2.1 Overall Description

To analyze lifetimes affected by the TDDB phenomenon, it is necessary to understand generated traps through various mechanisms ⁽²⁾, generated percolation paths using statistical techniques, and statistically analyze the lifetime prediction based on them. In this paper, we develop a TDDB simulator that modulates the traps and percolation paths, as shown in Fig. 1. When the user inputs the geometry parameters of the device, fresh-state electrical characteristics, and testing conditions, various lifetime prediction results, such as the bathtub curve, testing condition for initial failure detection, and lifetime expectancy in the chip operation condition are outputted. This TDDB analysis framework is implemented using the Python language.

2.2 Modeling of Trap Generation

The trap-generation mechanisms in the insulator, which is the cause of TDDB, consist of AHI, AHR (also known as HR), and TC mechanisms ⁽³⁾. According to previous papers describing the trap generation in the insulator by the different physical mechanisms, the dominant trap-generation mechanism depends on the conditions in which the insulator operates. In this section, the dominant trap-generation mechanism is classified according to the experimental results of previous studies, and the hybrid analysis method having the continuity of each mechanism is implemented for the first time, to the knowledge of the author, in the proposed framework. The physical description for each mechanism and the modeling procedures are as follows:

① TC: Polarization occurs when an electric field (E-field) is applied to a dielectric with polarity. This polarization causes the distortion of the lattice, and as a result, the molecules in the dielectric are affected not only by the external E-field ($E_{ox}$) but also by the polarization vector $\textit{P}$ (=$\chi \epsilon _{0}E_{ox}$). Finally, the magnitude of the local E-field ($\textit{E}$$_{loc}$) applied to the insulator molecules is

where $\chi $ is the electric susceptibility, and $\textit{L}$ is the Lorentz factor, which is approximately $\textit{L}$ = 1/3 for symmetrical cubic structures ⁽⁴⁾. Thus, $E_{loc}=$ $\left(\frac{2+k}{3}\right)E_{ox}$. In the SiO$_{2}$ system, when the dielectric constant $\textit{k}$ is 3.9, the local E-field is estimated to be approximately $2E_{loc}$. This electrical stress causes the breakage of the natural weak bond of the insulator (Si-Si bond, i.e., oxygen vacancy). The reduced activation energy for the breakage of the Si-Si bond induced by $E_{loc}$ in the insulator can be expressed as

where $Δ H_{0}$ is the field-free activation energy and $p_{\mathrm{eff}}$ is the effective dipole moment, which is determined by the polar bonding in the molecule and has a value of approximately 7–14 eÅ ⁽⁵⁾. According to a previous study ⁽⁶⁾, $Δ H_{0}\approx 1.0\,\,\mathrm{eV}$ and $p_{\mathrm{eff}}\approx 7\,\,\mathrm{eÅ }$ under a weak E-field, and $Δ H_{0}\approx 2.0\,\,\mathrm{eV}$ and $p_{\mathrm{eff}}\approx 13\,\,\mathrm{eÅ }$ under a strong E-field. In this study, we use the latter condition for insulator breakdown analysis under a strong E-field.

According to the TC model, the field-enhanced thermal bond breakage (\textit{dN/dt}) can be expressed by the first-order reaction rate equation as

where $\textit{N}$($\textit{t}$) is the number of traps per cm$^{3}$ under the given conditions (time, temperature, and E-field), and $\textit{k}$$_{break}$ is the bond-breakage rate. The solution of Eq. (3) is $N\left(t\right)=N_{0}\left(1- exp\left(- k_{\textit{break}}t\right)\right)$, where $\textit{N}$$_{0}$ is the initial number of weak bonds (number of oxygen vacancies). According to ⁽⁷⁾, $\textit{N}$$_{0}$ is 1/20 for number of SiO$_{2}$ bonding; thus, we use $N_{0}=1.4\times 10^{21}/\mathrm{cm}^{3}$⁽⁷⁾. The bond-breakage rate, $\textit{k}$$_{break}$, can be expressed using the Boltzmann probability.

Here, $\textit{v}$$_{0}$ is the lattice vibration frequency (~10$^{13}$/s),$Δ H$is the activation energy due to the local E-field, $\textit{k}$$_{b}$ is the Boltzmann constant, and $\textit{T}$ is the temperature ⁽⁸⁾. According to Eqs. (1-3), if the probability of trap generation for the unit cell constituting the insulator follows the Poisson distribution, the parameter $\textit{b}$ of the Poisson distribution is the product of the number of traps per unit volume ($\textit{N}$($\textit{t}$)) and volume of the unit cell (${a_{0}}^{3}$). Therefore, $b=N\left(t\right)\times {a_{0}}^{3}$.

② AHI: Unlike the TC model, which analyzes the TDDB according to the breakage of weak bonds in the insulator, the AHI model analyzes the TDDB according to the Fowler–Nordheim (FN) tunneling phenomenon. In comparison with the TC model, the AHI model is caused by a stronger E-field. In many previous studies, AHI modeling was performed according to the carrier fluence ⁽⁹⁾. However, in this study, the cell-based analytical percolation path model is used to analyze the TDDB.

Fig. 2 depicts the operation of the AHI model. The operation of the AHI model begins with injecting electrons from the cathode to the anode via FN tunneling. The injected electrons generate holes through the valence electrons and impact the ionization of the anode. These high-energy hot holes pass through the insulator through the tunneling and thermionic emission mechanism and generate defects in the insulator ⁽³⁾. The impact ionization is divided into majority ionization and minority ionization ⁽¹⁰⁾. Majority ionization is the case where the ionized hole has a kinetic energy that exceeds the$\mathrm{E}_{\mathrm{v}}$ of the insulator, as shown in Fig. 2. This occurs when the energy ($E_{IN}$) of the injected electrons is higher than the threshold energy ($E_{TH}$) of 6 eV; $E_{TH}=$ $\phi _{H}+E_{G}\left(Si\right)=4.8\,\,\mathrm{eV}+1.1\,\,\mathrm{eV}\approx 6\,\,\mathrm{eV}\,.$The minority ionization mechanism does not have enough energy to cause the generated hole to exceed the valence-band energy ($\mathrm{E}_{\mathrm{v}})$ of the insulator. Because more tunneling events are needed to pass through the insulator, the generation rate is significantly lower than that for majority ionization, and the minority ionization has a smaller effect on the TDDB ⁽¹¹⁾. Therefore, the AHI modeling in this study considers only the dominant majority ionization.

Assumptions are needed to perform AHI modeling, as shown in Table 1, in accordance with previous studies and the electrical characteristics of the SiO$_{2}$ system, which is the gate insulator in the semiconductor device considered in this study. AHI modeling begins with the FN tunneling current equation ⁽¹²⁾:

The impact ionization rate can be expressed as ⁽¹³⁾

where we can take parameter values such as $\text{Y} _{0}=2.15\times 10^{6}\,\mathrm{cm}^{- 1}$ and $\textit{H}$ = 82 MV/cm from a previous experiment ⁽¹⁴⁾. As the current density of the generated holes is $J_{H}=\text{Y} J_{FN}$⁽¹¹⁾, the charge of the generated hole is $J_{H}\times A_{ox}\times t$, where $\textit{A}$$_{ox}$ is the insulator area and $\textit{t}$ is the observation time. This value divided by $\textit{q}$ is the number of generated holes. As the generated holes drift to the insulator, they can generate defects. The defect-generation probability ($P_{\textit{defect}}$) is newly defined as a fitting parameter. This is the probability that a defect will be generated for each drifted hot hole. This parameter can have the field dependency, $P_{\textit{defect}}$ increases as $E_{ox}$ increases. Accordingly, the expected number of defects throughout the oxide is $\frac{J_{H}\times A_{ox}\times t}{q}\times P_{\textit{defect}}$.

If the number of defects generated in the insulator is divided by the insulator volume, the expected number of defects per unit volume in the insulator can be obtained as follows: number of defects generated in insulator ${\div}$ insulator volume = number of defects per unit volume of insulator [ea/cm$^{3}$]. Similar to the TC model, assuming that the defect occurrence of the unit cell follows the Poisson distribution, the Poisson parameter $\textit{b}$ can be calculated as follows: $\textit{b}$ = number of defects per unit volume of insulator$\times $ volume of insulator(${a_{0}}^{3}$).

③ AHR (HR): In semiconductor devices, such as metal–oxide semiconductor (MOS) transistors, the hydrogen passivation technique is widely used to improve the interface quality of SiO$_{2}$ insulators grown on a silicon substrate. The Si-H bonds at the Si-SiO$_{2}$ interface are broken, the generated hydrogen atoms/ions are released into the insulator, and TDDB occurs ⁽¹⁵⁾. This is mainly observed in hyper-thin insulators of $T_{ox}\leq 50\mathrm{Å }$⁽³⁾.

These hydrogen atoms/ions drift and diffuse into the insulator and act as traps ⁽¹¹⁾. The HR model is similar to the AHI model in that electrons are tunneled from the cathode. However, it differs in that direct tunneling (DT) is dominant over FN tunneling in thin insulators with $T_{ox}\leq 50\mathrm{Å }$. Therefore, in the HR modeling, the DT equation is used. Additionally, the hydrogen desorption mechanism of the HR model is divided into several categories, with the main ones being electrical excitation (EE) and vibrational excitation (VE). In EE, hydrogen desorption occurs via field emission, and in VE, hydrogen desorption occurs via phonons. The results of previous experiments show that the Si-H bond has a relatively long vibrational lifetime of $\sim 10^{- 8}$s in the Si-SiO2 system ⁽¹⁶⁾. Thus, VE mechanism is more important than EE mechanism. VE can be divided into three mechanisms: single electron coherent excitation, incoherent excitation ⁽¹⁷⁾, and multi-electron excitation ⁽¹⁸⁾. Incoherent excitation in a modern small semiconductor device that operate at high E-field is very rare, and the rate of occurrence of multi-electron excitation is lower than that of single electron excitation. Therefore, in this study, only single electron excitation, which has the highest occurrence rate among the three VE mechanisms, is considered, and the threshold energy required for hydrogen desorption is 2.5–3 eV. Fig. 3 depicts the operation of the HR model.

Assumptions are needed to perform HR modeling, as shown in Table 2. The HR modeling begins with the DT current equation ^(19,20)

where $\phi _{s}$ is the barrier height and $V_{ox}$ is the voltage across the insulator. Using Eq. (7), the charge amount of the injected electron can be calculated as $J_{DT}\times A_{ox}\times t$, where $\textit{A}$$_{ox}$ is the insulator area, and $\textit{t}$ is the observation time. This value divided by $\textit{q}$ is the number of injected electrons. The injected electrons can release hydrogen via Si-H bonding at the Si-SiO$_{2}$ interface. The hydrogen-release probability ($P_{\textit{release}}$) is newly defined as a fitting parameter. This is the probability that a hydrogen atom/ion will be released for each injected electron. This parameter can have the field dependency, $P_{\textit{release}}$ increases as $E_{ox}$ increases. The number of released hydrogen atoms/ions is calculated as $\frac{J_{DT}\times A_{ox}\times t}{q}~ \times P_{\textit{release}}$. If we assume that all the injected hydrogen operates as a defect inside the insulator, by dividing the number of hydrogen atoms or ions in the insulator by the volume of the insulator, the expected number of traps per unit volume of the insulator can be determined. Finally, the Poisson distribution parameter can be calculated in the same manner as for the AHI model.

The trap-generation mechanism is appropriately selected automatically in consideration of the semiconductor device insulator structure, and electrical characteristics (dynamic bias, E-field, etc.). By using the modeled trap-generation mechanisms as described above, it is possible to calculate the trap-generation probability (failure probability) in the unit cell in the oxide insulator by calculating the Poisson parameter, assuming the Poisson distribution.

2.3 Modeling of Percolation Path Estimator

We use the cell-based analytical percolation model to analyze the lifetime of dielectric breakdown caused by traps in the insulator, as described by the TC, AHI, and HR physical mechanisms. As shown in Fig. 4, in the percolation model, traps inside the insulator between two electrodes are randomly generated by stress and form a conductive path between the electrodes, causing electrical breakdown ^(21,22). The multiple percolation path model is a model that considers not only the shortest distance between the electrodes ⁽²³⁾ but also the path of the nearest cells. As shown in Fig. 4, the simulation was conducted with a model in which the nine nearest cells could form a path.

The two most widely used types of percolation models are the Monte Carlo (MC) model and the cell-based analytical model. The MC model generates traps randomly in the insulator to determine the failure probability of the device over time and has the disadvantage of a long simulation time. In comparison, the cell-based analytical model is advantageous because it requires less simulation time (<10% compared with the MC model) and can predict the lifetime efficiently even under various stress conditions ⁽²⁴⁾. Therefore, in this study, a cell-based analytical model is used to perform lifetime analysis.

First, the insulating layer is divided into unit cells. If the Poisson distribution parameter obtained in the previous step is $\textit{b}$, the probability of generating $\textit{k}$-traps in the unit cell can be obtained by using the PMF$P\left(X=k\right)=(b^{k}e^{- b})/k!\left(k=0,1,2,3,\cdots \right)$of the Poisson distribution ⁽²⁵⁾. If there is at least one trap in the unit cell, the cell is determined as defective, and $\textit{P}$(defective cell) = 1 - $\textit{P}$(no trap in the unit cell) = $1- e^{- b}$. The failure rate of a cell is denoted as$F_{cell}=\lambda \,.$ The probability that a column becomes the percolation path is $F_{perc}=9^{n- 1}\lambda ^{n}$, and the probability that the percolation path is not generated in all $\textit{N}$ columns is$1- F_{perc}$ $=\left(1- \frac{1}{9}\left(9\lambda \right)^{n}\right)^{N}.$Then, the Weibit ($\textit{W}$$_{BD}$) can be calculated as$W_{BD}=\ln \left[- \ln \left(1- F_{BD}\right)\right]=\ln \left[- N\ln \left(1- \frac{1}{9}\left(9\lambda \right)^{n}\right]\right.\,,$ and because $\lambda \leq 1$,$W_{BD}\approx \ln \left(N\right)+\ln \left(\frac{1}{9}\right)+n\ln \left(9\lambda \right)$^(23,26).

2.4 Modeling of Lifetime Analyzer

In this study, we use the Weibull distribution and temperature–nonthermal (T-NT) relationship for the lifetime distribution and lifetime/stress relationship, respectively, and the T-NT Weibull model is assumed as an accelerated-lifetime test model by combining the lifetime distribution and the lifetime/stress relationship. Note that the Weibull distribution is applicable to all cases, regardless of whether the probability of failure increases, decreases, or remains constant over time. This is why we use the Weibull distribution in this study. The suitability of the data for the Weibull distribution can be evaluated by using the Weibull plot with the Weibit ($\textit{W}$$_{BD}$, which was obtained in Sections 2.3) and \textit{ln}(\textit{time}) as the y and x axes, respectively. Fig. 5 shows an example of a Weibull plot in which the temperature and bias voltage in an MOS transistor device are varied ⁽²⁷⁾.

In the T-NT Weibull relation, the temperature and the bias voltage are considered separately as stress sources by considering the Arrhenius relationship and the inverse power law relationship simultaneously. The corres-ponding equation is ⁽²⁸⁾

where $\textit{t}$ is time, $\textit{V}$ is the voltage, and $\textit{T}$ is the temperature. If we have three parameters, such as $\textit{B}$, $\textit{C}$, and $\textit{n}$, the lifetime and stress relationship can be predicted. ${\beta}$ can be obtained from the slope of the $\textit{W}$$_{BD}$$\textit{-ln(t)}$ curve, which is used to check the validity of the data. The three parameters ($\textit{B}$, $\textit{C}$, and $\textit{n}$) are extracted from the lifetime and stress experimental data by using the linearization technique of the T-NT Weibull relationship. Once the three parameters are obtained, relationship between the lifetime and stress (such as the voltage and temperature) can be obtained according to the T-NT relationship, which is expressed by the following equation.

The model parameters are extracted automatically via regression analysis using Python code, and various lifetime and stress analyses (e.g., for a specific lifetime or predicting the lifetime at a specific failure rate) can be easily performed in the framework.