vision-language pre-training objectives, multimodal ai
Training tasks for VL models.
3,145 technical terms and definitions
Training tasks for VL models.
Tasks for learning multimodal representations.
Models that perceive and act.
Apply commonsense to visual scenes.
Determine if image entails text.
Determine if image entails text.
Localize objects from language.
Train models to follow visual instructions.
Use visual elements as prompts.
Answer questions about image content.
Answer complex questions about images.
Reason about visual content.
Generate stories from image sequences.
Generate stories from image sequences.
Vision Transformer (ViT) applies transformers to images. Split into patches, embed, self-attention.
High-throughput LLM serving.
High-throughput LLM serving system using PagedAttention.
vLLM and TGI are fast LLM inference engines with PagedAttention, continuous batching. Much faster than naive serving.
Vendor-Managed Inventory delegates inventory monitoring and replenishment to suppliers reducing buyer's inventory management burden.
Volatile organic compound abatement destroys or captures organic vapors from process exhaust.
Voltage contrast imaging in SEM reveals floating or improperly biased conductors through secondary electron emission differences.
Volume rendering integrates color and density along rays producing realistic images from volumetric data.
VQ-Diffusion for audio generates discrete tokens through denoising diffusion in codebook space.
VQ-VAE-2 uses hierarchical vector quantization for high-fidelity image generation.
VQGAN combines vector quantization with adversarial training for high-quality image synthesis.
Variational Recurrent Neural Network combines VAE with RNN for learning probabilistic latent dynamics.
Find security flaws.
Different latent spaces in StyleGAN.
W+ space allows per-layer style codes in StyleGAN providing greater editability.
Predict across-wafer variations.
Waiting waste occurs when resources idle between operations.
Idle time in process.
One-time approval for deviation.
Warm-starting neural architecture search initializes search with promising architectures from prior knowledge or searches.
Gradually increase learning rate at start of training for stability.
Warpage measurement quantifies package deformation using shadow moiré or laser profilometry.
Waste minimization reduces hazardous waste generation through process optimization and substitution.
Wastewater treatment removes contaminants from fab effluent through chemical neutralization precipitation filtration and biological processes.
Water footprint quantifies total water consumption and quality degradation from semiconductor manufacturing operations.
Water intensity measures water consumption per unit production quantifying manufacturing efficiency.
Water recycling systems treat and reuse process water reducing consumption through technologies like reverse osmosis and filtration.
Water reuse rate is percentage of water recycled for reuse reducing fresh water consumption.
Embed signatures in AI-generated text or images.
Embed signatures to prove ownership.
Wavelet pooling uses graph wavelets for multi-scale signal decomposition enabling hierarchical graph learning.
WaveNet architecture adapted for forecasting uses dilated causal convolutions to capture long-term dependencies.
Failures from aging mechanisms.
# Weather Forecasting: Mathematical Modeling Weather forecasting relies on sophisticated mathematical frameworks that attempt to predict the chaotic behavior of Earth's atmosphere. This document covers the fundamental equations, numerical methods, and modern approaches. ## 1. The Fundamental Equations (Primitive Equations) Weather models are built on the **primitive equations**—a set of nonlinear partial differential equations derived from fundamental physics: ### 1.1 Navier-Stokes Equations (Momentum Conservation) The momentum equations describe fluid motion in the atmosphere: $$ \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{v} = -\frac{1}{\rho}\nabla p - 2\mathbf{\Omega} \times \mathbf{v} + \mathbf{g} + \mathbf{F} $$ Where: - $\mathbf{v}$ = velocity vector $(u, v, w)$ - $t$ = time - $\rho$ = air density - $p$ = pressure - $\mathbf{\Omega}$ = Earth's angular velocity vector - $\mathbf{g}$ = gravitational acceleration - $\mathbf{F}$ = frictional forces **Component form (horizontal):** $$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z} = -\frac{1}{\rho}\frac{\partial p}{\partial x} + fv + F_x $$ $$ \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z} = -\frac{1}{\rho}\frac{\partial p}{\partial y} - fu + F_y $$ Where $f = 2\Omega\sin\phi$ is the **Coriolis parameter** at latitude $\phi$. ### 1.2 Continuity Equation (Mass Conservation) $$ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 $$ Or in expanded form: $$ \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0 $$ ### 1.3 Thermodynamic Energy Equation $$ \frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T = \frac{1}{c_p}\left(\frac{dQ}{dt} + \frac{RT}{p}\omega\right) $$ Where: - $T$ = temperature - $c_p$ = specific heat at constant pressure - $Q$ = diabatic heating rate - $R$ = gas constant for dry air - $\omega = \frac{dp}{dt}$ = vertical velocity in pressure coordinates ### 1.4 Equation of State (Ideal Gas Law) $$ p = \rho R T $$ Or equivalently: $$ p\alpha = RT $$ Where $\alpha = \frac{1}{\rho}$ is the specific volume. ### 1.5 Water Vapor Conservation $$ \frac{\partial q}{\partial t} + \mathbf{v} \cdot \nabla q = E - C + \frac{1}{\rho}\nabla \cdot (\rho K \nabla q) $$ Where: - $q$ = specific humidity - $E$ = evaporation rate - $C$ = condensation rate - $K$ = turbulent diffusion coefficient ## 2. Hydrostatic Approximation For large-scale motions, vertical accelerations are negligible: $$ \frac{\partial p}{\partial z} = -\rho g $$ This simplifies to the **hypsometric equation**: $$ Z_2 - Z_1 = \frac{R\bar{T}}{g}\ln\left(\frac{p_1}{p_2}\right) $$ ## 3. Numerical Methods ### 3.1 Spatial Discretization The continuous atmosphere is divided into a 3D grid. Derivatives are approximated using **finite differences**: **Forward difference:** $$ \frac{\partial u}{\partial x} \approx \frac{u_{i+1} - u_i}{\Delta x} $$ **Central difference (more accurate):** $$ \frac{\partial u}{\partial x} \approx \frac{u_{i+1} - u_{i-1}}{2\Delta x} $$ **Second derivative:** $$ \frac{\partial^2 u}{\partial x^2} \approx \frac{u_{i+1} - 2u_i + u_{i-1}}{(\Delta x)^2} $$ ### 3.2 Time Integration Schemes **Euler forward (first-order):** $$ u^{n+1} = u^n + \Delta t \cdot F(u^n) $$ **Leapfrog scheme (second-order):** $$ u^{n+1} = u^{n-1} + 2\Delta t \cdot F(u^n) $$ **Runge-Kutta 4th order:** $$ u^{n+1} = u^n + \frac{\Delta t}{6}(k_1 + 2k_2 + 2k_3 + k_4) $$ Where: - $k_1 = F(t^n, u^n)$ - $k_2 = F(t^n + \frac{\Delta t}{2}, u^n + \frac{\Delta t}{2}k_1)$ - $k_3 = F(t^n + \frac{\Delta t}{2}, u^n + \frac{\Delta t}{2}k_2)$ - $k_4 = F(t^n + \Delta t, u^n + \Delta t \cdot k_3)$ ### 3.3 CFL Stability Condition The **Courant-Friedrichs-Lewy** condition ensures numerical stability: $$ C = \frac{u \Delta t}{\Delta x} \leq C_{max} $$ Where $C_{max} \approx 1$ for explicit schemes. ### 3.4 Spectral Methods Transform variables into spherical harmonics: $$ \psi(\lambda, \phi) = \sum_{m=-M}^{M} \sum_{n=|m|}^{N} \psi_n^m Y_n^m(\lambda, \phi) $$ Where $Y_n^m$ are spherical harmonic functions. ## 4. Chaos and Predictability ### 4.1 Lorenz System Edward Lorenz's simplified convection model demonstrates chaos: $$ \frac{dx}{dt} = \sigma(y - x) $$ $$ \frac{dy}{dt} = x(\rho - z) - y $$ $$ \frac{dz}{dt} = xy - \beta z $$ Classic parameters: $\sigma = 10$, $\rho = 28$, $\beta = \frac{8}{3}$ ### 4.2 Lyapunov Exponent Quantifies the rate of separation of trajectories: $$ \lambda = \lim_{t \to \infty} \frac{1}{t} \ln\frac{|\delta \mathbf{x}(t)|}{|\delta \mathbf{x}(0)|} $$ For the atmosphere, $\lambda \approx 0.4 \text{ day}^{-1}$, giving a **doubling time** of: $$ \tau = \frac{\ln 2}{\lambda} \approx 1.7 \text{ days} $$ ### 4.3 Predictability Limit Error growth can be modeled as: $$ E(t) = E_0 \cdot e^{\lambda t} $$ Theoretical limit: **~2 weeks** for deterministic forecasts. ## 5. Data Assimilation ### 5.1 Optimal Interpolation $$ \mathbf{x}^a = \mathbf{x}^b + \mathbf{K}(\mathbf{y}^o - \mathbf{H}\mathbf{x}^b) $$ Where: - $\mathbf{x}^a$ = analysis (best estimate) - $\mathbf{x}^b$ = background (model forecast) - $\mathbf{y}^o$ = observations - $\mathbf{H}$ = observation operator - $\mathbf{K}$ = Kalman gain matrix ### 5.2 Kalman Gain $$ \mathbf{K} = \mathbf{B}\mathbf{H}^T(\mathbf{H}\mathbf{B}\mathbf{H}^T + \mathbf{R})^{-1} $$ Where: - $\mathbf{B}$ = background error covariance matrix - $\mathbf{R}$ = observation error covariance matrix ### 5.3 Variational Methods (4D-Var) Minimize the cost function: $$ J(\mathbf{x}_0) = \frac{1}{2}(\mathbf{x}_0 - \mathbf{x}^b)^T\mathbf{B}^{-1}(\mathbf{x}_0 - \mathbf{x}^b) + \frac{1}{2}\sum_{i=0}^{N}(\mathbf{y}_i - \mathbf{H}_i\mathbf{x}_i)^T\mathbf{R}_i^{-1}(\mathbf{y}_i - \mathbf{H}_i\mathbf{x}_i) $$ ## 6. Ensemble Forecasting ### 6.1 Ensemble Mean $$ \bar{\mathbf{x}} = \frac{1}{N}\sum_{i=1}^{N}\mathbf{x}_i $$ ### 6.2 Ensemble Spread (Uncertainty) $$ \sigma = \sqrt{\frac{1}{N-1}\sum_{i=1}^{N}(\mathbf{x}_i - \bar{\mathbf{x}})^2} $$ ### 6.3 Probability of Exceedance $$ P(X > \tau) = \frac{1}{N}\sum_{i=1}^{N}\mathbf{1}(x_i > \tau) $$ ## 7. Model Grid Specifications | Model | Organization | Horizontal Resolution | Grid Points | Vertical Levels | |-------|--------------|----------------------|-------------|-----------------| | GFS | NOAA/NCEP | ~13 km | ~6.6 million | 127 | | ECMWF IFS | ECMWF | ~9 km | ~18 million | 137 | | ICON | DWD | ~13 km | ~2.9 million | 90 | | NAM | NOAA/NCEP | ~3 km | ~1.9 million | 60 | ## 8. Machine Learning Approaches ### 8.1 Neural Network Weather Prediction Loss function for training: $$ \mathcal{L} = \frac{1}{N}\sum_{i=1}^{N}\|\hat{\mathbf{y}}_i - \mathbf{y}_i\|^2 + \lambda\|\mathbf{w}\|^2 $$ ### 8.2 Graph Neural Networks (GraphCast) Message passing on mesh: $$ \mathbf{h}_v^{(k+1)} = \phi\left(\mathbf{h}_v^{(k)}, \bigoplus_{u \in \mathcal{N}(v)} \psi(\mathbf{h}_v^{(k)}, \mathbf{h}_u^{(k)}, \mathbf{e}_{uv})\right) $$ ## 9. Physical Constants | Constant | Symbol | Value | |----------|--------|-------| | Gas constant (dry air) | $R_d$ | $287.05 \text{ J kg}^{-1}\text{K}^{-1}$ | | Specific heat (const. pressure) | $c_p$ | $1004 \text{ J kg}^{-1}\text{K}^{-1}$ | | Gravitational acceleration | $g$ | $9.81 \text{ m s}^{-2}$ | | Earth's angular velocity | $\Omega$ | $7.292 \times 10^{-5} \text{ rad s}^{-1}$ | | Earth's radius | $a$ | $6.371 \times 10^6 \text{ m}$ | | Stefan-Boltzmann constant | $\sigma$ | $5.67 \times 10^{-8} \text{ W m}^{-2}\text{K}^{-4}$ | | Latent heat of vaporization | $L_v$ | $2.5 \times 10^6 \text{ J kg}^{-1}$ | ## 10. Summary Weather forecasting mathematics involves: - **Governing equations**: Navier-Stokes, continuity, thermodynamics, equation of state - **Numerical methods**: Finite differences, spectral methods, time integration - **Chaos theory**: Lyapunov exponents, predictability limits (~2 weeks) - **Data assimilation**: Kalman filtering, variational methods (3D-Var, 4D-Var) - **Ensemble methods**: Probabilistic forecasting, uncertainty quantification - **Machine learning**: Neural networks complementing physics-based models
WebArena tests agents on realistic web navigation and interaction tasks.
# Weibull Distribution Mathematics in Semiconductor Manufacturing A comprehensive guide to the mathematical foundations and applications of Weibull distribution in semiconductor reliability engineering. ## 1. Fundamental Weibull Mathematics ### 1.1 The Core Equations **Two-parameter Weibull Probability Density Function (PDF):** $$ f(t) = \frac{\beta}{\eta} \left(\frac{t}{\eta}\right)^{\beta-1} \exp\left[-\left(\frac{t}{\eta}\right)^\beta\right] $$ **Cumulative Distribution Function (CDF) — probability of failure by time $t$:** $$ F(t) = 1 - \exp\left[-\left(\frac{t}{\eta}\right)^\beta\right] $$ **Reliability (Survival) Function:** $$ R(t) = \exp\left[-\left(\frac{t}{\eta}\right)^\beta\right] $$ **Parameter Definitions:** - $t \geq 0$ — random variable (typically time or stress cycles) - $\beta > 0$ — **shape parameter** (Weibull slope/modulus) - $\eta > 0$ — **scale parameter** (characteristic life, where $F(\eta) = 0.632$) ### 1.2 Three-Parameter Weibull Adding a location parameter $\gamma$ (threshold/minimum life): $$ F(t) = 1 - \exp\left[-\left(\frac{t-\gamma}{\eta}\right)^\beta\right], \quad t \geq \gamma $$ ### 1.3 The Hazard Function (Instantaneous Failure Rate) $$ h(t) = \frac{f(t)}{R(t)} = \frac{\beta}{\eta} \left(\frac{t}{\eta}\right)^{\beta-1} $$ **Physical Interpretation of Shape Parameter $\beta$:** | $\beta$ Value | Failure Rate | Physical Meaning | |---------------|--------------|------------------| | $\beta < 1$ | Decreasing | Infant mortality, early defects | | $\beta = 1$ | Constant | Random failures (exponential distribution) | | $\beta > 1$ | Increasing | Wear-out mechanisms | This directly models the semiconductor **bathtub curve**. ## 2. Semiconductor-Specific Applications ### 2.1 Time-Dependent Dielectric Breakdown (TDDB) Gate oxide breakdown follows Weibull statistics. The **area scaling law** derives from weakest-link theory: $$ \eta_2 = \eta_1 \left(\frac{A_1}{A_2}\right)^{1/\beta} $$ **Where:** - $A_1$ — reference test area - $A_2$ — target device area - $\eta_1$ — characteristic life at area $A_1$ - $\eta_2$ — predicted characteristic life at area $A_2$ **Typical $\beta$ values for oxide breakdown:** - Intrinsic breakdown: $\beta \approx 10$–$30$ (tight distribution) - Extrinsic/defect-related: $\beta \approx 1$–$5$ (broader distribution) ### 2.2 Electromigration Metal interconnect failure combines **Black's equation** with Weibull statistics: $$ MTF = A \cdot j^{-n} \cdot \exp\left(\frac{E_a}{k_B T}\right) $$ **Where:** - $MTF$ — median time to failure - $j$ — current density ($A/cm^2$) - $n$ — current density exponent (typically 1–2) - $E_a$ — activation energy (eV) - $k_B$ — Boltzmann constant ($8.617 \times 10^{-5}$ eV/K) - $T$ — absolute temperature (K) Typical $\beta$ values: **2–4** (wear-out behavior) ### 2.3 Hot Carrier Injection (HCI) Degradation follows power-law kinetics: $$ \Delta V_{th} = A \cdot t^n $$ **Where:** - $\Delta V_{th}$ — threshold voltage shift - $t$ — stress time - $n$ — time exponent (typically 0.3–0.5) ### 2.4 Negative Bias Temperature Instability (NBTI) For PMOS transistors: $$ \Delta V_{th} = A \cdot t^n \cdot \exp\left(-\frac{E_a}{k_B T}\right) $$ ## 3. Statistical Analysis Methods ### 3.1 Weibull Probability Plotting **Linearization transformation** — take double logarithm of CDF: $$ \ln\left[-\ln(1-F(t))\right] = \beta \ln(t) - \beta \ln(\eta) $$ **Plotting $\ln[-\ln(1-F)]$ vs $\ln(t)$:** - **Slope** = $\beta$ - **Intercept at $F = 0.632$** gives $t = \eta$ **Bernard's Median Rank Approximation** for ranking data: $$ \hat{F}(t_{(r)}) \approx \frac{r - 0.3}{n + 0.4} $$ **Where:** - $r$ — rank of the $r$-th ordered failure - $n$ — total sample size ### 3.2 Maximum Likelihood Estimation (MLE) **Log-likelihood function** for $n$ samples with $r$ failures and $(n-r)$ censored units: $$ \mathcal{L}(\beta, \eta) = \sum_{i=1}^{r} \left[\ln\beta - \beta\ln\eta + (\beta-1)\ln t_i - \left(\frac{t_i}{\eta}\right)^\beta\right] - \sum_{j=1}^{n-r}\left(\frac{t_j}{\eta}\right)^\beta $$ **MLE Estimator for $\eta$:** $$ \hat{\eta} = \left[\frac{1}{r}\sum_{i=1}^{n} t_i^{\hat{\beta}}\right]^{1/\hat{\beta}} $$ **MLE Equation for $\beta$** (solve numerically): $$ \frac{1}{\hat{\beta}} + \frac{\sum_{i=1}^{n} t_i^{\hat{\beta}} \ln t_i}{\sum_{i=1}^{n} t_i^{\hat{\beta}}} - \frac{1}{r}\sum_{i=1}^{r} \ln t_i = 0 $$ ## 4. Accelerated Life Testing Mathematics ### 4.1 Acceleration Factors **Arrhenius Model (Thermal Acceleration):** $$ AF = \exp\left[\frac{E_a}{k_B}\left(\frac{1}{T_{use}} - \frac{1}{T_{stress}}\right)\right] $$ **Exponential Voltage Acceleration:** $$ AF = \exp\left[\gamma(V_{stress} - V_{use})\right] $$ **Power-Law Voltage Acceleration:** $$ AF = \left(\frac{V_{stress}}{V_{use}}\right)^n $$ **Life Extrapolation:** $$ \eta_{use} = AF \times \eta_{stress} $$ ### 4.2 Combined Stress Models (Eyring) $$ AF = A \cdot \exp\left(\frac{E_a}{k_B T}\right) \cdot V^n \cdot (RH)^m $$ **Where:** - $RH$ — relative humidity - $m$ — humidity exponent - Additional stress factors can be included ## 5. Competing Failure Modes ### 5.1 Series (Competing Risks) Model Device fails when the **first** mechanism fails: $$ R(t) = \prod_{i=1}^{k} \exp\left[-\left(\frac{t}{\eta_i}\right)^{\beta_i}\right] = \exp\left[-\sum_{i=1}^{k}\left(\frac{t}{\eta_i}\right)^{\beta_i}\right] $$ **Combined CDF:** $$ F(t) = 1 - \exp\left[-\sum_{i=1}^{k}\left(\frac{t}{\eta_i}\right)^{\beta_i}\right] $$ ### 5.2 Mixture Model Different subpopulations with different failure characteristics: $$ F(t) = \sum_{i=1}^{k} p_i \cdot F_i(t) $$ **Where:** - $p_i$ — proportion in subpopulation $i$ - $\sum_{i=1}^{k} p_i = 1$ - $F_i(t)$ — CDF for subpopulation $i$ **PDF for mixture:** $$ f(t) = \sum_{i=1}^{k} p_i \cdot f_i(t) $$ ## 6. Key Derived Quantities ### 6.1 Moments of the Weibull Distribution **$k$-th Raw Moment:** $$ E[T^k] = \eta^k \cdot \Gamma\left(1 + \frac{k}{\beta}\right) $$ **Mean (MTTF — Mean Time To Failure):** $$ \mu = \eta \cdot \Gamma\left(1 + \frac{1}{\beta}\right) $$ **Variance:** $$ \sigma^2 = \eta^2 \left[\Gamma\left(1 + \frac{2}{\beta}\right) - \Gamma^2\left(1 + \frac{1}{\beta}\right)\right] $$ **Standard Deviation:** $$ \sigma = \eta \sqrt{\Gamma\left(1 + \frac{2}{\beta}\right) - \Gamma^2\left(1 + \frac{1}{\beta}\right)} $$ ### 6.2 Percentile Lives (B$X$ Life) Time by which $X\%$ have failed: $$ t_X = \eta \cdot \left[\ln\left(\frac{1}{1-X/100}\right)\right]^{1/\beta} $$ **Common Percentile Lives:** | Percentile | Formula | Application | |------------|---------|-------------| | B1 Life | $t_1 = \eta \cdot (0.01005)^{1/\beta}$ | High-reliability | | B10 Life | $t_{10} = \eta \cdot (0.1054)^{1/\beta}$ | Automotive/Aerospace | | B50 Life (Median) | $t_{50} = \eta \cdot (0.6931)^{1/\beta}$ | General reference | | B0.1 Life | $t_{0.1} = \eta \cdot (0.001001)^{1/\beta}$ | Critical systems | ### 6.3 Characteristic Life Significance At $t = \eta$: $$ F(\eta) = 1 - \exp(-1) = 1 - 0.368 = 0.632 $$ This means **63.2% of units have failed** by the characteristic life, regardless of $\beta$. ## 7. Confidence Bounds ### 7.1 Fisher Information Matrix Approach **Information Matrix:** $$ I(\beta, \eta) = -E\left[\frac{\partial^2 \mathcal{L}}{\partial \theta_i \partial \theta_j}\right] $$ **Asymptotic Variance-Covariance Matrix:** $$ \text{Var}(\hat{\theta}) \approx I^{-1}(\hat{\theta}) $$ **Fisher Matrix Elements:** $$ I_{\beta\beta} = \frac{r}{\beta^2}\left[1 + \frac{\pi^2}{6}\right] $$ $$ I_{\eta\eta} = \frac{r\beta^2}{\eta^2} $$ $$ I_{\beta\eta} = \frac{r}{\eta}(1 - \gamma_E) $$ Where $\gamma_E \approx 0.5772$ is the Euler-Mascheroni constant. ### 7.2 Likelihood Ratio Bounds (Preferred for Small Samples) $$ -2\left[\mathcal{L}(\theta_0) - \mathcal{L}(\hat{\theta})\right] \leq \chi^2_{\alpha, df} $$ **Approximate $(1-\alpha)$ Confidence Interval:** $$ \left\{\theta : -2\left[\mathcal{L}(\theta) - \mathcal{L}(\hat{\theta})\right] \leq \chi^2_{\alpha, p}\right\} $$ ## 8. Order Statistics ### 8.1 Expected Value of Order Statistics For $n$ samples, the expected value of the $r$-th order statistic: $$ E[t_{(r)}] = \eta \cdot \Gamma\left(1 + \frac{1}{\beta}\right) \cdot \sum_{j=0}^{r-1} \frac{(-1)^j \binom{r-1}{j}}{(n-r+1+j)^{1+1/\beta}} $$ ### 8.2 Plotting Positions **Bernard's Approximation (recommended):** $$ \hat{F}_i = \frac{i - 0.3}{n + 0.4} $$ **Hazen's Approximation:** $$ \hat{F}_i = \frac{i - 0.5}{n} $$ **Mean Rank:** $$ \hat{F}_i = \frac{i}{n + 1} $$ ## 9. Practical Example: Gate Oxide Qualification ### 9.1 Test Setup - **Sample size:** 50 oxide capacitors - **Stress conditions:** 125°C, 1.2× nominal voltage - **Test duration:** 1000 hours - **Failures:** 8 units at times: 156, 289, 412, 523, 678, 734, 891, 967 hours - **Censored:** 42 units still running at 1000h ### 9.2 Analysis Steps **Step 1: Calculate Median Ranks** | Rank ($i$) | Failure Time (h) | Median Rank $\hat{F}_i$ | |------------|------------------|-------------------------| | 1 | 156 | 0.0139 | | 2 | 289 | 0.0337 | | 3 | 412 | 0.0535 | | 4 | 523 | 0.0733 | | 5 | 678 | 0.0931 | | 6 | 734 | 0.1129 | | 7 | 891 | 0.1327 | | 8 | 967 | 0.1525 | **Step 2: MLE Results** $$ \hat{\beta} \approx 2.1, \quad \hat{\eta} \approx 1850 \text{ hours (at stress)} $$ **Step 3: Calculate Acceleration Factor** Given: $E_a = 0.7$ eV, voltage exponent $n = 40$ $$ AF_{thermal} = \exp\left[\frac{0.7}{8.617 \times 10^{-5}}\left(\frac{1}{298} - \frac{1}{398}\right)\right] \approx 85 $$ $$ AF_{voltage} = (1.2)^{40} \approx 1.8 $$ $$ AF_{total} \approx 85 \times 1.8 \approx 150 $$ **Step 4: Extrapolate to Use Conditions** $$ \eta_{use} = 1850 \times 150 = 277{,}500 \text{ hours} $$ **Step 5: Calculate B0.1 Life** $$ t_{0.1} = 277{,}500 \times (0.001001)^{1/2.1} \approx 3{,}200 \text{ hours} $$ ## 10. Key Equations ### 10.1 Quick Reference Table | Quantity | Formula | |----------|---------| | PDF | $f(t) = \frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1}\exp\left[-\left(\frac{t}{\eta}\right)^\beta\right]$ | | CDF | $F(t) = 1 - \exp\left[-\left(\frac{t}{\eta}\right)^\beta\right]$ | | Reliability | $R(t) = \exp\left[-\left(\frac{t}{\eta}\right)^\beta\right]$ | | Hazard Rate | $h(t) = \frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1}$ | | Mean Life | $\mu = \eta \cdot \Gamma(1 + 1/\beta)$ | | B10 Life | $t_{10} = \eta \cdot (0.1054)^{1/\beta}$ | | Area Scaling | $\eta_2 = \eta_1 (A_1/A_2)^{1/\beta}$ | | Linearization | $\ln[-\ln(1-F)] = \beta\ln t - \beta\ln\eta$ | ### 10.2 Why Weibull Works for Semiconductors 1. **Physical meaning of $\beta$** — directly indicates failure mechanism type 2. **Area/volume scaling** — derives from extreme value theory (weakest-link) 3. **Censored data handling** — essential since most test units don't fail 4. **Acceleration compatibility** — seamlessly integrates with physics-based models 5. **Competing risks framework** — models complex multi-mechanism devices ## Gamma Function Values Common values of $\Gamma(1 + 1/\beta)$ for mean life calculations: | $\beta$ | $\Gamma(1 + 1/\beta)$ | $\mu/\eta$ | |---------|------------------------|------------| | 0.5 | 2.000 | 2.000 | | 1.0 | 1.000 | 1.000 | | 1.5 | 0.903 | 0.903 | | 2.0 | 0.886 | 0.886 | | 2.5 | 0.887 | 0.887 | | 3.0 | 0.893 | 0.893 | | 3.5 | 0.900 | 0.900 | | 4.0 | 0.906 | 0.906 | | 5.0 | 0.918 | 0.918 | | 10.0 | 0.951 | 0.951 | ## Common Activation Energies | Failure Mechanism | Typical $E_a$ (eV) | Typical $\beta$ | |-------------------|---------------------|-----------------| | TDDB (oxide breakdown) | 0.6–0.8 | 1–3 | | Electromigration | 0.5–0.9 | 2–4 | | Hot Carrier Injection | 0.1–0.3 | 2–5 | | NBTI | 0.1–0.2 | 2–4 | | Corrosion | 0.3–0.5 | 1–3 | | Solder Fatigue | — | 2–6 |