( $N_\mathrm {it}$ ), and the hydrogen atom diffuses into the dielectric where also other $\ch{H}$ related species can be created like for example a $\ch{H2}$ molecule. Further, $\ch{H}$ can also recombine with the positively charged Si dangling bond, thus the bond is again passivated.

Figure 2.6: **Schematic illustration of the classical reaction-diffusion model of NBTI:** $\ch{Si}$ - $\ch{H}$ bonds at the interface between the substrate and the oxide are broken during NBTI stress. Neutral $\ch{H}$ , expressed by the $\ch{H}$ density $H(x,t)$ , diffuses into the oxide and leaves behind posi- tively charged interface states. $\ch{H}$ diffusion proceeds via shallow hopping sites in the oxide shown as a regular network of potential wells. Figure source: [31].

Figure 2.6: **Schematic illustration of the classical reaction-diffusion model of NBTI:** $\ch{Si}$ - $\ch{H}$ bonds at the interface between the substrate and the oxide are broken during NBTI stress. Neutral $\ch{H}$ , expressed by the $\ch{H}$ density $H(x,t)$ , diffuses into the oxide and leaves behind posi- tively charged interface states. $\ch{H}$ diffusion proceeds via shallow hopping sites in the oxide shown as a regular network of potential wells. Figure source: [31].

As soon as experimental characterization methods were improved and capable of measuring short-term effects a few inconsistencies between the reaction-diffusion model and the experimental data arose. In the context of the improvement of measurement methods, three aspects have to be mentioned: the characterization of the recoverable component of BTI, the continuous application of stress and the broadening of the experimental window. As an explanation, in Subsection 2.1.1 it is mentioned that as soon as the stress bias is removed a recovery effect of $\Delta V_{\mathrm {th}}$ is measured. As long as the stress and recovery in MSM measurements are interrupted in order to record a transfer characteristic for the extraction of $V_{\mathrm {th}}$ the overall degradation and recovery state will be different than if the stress and recovery biases are applied continuously. Therefore, innovative measurement methods have been introduced, which can obtain $\Delta V_{\mathrm {th}}$ without any interruptions of stress or recovery and additionally are capable of the characterization of short-term effects. An example in this regard is the OTF method capable of obtaining the $\Delta V_{\mathrm {th}}$ degradation without interruptions of the stress bias. However, realized with standard equipment the OTF has an integration time of more than 20 ms in order to achieve a statistical error of $\pm$ 1 mV in $\Delta V_{\mathrm {th}}$ [39], which is quite long in the context of short-term measurement. Soon after the OTF method had been proposed, the fast- $V_{\mathrm {th}}$ and fast- $I_\mathrm {D}$ methods were introduced [31, 39, 40]. Such fast methods are based on single point measurements of $\Delta V_{\mathrm {th}}$ during recovery instead of interruptions in order to record an $I_\mathrm {D}$ - $V_\mathrm {G}$ . Either $I_\mathrm {D}$ or $V_\mathrm {G}$ is measured at a point near $V_{\mathrm {th}}$ during the recovery phase in eMSM measurements and the corresponding $\Delta V_{\mathrm {th}}$ is extracted. Both will be discussed in detail in Section 3.5. These new methods extended the $t_\mathrm {str,min}$ and the lower limit of the experimental window during recovery ( $t_\mathrm {rec,min}$ ) to 100 µs or even 1 µs, which allows for a proper short-term characterization of BTI recovery.

The improvement of experimental setups and the broadening of the experimental window have led to different insights into degradation mechanisms and the understanding of BTI changed. Some important inconsistencies between the reaction-diffusion model and the experimental data are:

Figure 2.7: **Recovery traces of a nano-scale pMOSFET and the spectral map:** Two $\Delta V_{\mathrm {th}}$ recovery traces of a pMOSFET. **Top:** Recovery proceeds step-wise due to emission events of single defects in the oxide. The symbols mark the extracted emission times and step heights which are nearly unambiguous fingerprints of each defect. **Bottom:** The step heights and the emission time build the spectral map. Figure source: [41].

• A $\log ($ $t_\mathrm {str}$ $)$ behavior of $\Delta V_{\mathrm {th}}$ for $t_\mathrm {str}$ $<$ 1 s was found. Several investigations [31, 32, 39, 42] have led to the conclusion that not only the creation of interface states contributes to $\Delta V_{\mathrm {th}}$ during stress but also hole trapping in oxide defects near the interface between substrate and oxide. It has been found that there is evidence that these processes are highly coupled, which is not possible to be explained by the reaction-diffusion model.
• Even after long stress times ( $>$ 1 ks) $\Delta V_{\mathrm {th}}$ can recover more than 60 % within the first second [43]. Since the reaction-diffusion model assumes diffusion-limited degradation, which implies also a diffusion limited relaxation, 60 % of recovery means that 60 % of the hydrogen must diffuse back to the interface and passivate the positively charged dangling bonds within one second. This implies that the backward diffusion happens by orders of magnitude faster than the forward diffusion. This cannot be explained by the effect of diffusion since the diffusivity of $\ch{H}$ is a material constant.
• The overall recovery is considerably slower than the degradation during stress as obtained from eMSM measurements [44], which can be understood as an asymmetry of stress and recovery. In this context, the reaction-diffusion theory predicts a relatively short recovery phase (50 % relative recovery as soon as $t_\mathrm {rec}$ $=$ $t_\mathrm {str}$ ). Thus, the reaction-diffusion model is not capable to model the asymmetry.
• Recovery is a bias-dependent process (shown in Subsection 2.1.1). This is not in agreement with the reaction-diffusion model because it predicts recovery due to the back-diffusion of neutral $\ch{H2}$ and as such is bias-independent.
• It has been shown that interface-states nearly do not recover in the context of MSM measurements [45]. Therefore, the reaction-model, where recovery is explained as passivation of interface-states is not suitable to model recovery.

In order to overcome some of the mentioned inconsistencies, the reaction-diffusion model has been improved over the last two decades including the attempt to describe the diffusion as a dispersive process instead of a classical, i.e., Gaussian-like, process [31, 32, 46, 47].

These improvements notwithstanding, the complicated behavior of $\Delta V_{\mathrm {th}}$ during stress and recovery has led to the development of the non-radiative-multiphonon (NMP) model as a different approach to explain BTI. For the development of the NMP model two processes played a major role, the improvement of measurement setups as well as the downscaling of MOSFETs. As soon as devices were scaled to the nanometer regime it has been recognized that the recovery of MOSFETs proceeds not continuously as shown in Figure 2.2 but step-wise as shown in Figure 2.7 top.

2.1.3 Properties of Material Defects in Experiments

Back in the 80’s, it was found that random fluctuations in the terminal currents are caused by structural defects, also called states or traps, in the bulk oxide. Such fluctuations are due to defects randomly exchanging charge carriers with the substrate [13, 48]. The corresponding noise was introduced as random telegraph noise (RTN). The study of nano-scale devices has shown that device degradation and recovery is determined by single hole capture and emission processes in pMOSFETs, respectively, which is consistent with a first-order reaction-rate but not with a diffusion-limited process [40, 49, 50]. Soon it has been proposed that single oxide defects near the interface between oxide and substrate communicate with the inversion layer in the channel by exchanging charge carriers.

Such charge carrier exchange events, also called capture and emission events, of oxide defects near the interface are a serious perturbance of the electrostatic conditions in nano-scale devices. Considering the average impact of one capture or emission event due to the downscaled device dimensions and the inhomogeneous channel potential due to randomly placed dopants [51], such events cause step-wise measurable shifts in $\Delta V_{\mathrm {th}}$ experiments. Therefore, each individual defect leaves its fingerprint by the fact that it appears with a certain step height ( $d$ ), a certain mean value of the capture time ( $\tau _{\mathrm {c}}$ ) and a certain mean value of the emission time ( $\tau _{\mathrm {e}}$ ) in $\Delta V_{\mathrm {th}}$ traces, which depend on the position relative to the dopants, the depth of the defect in the oxide, the gate bias and the temperature. For example, as discussed in Chapter 1, the randomly placed dopants cause the current to flow inhomogeneously from source to drain. As soon as a defect is located right above the percolation path [52], its step height can be considerably larger than estimated from the charge-sheet approximation [53].

Figure 2.8: **Step height distribution of individual defects:** (a) Typical NBTI recovery traces in nano-scale devices. Each step corresponds to a single gate oxide defect discharge event. (b) The $\Delta V_{\mathrm {th}}$ step heights plotted on complementary cumulative distribution function (CCDF) plot. The step heights appear exponentially distributed. Figure source: [18].

In this context, based on the characterization of the step heights caused by single defects, an exponential step height distribution has been found [17, 18, 54–56] as shown in Figure 2.8. Since this distribution is based on a statistical characterization, the influence of device-to-device variation of the number of oxide defects and random dopants are considered. In detail, the probability density function (PDF) can be expressed as

$(2.1) \begin{equation} \label {eq:PDFshd} f(\Delta V_\mathrm {th},\eta )=\frac {1}{\eta }\mathrm {e}^{-\frac {\Delta V_\mathrm {th}}{\eta }} \end{equation}$

with

The corresponding CCDF is

$(2.2) \begin{equation} \label {eq:CCDFshd} F(\Delta V_\mathrm {th},\eta )=\mathrm {e}^{-\frac {\Delta V_\mathrm {th}}{\eta }} \textrm {.} \end{equation}$

The mean value of the $d$ distribution can be written as

$(2.3) \begin{equation} \label {eq:etameanshd} \eta =\frac {q}{C_{ox}}\cdot 2 \end{equation}$

with

(a) Stress: single oxide defects capture holes from the inversion layer. This process is measureable as single steps at stochastically distributed $\tau _{\mathrm {c}}$ around a characteristic mean value, which depends strongly on $V_{\mathrm {G}}^\mathrm {str}$ . With increasing $V_{\mathrm {G}}^\mathrm {str}$ , $\tau _{\mathrm {c}}$ decreases.

From experiments with the TDDS framework on nano-scale MOSFETs, it has been found that BTI degradation and recovery can be explained by capture and emission events of single oxide defects [16]. The TDDS framework, which will be introduced in Section 3.7 in detail, consists of several eMSM cycles followed by a postprocessing of the recorded data. $\Delta V_{\mathrm {th}}$ is obtained during the recovery phase by a single point measurement of $I_\mathrm {D}$ or $V_\mathrm {G}$ near $V_{\mathrm {th}}$ (also known as fast- $I_\mathrm {D}$ or fast- $V_{\mathrm {th}}$ methods in literature [31, 39, 40]). Figure 2.7 top shows typical recovery measurements of a nano-scale MOSFET, containing the steps of five defects enumerated with 1, 2, 3, 4 and 12. Due to the fact that the capture and emission events are assumed to be stochastic processes, TDDS requires a number, e.g., $100$ , of the same stress/recovery experiments in order to capture the statistics for a reliable characterization. The spectral map in Figure 2.7 bottom visualizes the individuality of each defect. It is built by entering $d$ and $\tau _{\mathrm {e}}$ of each emission event into a two-dimensional diagramm. From this, the spectral map can be obtained as the distribution of the numerical data [41].

$\tau _{\mathrm {e}}$ and $\tau _{\mathrm {c}}$ are strongly bias and temperature dependent. In this regard, the bias dependence is shown in Figure 2.9 where three defects characterized on nano-scale SiON pMOSFETs capture a hole during stress and emit it during recovery. The capture events (Subfigure 2.9a) cause an increase of the absolute value of $\Delta V_{\mathrm {th}}$ comparable to the degradation of large-area devices shown in Subfigure 2.4a while the emission events (Subfigure 2.9b) cause a decrease of $\Delta V_{\mathrm {th}}$ comparable to the recovery of large-area devices shown in Subfigure 2.4b. Both, capture and emission events happen at stochastically distributed $\tau _{\mathrm {c}}$ and $\tau _{\mathrm {e}}$ , respectively.

With increasing $V_{\mathrm {G}}^\mathrm {str}$ $\tau _{\mathrm {c}}$ decreases and with increasing $V_{\mathrm {G}}^\mathrm {rec}$ $\tau _{\mathrm {e}}$ increases. This leads to a different contribution of the three shown defects to degradation and recovery of $\Delta V_{\mathrm {th}}$ . At $V_{\mathrm {G}}^\mathrm {str}$ $=$ −1.4 V (blue trace in Subfigure 2.4a) $\tau _{\mathrm {c}}$ of defect 2 and 3 are larger than the upper limit of the experimental window during stress ( $t_\mathrm {str,max}$ ), which is 1 s. Thus, only defect 1 contributes to $\Delta V_{\mathrm {th}}$ . At $V_{\mathrm {G}}^\mathrm {str}$ $=$ −1.7 V (green trace in Subfigure 2.4a) $\tau _{\mathrm {c}}$ of defect 2 is decreased and shifted to times within the experimental window. Therefore, the capture event of defect 1 and defect 2 contribute to $\Delta V_{\mathrm {th}}$ . Finally at $V_{\mathrm {G}}^\mathrm {str}$ $=$ −1.9 V and $V_{\mathrm {G}}^\mathrm {str}$ $=$ −2.2 V also defect 3 contributes to $\Delta V_{\mathrm {th}}$ .

At recovery conditions, the process is quite similar. While at $V_{\mathrm {G}}^\mathrm {rec}$ $=$ −0.2 V all three defects contribute to recovery, at $V_{\mathrm {G}}^\mathrm {rec}$ $=$ −1 V only defect 3 contributes to $\Delta V_{\mathrm {th}}$ because $\tau _{\mathrm {e}}$ of the other two defects are higher than the upper limit of the experimental window during revery ( $t_\mathrm {rec,max}$ ). Quite comparable to the measurements in large-area devices (Figure 2.4) the evolution of $\Delta V_{\mathrm {th}}$ of nano-scale MOSFETs is accelerated during stress with increasing $V_\mathrm {G}$ and decelerated during recovery.

Figure 2.10: **Temperature dependence of the characteristic emission times:** Temperature accelerates degradation and recovery. Figure source: [33].

The capture and emission events are also highly affected by $T$ as shown in Figure 2.10 based on the temperature dependence of the emission events. In contrast to the discussion in Subsection 2.1.1 regarding the temperature dependence of recovery of large-area devices, which is negligible, in nano-scale devices the picture is a little more complicated. Basically, with increasing $T$ , both, $\tau _{\mathrm {c}}$ during stress and $\tau _{\mathrm {e}}$ during recovery decrease to lower values. In Figure 2.10 it can be seen that $\tau _{\mathrm {e}}$ decreases with increasing $T$ but the overall difference beween $\Delta V_{\mathrm {th}}$ at the lower limit of the experimental window and $\Delta V_{\mathrm {th}}$ at the upper limit is temperature independent because only the three defects can contribute to recovery.

The bias dependence of the capture and emission times leads to a crossing point of both, where $\tau _{\mathrm {e}}$ $($ $V_\mathrm {G}$ $)$ and $\tau _{\mathrm {c}}$ $($ $V_\mathrm {G}$ $)$ are equal, illustrated in Figure 2.16. Around this crossing point, defects cause random exchange events as shown in Subfigure 2.12b, which is called RTN. Since the whole $\tau _{\mathrm {e}}$ $($ $V_\mathrm {G}$ $)$ and $\tau _{\mathrm {c}}$ $($ $V_\mathrm {G}$ $)$ behavior is different for each defect, at a certain $V_\mathrm {G}$ some defects might capture charge carriers (because $\tau _{\mathrm {c}}$   $\gg$   $\tau _{\mathrm {e}}$ ), some defects might emit previously captured charge carriers (because $\tau _{\mathrm {c}}$   $\ll$   $\tau _{\mathrm {e}}$ ) and some defects might cause an RTN signal (because $\tau _{\mathrm {e}}$   $\approx$   $\tau _{\mathrm {c}}$ ). Finally, summing up the contribution of the transitions of all electrically active defects to $\Delta V_{\mathrm {th}}$ results in what is conventianally observed as BTI degradation and recovery.

Figure 2.11: **A capture/emission time:** The capture/emission time (CET) map is obtained by a variation of $V_{\mathrm {G}}^\mathrm {str}$ and $V_{\mathrm {G}}^\mathrm {rec}$ at NBTI conditions (left). The solid lines are obtained by integrating the CET map following and the dashed lines are the permanent component not visible in the CET map (right). The CET map summarizes the particular $\tau _{\mathrm {e}}$ $($ $V_\mathrm {G}$ $)$ and $\tau _{\mathrm {c}}$ $($ $V_\mathrm {G}$ $)$ behavior of many defects. Figure source: [16].

$ V_{\mathrm {G}}^\mathrm {str} $ — Figure 2.11: **A capture/emission time:** The capture/emission time (CET) map is obtained by a variation of $V_{\mathrm {G}}^\mathrm {str}$ and $V_{\mathrm {G}}^\mathrm {rec}$ at NBTI conditions (left). The solid lines are obtained by integrating the CET map following and the dashed lines are the permanent component not visible in the CET map (right). The CET map summarizes the particular $\tau _{\mathrm {e}}$ $($ $V_\mathrm {G}$ $)$ and $\tau _{\mathrm {c}}$ $($ $V_\mathrm {G}$ $)$ behavior of many defects. Figure source: [16].

Such a broad distribution of the characteristic times and their bias dependence can be illustrated as a CET map shown in Figure 2.11. The idea is that similar defects can be grouped together using a suitably defined density $g$ [57]. In order to draw such a map, the capture time is obtained at different stress conditions while the emission time is taken at different recovery conditions. With increasing $V_{\mathrm {G}}^\mathrm {str}$ and constant $V_{\mathrm {G}}^\mathrm {rec}$ , $\tau _{\mathrm {c}}$ decreases, while $\tau _{\mathrm {e}}$ is not affected. With increasing $V_{\mathrm {G}}^\mathrm {rec}$ and constant $V_{\mathrm {G}}^\mathrm {str}$ , $\tau _{\mathrm {e}}$ increases, while $\tau _{\mathrm {c}}$ is not affected. The particular behavior of $\tau _{\mathrm {e}}$ $($ $V_\mathrm {G}$ $)$ and $\tau _{\mathrm {c}}$ $($ $V_\mathrm {G}$ $)$ can be plotted as a density:

$g(\tau _\mathrm {c},\tau _\mathrm {e})\approx -\dfrac {\partial ^2\Delta V_\mathrm {th}(\tau _\mathrm {c},\tau _\mathrm {e})}{\partial \tau _\mathrm {c} \tau _\mathrm {e}}$

$\tilde {g}(\tau _\mathrm {c},\tau _\mathrm {e})=\tau _\mathrm {c}\tau _\mathrm {e}g(\tau _\mathrm {c},\tau _\mathrm {e})$

if it is represented on logarithmic axes.

Figure 2.11 shows how broad the distribution of the capture and emission times can be, which cannot be explained by models like SRH or the reaction-diffusion model. Thus, new perspectives were needed.

2.1.4 Two-State Model Including Non-Radiative Transitions

Due to several inconsistencies of the reaction-diffusion model with the experimental observations, single donor-like defects in the oxide have been taken into account as the main physical cause of NBTI. The random exchange of charge carriers between an oxide defect and the substrate, which occurs if the defect energy is approximately equal to the Fermi level, produces an RTN signal in $\Delta V_{\mathrm {th}}$ measurements as shown in Subfigure 2.12b. One simple modeling attempt to describe charge transfer reactions in RTN signals as capture and emission events is provided by the two-state model illustrated in Subfigure 2.12a. Therefore, the charge state of the defect was described by either the neutral state $1$ or the charged state $2$ with the defect occupancy of the state $i$ ( $X_{i}$ ) being $1$ when the defect is in state $i$ and $0$ otherwise [41]. The transition probabilities are obtained as a Markov process. The probability for the transition from $i$ to $j$ within the next infinitesimally small time interval $h$ is given as

(a) State transition rate diagram for a two-state model

$(2.4) \begin{equation} \label {eq:transitionprobij} P\left \{X_{j}(t+h)=1\vert X_{i}(t)=1\right \}=k_{ij}h+O(h) \end{equation}$

with $\lim _{h\to 0}O(h)/h=0$ and

The probability that no transitions from $j$ to $i$ occurs is given as

$(2.5) \begin{equation} \label {eq:transitionprobjinot} P\left \{X_{j}(t+h)=1\vert X_{j}(t)=1\right \}=1-k_{ji}h+O(h)\textrm {.} \end{equation}$

If $h$ is assumed to be so small that all higher-order terms are negligible and that the defect is currently in state $i$ , $p_{i}(t)=P\left \{X_{i}(t)=1\right \}$ the probability that $X_{j}(t+h)=1$ is given as

$(2.6) \begin{equation} \label {eq:probstatej1} p_{j}(t+h)=P\left \{X_{j}(t+h)=1\vert X_{i}(t)=1\right \}p_{i}(t)+P\left \{X_{j}(t+h)=1\vert X_{j}(t)=1\right \}p_{j}(t) \end{equation}$

By replacing the conditional probabilities in Equation 2.6 by the rates in Equation 2.4 and Equation 2.5, rearrangement and inserting $1-p_{j}(t)$ for $p_{i}(t)$ , the differential equation can be obtained:

$(2.7) \begin{equation} \label {eq:diffequtwostates} \dfrac {dp_{j}(t)}{dt}=k_{ij}(1-p_{j}(t))-k_{ji}p_{j}(t)\textrm {.} \end{equation}$

This is an ordinary differential equation with the solution

$(2.8) \begin{equation} \label {eq:probdiffsolution} p_{j}(t)=p_{j}(\infty )+(p_{j}(0)-p_{j}(\infty ))\mathrm {e}^{-t/\tau } \end{equation}$

with

$ p_{j}(\infty )=k_{ij}/(k_{ij}+k_{ji}) $

This solution describes the probability of the defect being in state $j$ as an exponential transition from its inital value $p_{j}(0)$ to its final stationary value $p_{j}(\infty )$ .

The probability that the defect is in the other state $i$ can be calculated as $1-p_{j}(t)$ . If it is assumed that $i=1$ is the neutral state and $j=2$ is the charged state, the capture time $\tau _{\mathrm {c}}$ and emission time $\tau _{\mathrm {e}}$ would be the transition times from $i$ to $j$ ( $\tau _\mathrm {12}$ ) and from $j$ to $i$ ( $\tau _\mathrm {21}$ ), respectively. $\tau _\mathrm {12}$ and $\tau _\mathrm {21}$ are stochastic variables. For the calculation of the transition times, the initial defect state is assumed to be neutral, thus, in state $1$ ( $p_\mathrm {1}(0)=1$ ). In this case, the time until the defect transits to state $2$ is independent of the backward rate $k_\mathrm {21}$ . The probability that the defect is in state $1$ at a certain time ( $t$ ) is given as $p_\mathrm {1}(t)=\exp (-k_\mathrm {12}t)$ and that it is in state $2$ as $p_\mathrm {2}(t)=1-p_\mathrm {1}(t)$ . If $\tau _\mathrm {12}<t$ , the probability that the defect is in state $2$ can be written as $p_\mathrm {2}(\tau _\mathrm {12})=1-\exp (-k_\mathrm {12}\tau _\mathrm {12})$ and the PDF can be expressed as $g(\tau _\mathrm {12})=dp_\mathrm {2}(\tau _\mathrm {12})/d\tau _\mathrm {12}=k_\mathrm {12}\exp (-k_\mathrm {12}\tau _\mathrm {12})$ . The mean value of $\tau _{\mathrm {c}}$ is the expectation value of the exponential distribution, which is given by

$(2.9) \begin{equation} \label {eq:twostatecapturetime} \bar {\tau _\mathrm {c}}=\int _{0}^{\infty }\tau _\mathrm {c}g(\tau _\mathrm {c})d\tau _\mathrm {c}=\dfrac {1}{k_\mathrm {12}}\textrm {.} \end{equation}$

Similar arguments hold also for the emission time:

$(2.10) \begin{equation} \label {eq:twostateemissiontime} \bar {\tau _\mathrm {e}}=\int _{0}^{\infty }\tau _\mathrm {e}g(\tau _\mathrm {e})d\tau _\mathrm {e}=\dfrac {1}{k_\mathrm {21}} \end{equation}$

with

A proper RTN analysis is only possible if $\bar {\tau _\mathrm {c}}\approx \bar {\tau _\mathrm {e}}$ (energy level of the defect is located near the Fermi level), ( $\tau _{\mathrm {e}}$ , $\tau _{\mathrm {c}}$ )  $\in [$ $t_\mathrm {rec,min}$ , $t_\mathrm {rec,max}$ $]$ and $\tau _{\mathrm {e}}$ , $\tau _{\mathrm {c}}$   $\ll$   $t_\mathrm {rec,max}$ at certain $V_\mathrm {G}$ and $T$ . Only RTN signals caused by defects with characteristic times, which fullfil these prerequisites, can be characterized because of a statistically meaningful number of transitions.

If $\bar {\tau _\mathrm {c}}\ll \bar {\tau _\mathrm {e}}$ or $\bar {\tau _\mathrm {c}}\gg \bar {\tau _\mathrm {e}}$ , eMSM measurements are required where the gate bias is switched from $V_{\mathrm {G}}^\mathrm {rec}$ to $V_{\mathrm {G}}^\mathrm {str}$ and back to $V_{\mathrm {G}}^\mathrm {rec}$ in order to obtain the probability transition. For this it is assumed that the Markov process is stationary before each change of the gate voltage. Before switching from $V_{\mathrm {G}}^\mathrm {rec}$ to $V_{\mathrm {G}}^\mathrm {str}$ the probability that the defect is in its state $2$ is given as $p_\mathrm {2}(t)=p_\mathrm {2}(V_\mathrm {G}^\mathrm {rec})$ . With Equation 2.8 the probability after switching the gate voltage from $V_{\mathrm {G}}^\mathrm {rec}$ to $V_{\mathrm {G}}^\mathrm {str}$ can be calculated as

$(2.11) \begin{equation} \label {eq:biasswitchprob} p_\mathrm {2}(t)=p_\mathrm {2}(V_\mathrm {G}^\mathrm {str})+\left (p_\mathrm {2}(V_\mathrm {G}^\mathrm {rec})-p_\mathrm {2}(V_\mathrm {G}^\mathrm {str})\right )\mathrm {e}^{-t_\mathrm {str}/\tau _\mathrm {c}} \end{equation}$

and the probability after switching the gate voltage from $V_{\mathrm {G}}^\mathrm {str}$ to $V_{\mathrm {G}}^\mathrm {rec}$ as

$(2.12) \begin{equation} \label {eq:biasswitchprobback} p_\mathrm {2}(t)=p_\mathrm {2}(V_\mathrm {G}^\mathrm {rec})+\left (p_\mathrm {2}(t_\mathrm {str,max})-p_\mathrm {2}(V_\mathrm {G}^\mathrm {rec})\right )\mathrm {e}^{-t_\mathrm {rec}/\tau _\mathrm {e}} \end{equation}$

with

The corresponding transition time constants are given as

$(2.13) \begin{equation} \label {eq:biasswitchtauc} \tau _\mathrm {c}=\dfrac {1}{\dfrac {1}{\tau _\mathrm {c}(V_\mathrm {G}^\mathrm {str})}+\dfrac {1}{\tau _\mathrm {e}(V_\mathrm {G}^\mathrm {str})}} \end{equation}$

and

$(2.14) \begin{equation} \label {eq:biasswitchtaue} \tau _\mathrm {e}=\dfrac {1}{\dfrac {1}{\tau _\mathrm {c}(V_\mathrm {G}^\mathrm {rec})}+\dfrac {1}{\tau _\mathrm {e}(V_\mathrm {G}^\mathrm {rec})}}\textrm {.} \end{equation}$

The corresponding occupancies are given as

$(2.15) \begin{equation} \label {eq:biasswitchoccstr} f(V_\mathrm {G}^\mathrm {str})=\dfrac {\tau _\mathrm {e}(V_\mathrm {G}^\mathrm {str})}{\tau _\mathrm {e}(V_\mathrm {G}^\mathrm {str})+\tau _\mathrm {c}(V_\mathrm {G}^\mathrm {str})} \end{equation}$

and

$(2.16) \begin{equation} \label {eq:biasswitchoccrec} f(V_\mathrm {G}^\mathrm {rec})=\dfrac {\tau _\mathrm {e}(V_\mathrm {G}^\mathrm {rec})}{\tau _\mathrm {e}(V_\mathrm {G}^\mathrm {rec})+\tau _\mathrm {c}(V_\mathrm {G}^\mathrm {rec})}\textrm {.} \end{equation}$

According to Equation 2.11, the occupancy for the transition from $f(V_\mathrm {G}^\mathrm {rec})$ to $f(V_\mathrm {G}^\mathrm {str})$ is

$f(t_\mathrm {str})=f(V_\mathrm {G}^\mathrm {str})+\left (f(V_\mathrm {G}^\mathrm {rec})-f(V_\mathrm {G}^\mathrm {str})\right )\mathrm {e}^{-t_\mathrm {str}/\tau _\mathrm {c}}$

while according to Equation 2.12, the transition from $f(V_\mathrm {G}^\mathrm {str})$ to $f(V_\mathrm {G}^\mathrm {rec})$ is

$f(t_\mathrm {str},t_\mathrm {rec})=f(V_\mathrm {G}^\mathrm {rec})+\left (f(t_\mathrm {str})-f(V_\mathrm {G}^\mathrm {rec})\right )\mathrm {e}^{-t_\mathrm {str}/\tau _\mathrm {e}}\textrm {.}$

Experimentally the differences $f(t_\mathrm {str})-f(V_\mathrm {G}^\mathrm {rec})$ is seen when switching from recovery to stress conditions and $f(t_\mathrm {str},t_\mathrm {rec})-f(V_\mathrm {G}^\mathrm {rec})$ when switching from stress to recovery conditions which results in

$(2.17) \begin{equation} \label {eq:deltaocc} \Delta f(t_\mathrm {str})=\left (f(V_\mathrm {G}^\mathrm {str})-f(V_\mathrm {G}^\mathrm {rec})\right )\left (1-\mathrm {e}^{-t_\mathrm {str}/\tau _\mathrm {c}}\right ) \end{equation}$

and

$(2.18) \begin{equation} \label {eq:deltaoccback} \Delta f(t_\mathrm {str},t_\mathrm {rec})=\left (f(V_\mathrm {G}^\mathrm {str})-f(V_\mathrm {G}^\mathrm {rec})\right )\left (1-\mathrm {e}^{-t_\mathrm {str}/\tau _\mathrm {c}}\right )\mathrm {e}^{-t_\mathrm {str}/\tau _\mathrm {e}}\textrm {,} \end{equation}$

respectively.

The overall degradation can be obtained as

$(2.19) \begin{equation} \label {eq:overalldegradation} -\Delta V_\mathrm {th}(t_\mathrm {str},t_\mathrm {rec})=\sum _k^N d_\mathrm {k}\left (f_\mathrm {k}(V_\mathrm {G}^\mathrm {str})-f_\mathrm {k}(V_\mathrm {G}^\mathrm {rec})\right )h_\mathrm {k}(t_\mathrm {str},t_\mathrm {rec};\tau _\mathrm {c,k},\tau _\mathrm {e,k}) \end{equation}$

with

Figure 2.13: **Configuration coordinate diagram for a two-state model using a non-radiative multiphonon theory:** The transition from $i$ to $j$ proceeds either radiatively or non-radiatively. In typi- cal semiconductor devices, the radiative transition can be excluded as no photons are available during the regular operation. Therefore, the energy needed to overcome the difference between the minimum point and the crossing point of the parabolas hasto be supplied by phonons. Figure source: [58].

$ i $ — Figure 2.13: **Configuration coordinate diagram for a two-state model using a non-radiative multiphonon theory:** The transition from $i$ to $j$ proceeds either radiatively or non-radiatively. In typi- cal semiconductor devices, the radiative transition can be excluded as no photons are available during the regular operation. Therefore, the energy needed to overcome the difference between the minimum point and the crossing point of the parabolas hasto be supplied by phonons. Figure source: [58].

So far, the stochastic process of capture and emission events has been formulated based on a two-state model. In order to be able to calculate the transition rates between the state $i$ and $j$ , it has to be considered that when electrons or holes are captured or emitted from oxide defects the whole surrounding (electrons and nuclei) is influenced. In each state, neutral and charged, the total energy consists of contributions from the ionic system, the electronic system, and a coupling term. In a simplified way, the atomic positions are reduced to one-dimensional configuration coordinates and the adiabatic energy surface, which would be obtained by solving the Schrödinger equation, is approximated by a harmonic oscillator. The total energy of each charge state $i$ can be written as

$(2.20) \begin{equation} \label {eq:totalenergystate} V_{i}=\dfrac {1}{2}M\omega _{i}^2(q-q_{i})^2+E_{i} \end{equation}$

with

This parabolic approximation is shown in the configuration coordinate diagram in Figure 2.13 for both states $i$ and $j$ . For the calculation of the transition rates of such a defect, two transition possibilities between the neutral and the charged state can be considered as shown in Figure 2.13, the radiative transition and the non-radiative transition. The radiative, or optical, transition occurs around the minima of the parabolas according to the Franck-Condon principle. During the transition from state $1$ to state $2$ the lattice coordinate $q$ does not change and a photon would have to supply the energy to the system. However, no photons are available for such direct transitions during regular operation in typical semiconductor devices. Therefore, the transition between $1$ and $2$ has been described by non-radiative multiphonon processes [13–15], where the energy to overcome the barrier has to be supplied by phonons. This means that the system has to overcome the difference between the minimum points $E_\mathrm {1}$ or $E_\mathrm {2}$ and the crossing point of the parabolas $\mathcal {E}_\mathrm {12}$ as illustrated in Figure 2.14.

The NMP transition describes the charge carrier transfer between the conduction or the valence band of a semiconductor and the oxide trap. The corresponding rates for continuously distributed charge carrier energies can be written as [59, 60]

$(2.21–2.24) \{begin}{align} \label {eq:transratesijC} k_{ij}^\mathrm {C}(E,E_\mathrm {D})&=\int _{E_\mathrm {C}}^\infty D_\mathrm {n}(E) f_\mathrm {p}(E)A_{ij}(E,E_\mathrm {D})f_{ij}(E,E_\mathrm {D})\mathrm {d}E\\ \label {eq:transratesijV} k_{ij}^\mathrm {V}(E,E_\mathrm {D})&=\int _{-\infty }^{E_\mathrm {V}} D_\mathrm {p}(E) f_\mathrm {p}(E)A_{ij}(E,E_\mathrm {D})f_\mathrm {ij}(E,E_\mathrm {D})\mathrm {d}E\\ \label {eq:transratesjiC} k_{ji}^\mathrm {C}(E,E_\mathrm {D})&=\int _{E_\mathrm {C}}^\infty D_\mathrm {n}(E) f_\mathrm {n}(E)A_{ji}(E,E_\mathrm {D})f_{ji}(E,E_\mathrm {D})\mathrm {d}E\\ \label {eq:transratesjiV} k_{ji}^\mathrm {V}(E,E_\mathrm {D})&=\int _{-\infty }^{E_\mathrm {V}} D_\mathrm {p}(E) f_\mathrm {n}(E)A_{ji}(E,E_\mathrm {D})f_{ji}(E,E_\mathrm {D})\mathrm {d}E \{end}{align}$

with

Figure 2.14: **The field-dependence of the NMP transition:** As a consequence of the electrostatic shift of the defect level, the relative position of the parabolas change. Figure source: [16].

Under homogeneous bias conditions the carriers in the channel are in equilibrium and thus the carrier distribution functions, $f_\mathrm {n}$ and $f_\mathrm {p}$ , are properly described by the Fermi-Dirac distribution. Substituting $f_\mathrm {n}$ and $f_\mathrm {p}$ by the Fermi-Dirac distribution and considering a linear electron-phonon coupling, the transition time constants for hole capture and emission are given by

$(2.25–2.26) \{begin}{align} \label {eq:tauctwostatenrm} \tau _\mathrm {c}&=\dfrac {1}{k_\mathrm {12}}=\dfrac {1}{k_\mathrm {0}}\mathrm {e}^{\mathcal {E}_\mathrm {12}/(k_\mathrm {B}T)}\\ \label {eq:tauetwostatenrm} \tau _\mathrm {e}&=\dfrac {1}{k_\mathrm {21}}=\dfrac {1}{k_\mathrm {0}}\mathrm {e}^{(\mathcal {E}_\mathrm {21}+E_\mathrm {1}+E_\mathrm {2})/(k_\mathrm {B}T)} \{end}{align}$

with

(a) The energy level $E_\mathrm {1}$ of the defect at the depth $x$ is located below or above the Fermi level if recovery conditions $E_\mathrm {OX}^\mathrm {L}$ or stress conditions $E_\mathrm {OX}^\mathrm {H}$ are applied, respectively.

$ E_\mathrm {1} $ — (a) The energy level $E_\mathrm {1}$ of the defect at the depth $x$ is located below or above the Fermi level if recovery conditions $E_\mathrm {OX}^\mathrm {L}$ or stress conditions $E_\mathrm {OX}^\mathrm {H}$ are applied, respectively.

The relative position of the parabolas depends on $E_\mathrm {OX}$ , which leads to a field- and temperature-dependence of the NMP transition as shown in Figure 2.14. At recovery conditions, the minimum of the neutral state parabola $E_\mathrm {1}$ is located energetically below the minimum of the charged state parabola $E_\mathrm {2}$ . If the defect was previously charged, it will transit from the charged state to the neutral state. At stress conditions, $E_\mathrm {2}>E_\mathrm {1}$ and a previously neutral state will transit to the charged state. According to Equations 2.9 and 2.10, the field- and temperature-dependence of the transitions rates leads to a field- and temperature-dependence of $\tau _{\mathrm {e}}$ and $\tau _{\mathrm {c}}$ , which reflects the experimentally observed behavior of single oxide defects discussed in Subsection 2.1.3.

Figure 2.14 already illustrates that in eMSM measurements only defects in a certain energy region contribute to $\Delta V_{\mathrm {th}}$ . This area is the AER and is shown in Figure 2.15 [16]. As an explanation, a defect energy level $E_\mathrm {1}$ , which is below the Fermi level $E_\mathrm {F}$ at recovery conditions (low level is abbreviated with L, $V_{\mathrm {G}}^\mathrm {rec}$   $\approx$   $V_{\mathrm {th}}$ ) and above the Fermi level at stress conditions typically above nominal operating conditions (high level is abbreviated with H) is withing the AER. Following the switch from the recovery to the stress voltage, defects with energy levels within the AER are moved above the Fermi level and capture charge carriers. Therefore, as time progresses, the AER determines the maximum possible degradation. Back at recovery voltage, the defects are moved back below the Fermi level and emit the charge carriers again.

The boundaries of the AER can be defined as shown in the following. $E_\mathrm {1}$ depends on the bias via the depth-dependent electrostatic potential $\varphi (x)$ : $E_\mathrm {1}(x)=E_\mathrm {10}-q \varphi (x)$ . At low defect concentration, it can be assumed to first-order that the charged defects inside the oxide do not significantly impact the electrostatic potential. With this assumption, the potential can be written as

$\varphi (x) = \varphi _\mathrm {s} - x E_\mathrm {OX},$

where $\varphi _\mathrm {s}$ is the potential at the interface. In a simple approximation, the trap level depends on the applied bias via

$E_\mathrm {1}=E_\mathrm {10}-q\varphi _\mathrm {s}+q x E_\mathrm {OX}.$

Figure 2.16: **Switching and fixed oxide defects:** TDDS data of a fixed positive charge trap (left) and a switching trap (right). Figure source: [61].

With this, the lower boundary ( $E_\mathrm {L}$ ) of the AER can be written as Equation 2.27 and the upper boundary ( $E_\mathrm {U}$ ) of the AER can be written as Equation 2.28

$(2.27) \begin{equation} \label {eq:lowerboundary} E_\mathrm {L}(x)=E_\mathrm {F}+q \varphi _\mathrm {s}^\mathrm {H}-q x E_\mathrm {OX}^\mathrm {H} \end{equation}$

$(2.28) \begin{equation} \label {eq:upperboundary} E_\mathrm {U}(x)=E_\mathrm {F}+q \varphi _\mathrm {s}^\mathrm {L}-q x E_\mathrm {OX}^\mathrm {L} \end{equation}$

with

Although the two-state model can explain a field- and temperature-dependence of the characteristic capture and emission times, important details seen in experiments are still missing in this model. For example:

• The predicted field-dependence of $\tau _{\mathrm {c}}$ $($ $V_\mathrm {G}$ $)$ is nearly linear, which is not the case in experiments. TDDS data shown in Figure 2.16 illustrates that $\tau _{\mathrm {c}}$ $($ $V_\mathrm {G}$ $)$ shows some curvature on a logarithmic scale.
• Defects have been observed which show an interrupted RTN signal [40, 41, 62]. Figure 2.17 illustrates a defect, which produces an RTN signal only for a limited amount of time after NBTI stress. Such observations have led to the conclusion that in addition to a neutral and a charged state also a metastable state must exist. This metastable state can either be neutral or charged.
• $\tau _{\mathrm {e}}$ $($ $V_\mathrm {G}$ $)$ cannot be explained properly by a two-state model. As shown in Figure 2.16 two types of defects occur, which have been named switching and fixed oxide defects [61]. In the first case $\tau _{\mathrm {e}}$ $($ $V_\mathrm {G}$ $)$ is nearly constant and shows only a slight bias-dependence. In the second case $\tau _{\mathrm {e}}$ $($ $V_\mathrm {G}$ $)$ is nearly bias-independent above $V_{\mathrm {th}}$ and drops below $V_{\mathrm {th}}$ . Such a behavior could be captured by two different paths for the emission process.

Figure 2.17: **Interrupted RTN:** After NBTI stress the defect produces RTN for a limited amount of time and is then neutral. Figure source: [16].

Especially the last two findings have led to the consideration of additional states. As a result, the four-state NMP model has been proposed [40].

2.1.5 Four-State Non-Radiative-Multiphonon Model

In order to reflect experimental observations like different $\tau _{\mathrm {e}}$ $($ $V_\mathrm {G}$ $)$ behavior and interrupted RTN signals, the two-state model has been extended as shown in the diagram in Figure 2.18 [16]. The potential energy surface is illustrated in Figure 2.19. The resulting four-state model consists of the two stable states $1$ (neutral) and $2$ (charged), which correspond to the two states in the two-state model, and additionally two meta-stable states, $1^\prime$ (neutral) and $2^\prime$ (charged). The transitions between $1$ and $2^\prime$ and vice versa as well as $2$ and $1^\prime$ correspond to NMP transitions as discussed in the previous subsection. The transitions from $2^\prime$ to $2$ and vice versa as well as from $1^\prime$ to $1$ proceed via a thermal barrier. This thermal barrier is associated with a structural relaxation of the defect and is described by transition state theory [63, 64] as

$(2.29–2.32) \{begin}{align} \label {eq:thermalbarrier} k_\mathrm {2^\prime 2}&=\nu \mathrm {e}^{-\mathcal {E}_\mathrm {2^\prime 2}/(k_\mathrm {B} T)}\textrm {,}\\ \label {eq:thermalbarrier1} k_\mathrm {1^\prime 1}&=\nu \mathrm {e}^{-\mathcal {E}_\mathrm {1^\prime 1}/(k_\mathrm {B} T)}\textrm {,}\\ \label {eq:thermalbarrier2} k_\mathrm {2 2^\prime }&=\nu \mathrm {e}^{-\mathcal {E}_\mathrm {2 2^\prime }/(k_\mathrm {B} T)}\textrm {,}\\ \label {eq:thermalbarrier3} k_\mathrm {1 1^\prime }&=\nu \mathrm {e}^{-\mathcal {E}_\mathrm {1 1^\prime }/(k_\mathrm {B} T)} \{end}{align}$

with

Figure 2.18: **Four-state diagram:** The four-state NMP model consists of two stable ( $1$ and $2$ ), two metastable states ( $1^\prime$ and $2^\prime$ ), two neutral ( $1$ and $1^\prime$ ) and two charged ( $2$ and $2^\prime$ ) states.

$ 1 $ — Figure 2.18: **Four-state diagram:** The four-state NMP model consists of two stable ( $1$ and $2$ ), two metastable states ( $1^\prime$ and $2^\prime$ ), two neutral ( $1$ and $1^\prime$ ) and two charged ( $2$ and $2^\prime$ ) states.

Assuming the defect is in its neutral state $1$ , the capturing of a charge carrier proceeds via $1 \rightarrow 2^\prime \rightarrow 2$ . The NMP transition $1 \rightarrow 2^\prime$ reflects the charging of the defect and corresponds to what is experimentally measurable. This transition is followed by a transition to the stable state $2^\prime \rightarrow 2$ via the thermal barrier.

The emission process may proceed via two different paths, either $2 \rightarrow 2^\prime \rightarrow 1$ (fixed oxide traps defects) or $2 \rightarrow 1^\prime \rightarrow 1$ (switching oxide defects). Figure 2.19 shows the case of a switching defect. If the defect is in the stable and charged state $2$ , the transition to the metastable state $1^\prime$ is an NMP transition, which is followed by a thermal transition to the stable state $1$ . The experimentally measurable $\Delta V_{\mathrm {th}}$ step in a recovery trace corresponds to the transitions $2 \rightarrow 1^\prime$ and $2^\prime \rightarrow 1$ .

Figure 2.19: **Configuration coordinate diagram for a four-state model using a non-radiative multiphonon theory:** Schematic cross-section of the potential energy surface of the four-state NMP model. The energy parameters needed for calculating all transition rates are shown. Figure source: [65].

The NMP transition rates can be calculated using the Equations 2.21 to 2.24:

$\{begin}{align*} k_\mathrm {1 2^\prime } &= k_\mathrm {1 2^\prime }^C + k_\mathrm {1 2^\prime }^V\\ k_\mathrm {2^\prime 1} &= k_\mathrm {2^\prime 1}^C + k_\mathrm {2^\prime 1}^V\\ k_\mathrm {2 1^\prime } &= k_\mathrm {2 1^\prime }^C + k_\mathrm {2 1^\prime }^V\\ k_\mathrm {1^\prime 2} &= k_\mathrm {1^\prime 2}^C + k_\mathrm {1^\prime 2}^V \{end}{align*}$

The four-state model can be written in a similar manner to the one of the two-state model. As a result, the overall capture and emission times (transition from one stable state to the other via a meta-stable state) can be written as

$(2.33) \begin{equation} \label {eq:capturetimefourstates} \tau _{\mathrm {c}}=\dfrac {1}{\dfrac {k_\mathrm {1 1^\prime } k_\mathrm {1^\prime 2}}{k_\mathrm {1 1^\prime }+k_\mathrm {1^\prime 1}+k_\mathrm {1^\prime 2}}+\dfrac {k_\mathrm {1 2^\prime } k_\mathrm {2^\prime 2}}{k_\mathrm {1 2^\prime }+k_\mathrm {2^\prime 1}+k_\mathrm {2^\prime 2}}} \end{equation}$

and

$(2.34) \begin{equation} \label {eq:emissiontimefourstates} \tau _{\mathrm {e}}=\dfrac {1}{\dfrac {k_\mathrm {1^\prime 1} k_\mathrm {2 1^\prime }}{k_\mathrm {1^\prime 1}+k_\mathrm {1^\prime 2}+k_\mathrm {2 1^\prime }}+\dfrac {k_\mathrm {2^\prime 1} k_\mathrm {2 2^\prime }}{k_\mathrm {2^\prime 1}+k_\mathrm {2^\prime 2}+k_\mathrm {2 2^\prime }}} \textrm {,} \end{equation}$

respectively where $\tau _{\mathrm {c}}$ corresponds to $\tau _\mathrm {12}$ and $\tau _{\mathrm {e}}$ to $\tau _\mathrm {21}$ .

The four-state model captures the bias- and temperature-dependence of the capture and emission times quite well. It can also explain the interruption of RTN signals, where the RTN is, for example, the hopping between $1$ and $2^\prime$ and the interruption is the transition to the stable state $2$ .

However, at this point the question which structural oxide defect would cause measurable steps in the $\Delta V_{\mathrm {th}}$ traces remains open. Promising defect candidates suitable for the four-state model are the oxygen vacancy and the hydrogen bridge defects [66, 67]. Moreover, the hydroxyl- $\mathrm {E}^\prime$ center has also been proposed as a possible defect candidate [65]. The atomic configurations of all three defect candidates are shown in Figure 2.20. However, by a statistical analysis of the experimentally found NMP parameter distributions (shown in Figure 2.19) for numerous defects and comparison with the parameters extracted from density-functional-theory (DFT) calculations [68] show that the oxygen vacancy is a very unlikely defect candidate [65]. By contrast, the hydrogen bridge and the hydroxyl- $\mathrm {E}^\prime$ give a good match for the majority of the parameters of the four-state model. Therefore, both of them are promising defect candidates for the four-state model.

Figure 2.20: **Possible defect candidates:** Atomic configurations corresponding to the states $1$ , $1’$ , $2’$ and $2$ for three possible oxide defects [65]. **Top:** Oxygen vacancy. **Center:** Hydrogen bridge. **Bottom:** Hydroxyl- $\mathrm {E}^\prime$ center. H atoms are shown as silver, Si atoms are yellow and O atoms are red. The blue bubbles represent the localized highest occupied orbitals for the neutral charge states and the lowest unoccupied orbital for the positive charge states. Figure source: [65].

$ 1 $ — Figure 2.20: **Possible defect candidates:** Atomic configurations corresponding to the states $1$ , $1’$ , $2’$ and $2$ for three possible oxide defects [65]. **Top:** Oxygen vacancy. **Center:** Hydrogen bridge. **Bottom:** Hydroxyl- $\mathrm {E}^\prime$ center. H atoms are shown as silver, Si atoms are yellow and O atoms are red. The blue bubbles represent the localized highest occupied orbitals for the neutral charge states and the lowest unoccupied orbital for the positive charge states. Figure source: [65].

Although the extension of the two-state model to a four-state model reflects the experimental observations quite well, the whole picture is still not completed. As will be discussed in the following, observations like the permanent component and volatility of defects need for further extensions.

2.1.6 Defect Volatility

The four-state model, as discussed so far, has been developed for active defects, which basically can be characterized in TDDS measurements ( $\tau _{\mathrm {c}}$ $<$ $t_\mathrm {str,max}$ and $\tau _{\mathrm {e}}$ $\in [$ $t_\mathrm {rec,min}$ , $t_\mathrm {rec,max}$ $]$ for any $V_{\mathrm {G}}^\mathrm {str}$ , $V_{\mathrm {G}}^\mathrm {rec}$ and $T$ ). However, it has been found that oxide defects can all of a sudden disappear from one to the other measurement from the measurement window ( $\tau _{\mathrm {c}}$ $\notin [$ $t_\mathrm {str,min}$ , $t_\mathrm {str,max}$ $]$ and $\tau _{\mathrm {e}}$ $\notin [$ $t_\mathrm {rec,min}$ , $t_\mathrm {rec,max}$ $]$ ) as it is shown in Figure 2.21 based on spectral maps for four different TDDS measurements. Additionally, the disappeared defects can also reappear. An electrically active state can be modeled using the previously discussed four-state NMP model. The disappearing of a defect in TDDS measurements can be described as a transition from one of the four states to an inactive state. This transition from an active state to the inactive state as well as the backward transition have been formulated as volatility [65, 67, 69, 70].

Figure 2.21: **Defect volatility in spectral maps:** Defects can dis- and reappear in TDDS measurements. Here this is shown based on spectral maps of different measurements on the same nano-scale device. Figure source: [33, 69].

As it is shown in Figure 2.22 the dis- and reappearing can occur several times during TDDS measurements. The corresponding time constant for the transition from the active into the inactive state is typically in the range of hours to weeks. It has been proposed that the transitions between active and inactive states can be described as thermally activated rearrangement of the atomic structure. The reaction barrier can be estimated as [65, 70]

$(2.35) \begin{equation} \label {eq:thermalactvol} \dfrac {t}{\tau _\mathrm {v}}=\nu \mathrm {e}^{-E_\mathrm {B}/(k_\mathrm {B} T)} \end{equation}$

with

Figure 2.22: **Monitoring of volatile defects over three months:** The shown defects A7, A8 and A9 can be both, active and inactive. Figure source: [65].

The possible potential energy surface of this thermal transition into an inactive state is shown in Figure 2.23 and the corresponding atomic structure on the right hand side of Figure 2.26. Both figures show the transition $2^\prime \rightarrow 0^+$ of a hydroxyl- $\mathrm {E}^\prime$ center, where $2^\prime$ is the charged metastable state of the four-state model. As a general explanation, for the atomic structure the transition from activity to inactivity means that a hydrogen atom is released from the hydrogen bridge or the hydroxyl- $\mathrm {E}^\prime$ center and relocated to a neighboring atom. This corresponds to either a neutral $\ch{H}$ atom moving away from the neutral defect state, which is associated with a transition to the neutral volatile state $0^\mathrm {n}$ , or to a proton from the positive defect state, which is associated with a transition to a positive volatile state $0^+$ . This results in four possible purely thermally activated transitions starting from a four-state model: $2^\prime \rightarrow 0^+$ , $2 \rightarrow 0^+$ , $1 \rightarrow 0^\mathrm {n}$ and $1^\prime \rightarrow 0^\mathrm {n}$ . From these four, it has been found that the transitions starting from $1$ and $2^\prime$ are always lower in energy [70], especially $2^\prime \rightarrow 0^+$ is a quite promising transition for the explanation of volatility. Moreover, it has been found that the hydrogen bridge is not a suitable candidate to explain volatility since the transition barriers are too high to explain the experimental characterizations. By contrast, the hydroxyl- $\mathrm {E}^\prime$ center has been considered as the suitable defect candidate for the explanation of volatility.

Figure 2.23: **Example of the potential energy surface of a hydroxyl- $\mathrm {E}^\prime$ center including volatile states:** The defect can become volatile starting from a positive charge state, which is one of the four (active) NMP states. As soon as it overcomes the barrier $\mathrm {E}_\mathrm {B}$ it is inactive and not visible in measurements. Figure source: [65].

Figure 2.23 shows the hydroxyl- $\mathrm {E}^\prime$ potential energy surface of the extension of the four-state model based the transition $2^\prime \rightarrow 0^+$ . As discussed in the previous paragraph, two additional states, a positive $0^+$ and a neutral $0^\mathrm {n}$ and their respective barriers are added. Theoretically, in such a configuration, it would be possible that transitions between $0^+$ and $0^\mathrm {n}$ cause RTN since they include a hole capture or emission. However, this has been discarded because it has been shown that due to typical transition barrier heights $E_\mathrm {B}$ , $E_\mathrm {Br}$ and $E_\mathrm {f}$ charge capture or emission events in the volatile states would occur most probably with a similar or lower frequency as volatility itself [58].

Basically, transitions from other active states are also possible, for example, the transition $1^\prime \rightarrow 0^\mathrm {n}$ via a hydrogen hopping process. However, it has been found that this transition is associated with the permanent component of degradation as will be discussed in the next subsection.

2.1.7 Recoverable and Permanent Component

Oxide defects contribute differently to $\Delta V_{\mathrm {th}}$ during recovery and thus can be classified into defects contributing to the recoverable component and defects contributing to the permanent component of degradation [71, 72]. Those with characteristic capture and emission times lying within the measurement window are typically associated with the recoverable component while those with $\tau _{\mathrm {c}}$ $<$ $t_\mathrm {str,max}$ but slowly-relaxing, $\tau _{\mathrm {e}}$   $>$   $t_\mathrm {rec,max}$ , are associated with the permanent component. However, properties like $\tau _{\mathrm {c}}$ and $\tau _{\mathrm {e}}$ and thus their assignment to the recoverable or the permanent component is highly alterable due to reactions with hydrogen [69]. Defects can also be created or annealed, whereby these terms are often related to the transition from the active to inactive states or to precursor states [62, 65, 69, 72]. The particular properties and their contribution to degradation can be spread widely.

In order to explain the permanent component of $\Delta V_{\mathrm {th}}$ in NBTI measurements a gate-side hydrogen release model has been proposed [73, 74]. The idea is based on the fact that in amorphous $\ch{SiO2}$ hydrogen can bind to a bridging oxygen, as calculated in recent DFT calculations [75, 76]. Throughout the binding process, the hydrogen can release its electron and bind to the oxygen or the hydrogen breaks one of the $\ch{Si}$ - $\ch{O}$ bonds and forms a hydroxyl group, which faces the dangling bond of the other $\ch{Si}$ . Such a defect is quite similar to the $\mathrm {E}^\prime$ center and is additionally to the previously discussed defects a proper candidate for the explanation of volatility.

A schematic illustration of the hydrogen release model is shown in Figure 2.24. During stress, the energy level of a trapped $H^+$ near the gate can move below the Fermi level. Consequently, it is neutralized and emitted over a thermal barrier. The $H^\mathrm {0}$ moves quickly towards the channel where it can become trapped in a preexisting hydrogen trapping site, again releases an electron and causes a $\Delta V_{\mathrm {th}}$ shift. The gate side can be interpreted as a hydrogen reservoir.

Figure 2.24: **Schematic Hydrogen release mechanism:** At the gate side a proton is trapped. During stress, the trap level can be shifted below the Fermi level, which makes it possible for the proton to be neutralized. This neutrally charged hydrogen atom can now be released by overcoming a barrier and move towards the channel side. The empty trap site can potentially be refilled by $\ch{H}$ released from the gate. This process should be strongly temperature dependent. Figure source: [74].

Figure 2.24: **Schematic Hydrogen release mechanism:** At the gate side a proton is trapped. During stress, the trap level can be shifted below the Fermi level, which makes it possible for the proton to be neutralized. This neutrally charged hydrogen atom can now be released by overcoming a barrier and move towards the channel side. The empty trap site can potentially be refilled by $\ch{H}$ released from the gate. This process should be strongly temperature dependent. Figure source: [74].

Figure 2.25 shows schematically that the hydrogen release model can be formulated by assuming that the oxide consists of discrete interstitial sites $i$ at which hydrogen can occur in a neutral position. From this interstitial sites, it can be either trapped in a neutral configuration or in a positive configuration. The rate equation for the interstitial hydrogen species can be written as

$(2.36) \begin{equation} \label {eq:hydrogenrelease} \dfrac {\partial H_{i}}{\partial t}=-\sum _j k_\mathrm {h}(H_{i}-H_{j})+\sum _n T_{i,n} \end{equation}$

with

Thereby, one spatial interstitial site $i$ is allowed to interact with several trapping sites $n$ . The corresponding trapping rate is

$(2.37) \begin{equation} \label {eq:trappingrateinterstitial} T_{i,n}=k_\mathrm {01}H_{i}\left (H_\mathrm {max}^\mathrm {T}-(H_{i,n}^\mathrm {0}+H_{i,n}^\mathrm {+})\right )-k_\mathrm {10}H_{i,n}^\mathrm {0} \end{equation}$

with

The temporal change in the number of neutral and positive trapped hydrogen atoms is given by

$(2.38–2.39) \{begin}{align} \label {eq:trappingrateneutral} \dfrac {\partial H_{i}^\mathrm {0}}{\partial t}&=-k_\mathrm {12}H_{i}^\mathrm {0}+k_\mathrm {21} H_{i}^\mathrm {+}+T_{i}\\ \label {eq:trappingratepositive} \dfrac {\partial H_{i}^\mathrm {+}}{\partial t}&=k_\mathrm {12}H_{i}^\mathrm {0}-k_\mathrm {21} H_{i}^\mathrm {+} \{end}{align}$

with

Figure 2.25: **One dimensional schematic of the H-release model:** The oxide of a MOSFET consists of potential trapping sites for hydrogen. Hydrogen can either occur in a neutral interstitial position (grey) or as a trapped neutral configuration $H^\mathrm {0}$ (blue) or in a trapped positive configuration $H^+$ (red). The gate side acts as an additional hydrogen reservoir. Due to the high diffu- sivity of hydrogen the exchange to a new trapping site can ocur very fast and is therefore not rate limiting. Figure source: [74].

$ H^\mathrm {0} $ — Figure 2.25: **One dimensional schematic of the H-release model:** The oxide of a MOSFET consists of potential trapping sites for hydrogen. Hydrogen can either occur in a neutral interstitial position (grey) or as a trapped neutral configuration $H^\mathrm {0}$ (blue) or in a trapped positive configuration $H^+$ (red). The gate side acts as an additional hydrogen reservoir. Due to the high diffu- sivity of hydrogen the exchange to a new trapping site can ocur very fast and is therefore not rate limiting. Figure source: [74].

The trapping rates $T_{i,n}$ describe the transition from the interstitial to the neutral trapped configuration. The corresponding transition rates $k_\mathrm {01}$ and backwards $k_\mathrm {10}$ are modeled as a thermal activation using an Arrhenius law for each trapping site $(i,n)$ . The number of neutral and positive trapped hydrogen can be calculated using rates with standard non-radiative multiphonon theory.

(image) — Figure 2.26: **Extended four-state NMP model:** Illustrated for a promising defect candidate, the hydroxyl- $\mathrm {E}^\prime$ center. Still, the core of this model (middle) is build around the bistable defect with four states ( $1$ , $1^\prime$ , $2$ , $2^\prime$ ) and describes the active defect, which is capable of capturing and emitting charge carriers. However, the extended variant of the model also accounts for the inactive phases of the defect via transitions to the precursor states $0$ and $0^2$ (left) and the inactive states $0^{\!+}$ and $0^{\mathrm {n}}$ (right).

Figure 2.26 shows a complete picture of all defect states being capable to describe the recoverable and the permanent component of NBTI degradation as well as the temporary inactivity of single oxide defects. This extended four-state model includes the four active states (center), the transition to the inactive states starting at state $2^\prime$ (right) and the transition to the precursor states starting at state $1$ (left) of a hydroxyl- $\mathrm {E}^\prime$ center.

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$f$	probability density function
$\Delta V_\mathrm {th}$	threshold voltage shift
$\eta$	mean value of the step height distribution.

$q$	elementary charge
$C_{ox}$	oxide capacitance.

$X_{j}$ / $X_{i}$	defect occupancies of the states $j$ / $i$
$t$	time
$k_{ij}$ / $k_{ji}$	rate for the transition from state $i$ to state $j$ /
	$j$ to state $i$ (probability per unit time)
$h$	infinitesimally small time interval
$O(h)$	higher-order terms of $h$ .

$p_{j}(\infty )=k_{ij}/(k_{ij}+k_{ji})$	probability that the defect is in the state $j$ for $t\gg \tau$
$\tau =1/(k_{ij}+k_{ji})$	transition time constant.

$\bar {\tau _\mathrm {c}}$ / $\bar {\tau _\mathrm {e}}$	mean value of the capture time $\tau _\mathrm {c}$ / emission time $\tau _\mathrm {e}$
$g$	probability density function.

$\tau _\mathrm {c}$ / $\tau _\mathrm {e}$	time constant for the transition from recovery to stress / from
	stress to recovery conditions
$\tau _\mathrm {c}(V_\mathrm {G})$ / $\tau _\mathrm {e}(V_\mathrm {G})$	defect time constant at different gate bias
$V_\mathrm {G}^\mathrm {str}$ / $V_\mathrm {G}^\mathrm {rec}$	gate bias at stress/recovery conditions
$t_\mathrm {str}$ / $t_\mathrm {rec}$	stress / recovery time
$t_\mathrm {str,max}$	point in time when the bias conditions are switched from
	stress to recovery
$f$	occupancy.

$k$	defect index
$N$	total amount of active defects
$d_{k}$	step height of defect $k$ , induced shift in $\Delta V_{\mathrm {th}}$
$h_{k}$	$h(t_\mathrm {str},t_\mathrm {rec};\tau _\mathrm {c},\tau _\mathrm {e})=\Delta f(t_\mathrm {str},t_\mathrm {rec};\tau _\mathrm {c},\tau _\mathrm {e})/\left (f(V_\mathrm {G}^\mathrm {str})-f(V_\mathrm {G}^\mathrm {rec})\right )$ .

$V_{i}$	total energy of the charge state $i$
$M$	effective mass
$q$	reaction coordinate
$q_{i}$	local equilibrium position
$\omega _{i}$	vibrational frequency in minimum $i$
$E_{i}$	potential energy.

$k_{ij}^\mathrm {C}$ / $k_{ij}^\mathrm {V}$	transition rate from $i$ to $j$ for the conduction / valence band
$k_{ji}^\mathrm {C}$ / $k_{ji}^\mathrm {V}$	transition rate from $j$ to $i$ for the conduction / valence band
$E$	energy
$E_\mathrm {D}$	trap level
$E_\mathrm {C}$ / $E_\mathrm {V}$	conduction / valence band edge energy
$D_\mathrm {n}$ / $D_\mathrm {p}$	density of states of the conduction / valence band
$f_\mathrm {n}$ / $f_\mathrm {p}$	carrier distribution functions for electrons / holes
$A_{ji}$ / $A_{ji}$	electron wave functions of both states and accounts
	for possible tunneling processes
$f_{ji}$ / $f_{ji}$	lineshape function

$\mathcal {E}_\mathrm {1}$ / $\mathcal {E}_\mathrm {2}$	minimum of the total energy of the charge state $1$ / $2$
$\mathcal {E}_\mathrm {12}$ / $\mathcal {E}_\mathrm {21}$	energy barrier height for the transitions $1 \rightarrow 2$ / $2 \rightarrow 1$ :
	difference between $E_\mathrm {1}$ / $E_\mathrm {2}$ and the crossing point of the
	parabolas (Figure 2.14)
$k_\mathrm {B}$	Boltzmann constant
$T$	temperature.

$E_\mathrm {L}$ / $E_\mathrm {U}$	lower / upper energy boundary
$q$	elementary charge
$\varphi _\mathrm {s}^\mathrm {H}$ / $\varphi _\mathrm {s}^\mathrm {L}$	potential at the interface at stress / recovery conditions
$E^\mathrm {H}_\mathrm {OX}$ / $E^\mathrm {L}_\mathrm {OX}$	electric field across the oxide at stress / recovery conditions
$x$	depth

$k_\mathrm {ij}$	transition rate $i \rightarrow j$ ,
$\nu$	attempt frequency,
$\mathcal {E}_\mathrm {ij}$	energy barrier height,
$k_\mathrm {B}$	Boltzmann constant,
$T$	temperature.

$\tau _\mathrm {v}$	transition time constant for volatile transitions
$\nu$	attempt frequency
$E_\mathrm {B}$	energy barrier height.

$H_{i}$	expectation value of hydrogen in a neutral interstitial
	position at site $i$
$t$	time
$k_\mathrm {h}$	hopping rate from interstitial site $i$ to $j$ ,
	thermal transition
$i$	defect site where the hydrogen can be trapped
$j$	neighbouring sites
$T_{i,n}$	trapping rates for interstitial site $i$ interacting
	with several trapping sites $n$ .

$k_\mathrm {01}$ / $k_\mathrm {10}$	transition rate from the interstitial to the neutral
	trapped configuration / from the neutral to the
	interstitial, thermal transition
$H_\mathrm {max}^\mathrm {T} \geq H_{i,n}^\mathrm {0}+H_{i,n}^\mathrm {+}$	maximum number of trapped hydrogen atoms
$H_{i,n}^\mathrm {0}$ / $H_{i,n}^\mathrm {+}$	expectation value of trapped neutral / positive
	hydrogen trapped.