cost performance index, quality & reliability
Cost performance index compares earned value to actual cost.
1,005 technical terms and definitions
Cost performance index compares earned value to actual cost.
Plan for lowering costs.
Balance cost and speed.
Assign different costs to errors.
LLM cost = tokens x price/token x overhead. Cut cost by: using smaller models, smart routing, shorter prompts, aggressive caching, and RAG.
Combine reasoning and voting.
Collaborative Transformer for Session-based Recommendation combines self-attention with collaborative filtering.
Simple molecular descriptor.
Scattering by charged centers.
Explore less-visited states.
Count-based exploration methods provide intrinsic rewards inversely proportional to state visitation counts to encourage visiting novel states.
Counterfactual augmentation creates synthetic examples with modified sensitive attributes.
Create examples swapping protected attributes.
Find minimal changes to flip prediction.
Counterfactual explanations identify minimal input changes producing different predictions.
Show minimal input changes that would flip the model's decision.
Counterfactual fairness ensures predictions unchanged if sensitive attributes differ.
Output doesn't change if you swap demographic attributes.
Consider what-if scenarios.
Counterfactual recommendation estimates what would have happened under alternative item exposures for policy evaluation.
Counterfactual explanations show minimal input change to flip prediction. Actionable insights.
Manufacturing location marking.
Measure module dependencies.
Stanford CS224N for NLP, fast.ai for practical deep learning, Karpathy for LLMs. Many free resources.
Input distribution changes but P(Y|X) stays same.
Generate cover letters. Personalized, role-specific.
Seals components in carrier.
Multiplier for confidence interval.
Formal promise that true value in prediction set.
Generate tests to maximize code coverage.
TSMC's 2.5D packaging technology using silicon interposer.
Process capability indices.
CP decomposition represents weight tensors as sums of rank-one tensors.
CANDECOMP/PARAFAC decomposes tensors into sum of rank-one tensors for recommendations.
Cp index compares six-sigma spread to specification width assuming centered process.
Ratio of spec width to process spread.
Alternative power intent format.
Collaborative Planning Forecasting and Replenishment integrates buyer and supplier planning processes.
Various ways to compute Cpk.
Cpk index accounts for both spread and centering relative to specifications.
# Semiconductor Manufacturing Process Capability Analysis ## Mathematical Framework for Statistical Process Control ## 1. Foundational Capability Indices ### 1.1 Basic Indices **Process Capability ($C_p$)** — measures process spread relative to specifications: $$ C_p = \frac{USL - LSL}{6\sigma} $$ Where: - $USL$ = Upper Specification Limit - $LSL$ = Lower Specification Limit - $\sigma$ = Process standard deviation (within-subgroup) **Centered Capability ($C_{pk}$)** — accounts for process centering: $$ C_{pk} = \min\left(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma}\right) $$ Alternative formulation: $$ C_{pk} = C_p(1 - k) $$ Where the centering factor $k$ is: $$ k = \frac{|\text{Target} - \mu|}{(USL - LSL)/2} $$ ### 1.2 Performance Indices **Process Performance ($P_p$)** — uses overall standard deviation: $$ P_p = \frac{USL - LSL}{6s_{overall}} $$ **Centered Performance ($P_{pk}$)**: $$ P_{pk} = \min\left(\frac{USL - \mu}{3s}, \frac{\mu - LSL}{3s}\right) $$ Key distinction: - $C_p$, $C_{pk}$ use **within-subgroup** variation ($\sigma$) - $P_p$, $P_{pk}$ use **overall** variation ($s$), including between-subgroup effects ## 2. Semiconductor Industry Requirements ### 2.1 Capability Targets Semiconductor manufacturing demands exceptional precision due to: - Nanometer-scale feature sizes (3nm, 5nm, 7nm nodes) - Hundreds of sequential process steps - Extremely tight tolerances - High cost of defects | $C_{pk}$ Value | Sigma Level | DPMO | Typical Application | |----------------|-------------|------|---------------------| | 1.00 | $3\sigma$ | 2,700 | Unacceptable for production | | 1.33 | $4\sigma$ | 63 | Minimum for established processes | | 1.67 | $5\sigma$ | 0.57 | Critical parameters | | 2.00 | $6\sigma$ | 0.002 | Most critical dimensions (CD, overlay) | ### 2.2 Defect Rate Calculation Assuming normal distribution: $$ \text{DPMO} = 10^6 \times 2\Phi(-3C_{pk}) $$ Where $\Phi$ is the standard normal cumulative distribution function. For one-sided specifications: $$ \text{DPMO}_{upper} = 10^6 \times \Phi\left(-\frac{USL - \mu}{\sigma}\right) $$ $$ \text{DPMO}_{lower} = 10^6 \times \Phi\left(-\frac{\mu - LSL}{\sigma}\right) $$ ## 3. Variance Component Decomposition ### 3.1 Hierarchical Variation Model Semiconductor processes exhibit hierarchical variation: $$ \sigma^2_{total} = \sigma^2_{W} + \sigma^2_{W2W} + \sigma^2_{L2L} + \sigma^2_{T2T} $$ Where: - $\sigma^2_{W}$ = Within-wafer variation - $\sigma^2_{W2W}$ = Wafer-to-wafer variation - $\sigma^2_{L2L}$ = Lot-to-lot variation - $\sigma^2_{T2T}$ = Tool-to-tool variation ### 3.2 ANOVA-Based Estimation For nested random effects model: $$ x_{ijkl} = \mu + \alpha_i + \beta_{j(i)} + \gamma_{k(ij)} + \epsilon_{l(ijk)} $$ Variance component estimates: $$ \hat{\sigma}^2_{between} = \frac{MS_{between} - MS_{within}}{n} $$ Expected Mean Squares: $$ E[MS_{lots}] = \sigma^2_W + n_w \sigma^2_{W2W} + n_w n_{wafer} \sigma^2_{L2L} $$ $$ E[MS_{wafers}] = \sigma^2_W + n_w \sigma^2_{W2W} $$ $$ E[MS_{within}] = \sigma^2_W $$ ### 3.3 Practical Implications | Variation Source | Root Cause | Improvement Strategy | |------------------|------------|---------------------| | Within-wafer ($\sigma^2_W$) | Equipment uniformity | Hardware tuning, flow optimization | | Wafer-to-wafer ($\sigma^2_{W2W}$) | Process stability | Run-to-run control, PM schedules | | Lot-to-lot ($\sigma^2_{L2L}$) | Material variation | Incoming inspection, supplier control | | Tool-to-tool ($\sigma^2_{T2T}$) | Equipment matching | Tool qualification, offset adjustment | ## 4. Non-Normal Distributions ### 4.1 Common Non-Normal Parameters Many semiconductor parameters violate normality assumptions: | Parameter | Typical Distribution | Characteristics | |-----------|---------------------|-----------------| | Particle counts | Poisson | Discrete, bounded below by zero | | Contamination levels | Log-normal | Right-skewed, multiplicative effects | | Defect density | Negative binomial | Overdispersed counts | | Overlay errors | Potentially bimodal | Multiple systematic components | | Line-edge roughness | Often skewed | Physical boundary constraints | ### 4.2 Box-Cox Transformation $$ y^{(\lambda)} = \begin{cases} \frac{y^\lambda - 1}{\lambda} & \text{if } \lambda \neq 0 \\[8pt] \ln(y) & \text{if } \lambda = 0 \end{cases} $$ Optimal $\lambda$ found by maximizing the log-likelihood: $$ \ell(\lambda) = -\frac{n}{2}\ln\left(\frac{SS_E(\lambda)}{n}\right) + (\lambda - 1)\sum_{i=1}^n \ln(y_i) $$ Common transformations: - $\lambda = 1$: No transformation - $\lambda = 0.5$: Square root (count data) - $\lambda = 0$: Natural logarithm (multiplicative) - $\lambda = -1$: Reciprocal ### 4.3 Johnson Transformation System Three families covering all continuous distributions: **$S_B$ (Bounded):** $$ z = \gamma + \delta \ln\left(\frac{x - \xi}{\lambda + \xi - x}\right) $$ **$S_L$ (Log-normal):** $$ z = \gamma + \delta \ln(x - \xi) $$ **$S_U$ (Unbounded):** $$ z = \gamma + \delta \sinh^{-1}\left(\frac{x - \xi}{\lambda}\right) $$ ### 4.4 Percentile-Based Capability (Distribution-Free) $$ C_{np} = \frac{USL - LSL}{X_{99.865} - X_{0.135}} $$ $$ C_{npk} = \min\left(\frac{USL - \tilde{x}}{X_{99.865} - \tilde{x}}, \frac{\tilde{x} - LSL}{\tilde{x} - X_{0.135}}\right) $$ Where $\tilde{x}$ is the median. ### 4.5 Clements' Method (Pearson Distributions) $$ C_p = \frac{USL - LSL}{U_p - L_p} $$ $$ C_{pk} = \min\left(\frac{USL - M}{U_p - M}, \frac{M - LSL}{M - L_p}\right) $$ Where: - $U_p$ = 99.865th percentile - $L_p$ = 0.135th percentile - $M$ = Median ## 5. Spatial Process Capability ### 5.1 Spatial Variation Models Wafers exhibit systematic spatial patterns requiring decomposition: **General Model:** $$ x(r, \theta) = \mu + f(r, \theta) + \epsilon $$ **Zernike Polynomial Expansion:** $$ x(r, \theta) = \mu + \sum_{n=0}^{N} \sum_{m=-n}^{n} a_{nm} Z_n^m(r, \theta) + \epsilon $$ Where $Z_n^m(r, \theta)$ are Zernike polynomials. ### 5.2 Practical Spatial Model **Radial Model:** $$ x_{ij} = \mu + \beta_1 r_i + \beta_2 r_i^2 + \epsilon_{ij} $$ **Radial + Angular Model:** $$ x_{ij} = \mu + \beta_1 r_i + \beta_2 r_i^2 + \beta_3 \cos(\theta_j) + \beta_4 \sin(\theta_j) + \epsilon_{ij} $$ ### 5.3 Spatial Capability Index $$ C_{pk,spatial} = \min_{(r,\theta) \in \text{wafer}} \left[ \frac{USL - \hat{\mu}(r,\theta)}{3\hat{\sigma}(r,\theta)}, \frac{\hat{\mu}(r,\theta) - LSL}{3\hat{\sigma}(r,\theta)} \right] $$ ### 5.4 Within-Wafer Non-Uniformity (WIWNU) $$ \text{WIWNU} = \frac{\sigma_{within-wafer}}{\bar{x}} \times 100\% $$ Range-based uniformity: $$ \text{Uniformity}_{\%} = \frac{x_{max} - x_{min}}{2 \bar{x}} \times 100\% $$ ## 6. Multivariate Process Capability ### 6.1 Motivation Critical for correlated parameters: - CD (Critical Dimension) and sidewall angle - Film thickness and uniformity - Overlay X and Y components - Etch depth and profile ### 6.2 Multivariate Capability Indices For $p$-dimensional quality vector $\mathbf{X} \sim N(\boldsymbol{\mu}, \boldsymbol{\Sigma})$: **Taam's Index ($MC_{pm}$):** $$ MC_{pm} = \frac{C_p^*}{d(\boldsymbol{\mu}, \mathbf{T})} $$ Where $d$ is the Mahalanobis distance from process mean to target. ### 6.3 Geometric Approach $$ MC_p = \left[\frac{V(\text{Specification Region})}{V(\text{Process Region})}\right]^{1/p} $$ For ellipsoidal regions: $$ MC_p = \frac{|\mathbf{T}|^{1/(2p)}}{|\boldsymbol{\Sigma}|^{1/(2p)} \cdot (\chi^2_{p, 0.9973})^{1/2}} $$ Where: - $|\mathbf{T}|$ = Determinant of tolerance matrix - $|\boldsymbol{\Sigma}|$ = Determinant of covariance matrix - $\chi^2_{p, 0.9973}$ = Chi-squared critical value ### 6.4 Principal Component Analysis (PCA) Approach Transform correlated variables to uncorrelated principal components: $$ \mathbf{Z} = \mathbf{P}^T(\mathbf{X} - \boldsymbol{\mu}) $$ Where $\mathbf{P}$ contains eigenvectors of $\boldsymbol{\Sigma}$. Individual PC capability: $$ C_{pk,i} = \min\left(\frac{USL_{z_i} - 0}{3\sqrt{\lambda_i}}, \frac{0 - LSL_{z_i}}{3\sqrt{\lambda_i}}\right) $$ ## 7. Yield Models Integration ### 7.1 Defect-Limited Yield Models **Poisson Model:** $$ Y = e^{-D_0 A} $$ **Murphy's Model (Clustered Defects):** $$ Y = \left(\frac{1 - e^{-D_0 A}}{D_0 A}\right)^2 $$ **Seeds' Compound Poisson (Negative Binomial):** $$ Y = \left(1 + \frac{D_0 A}{\alpha}\right)^{-\alpha} $$ Where: - $D_0$ = Average defect density (defects/cm²) - $A$ = Chip area (cm²) - $\alpha$ = Clustering parameter ### 7.2 Parametric Yield For Gaussian parameters: $$ Y_{parametric} = \Phi\left(\frac{USL - \mu}{\sigma}\right) - \Phi\left(\frac{LSL - \mu}{\sigma}\right) $$ Relationship to $C_{pk}$: $$ Y_{parametric} = 2\Phi(3C_{pk}) - 1 $$ ### 7.3 Combined Yield For $n$ independent parameters: $$ Y_{total} = Y_{defect} \times \prod_{i=1}^n Y_{parametric,i} $$ With correlation (multivariate normal): $$ Y_{total} = Y_{defect} \times P(\mathbf{X} \in \text{Spec Region}) $$ ## 8. Measurement System Analysis ### 8.1 Gauge R&R Components $$ \sigma^2_{observed} = \sigma^2_{actual} + \sigma^2_{measurement} $$ $$ \sigma^2_{measurement} = \sigma^2_{repeatability} + \sigma^2_{reproducibility} $$ Expanded: $$ \sigma^2_{reproducibility} = \sigma^2_{operator} + \sigma^2_{operator \times part} $$ ### 8.2 Key Metrics **Precision-to-Tolerance Ratio (P/T):** $$ P/T = \frac{6\sigma_{measurement}}{USL - LSL} \times 100\% $$ Requirement: $P/T < 10\%$ **%GRR:** $$ \%GRR = \frac{\sigma_{measurement}}{\sigma_{total}} \times 100\% $$ **Discrimination Ratio (DR):** $$ DR = \frac{\sigma_{parts}}{\sigma_{gauge}} \times \sqrt{2} $$ Requirement: $DR \geq 4$ (can distinguish 4+ categories) **Number of Distinct Categories (ndc):** $$ ndc = 1.41 \times \frac{\sigma_{parts}}{\sigma_{gauge}} $$ Requirement: $ndc \geq 5$ ### 8.3 True Process Capability $$ \sigma^2_{actual} = \sigma^2_{observed} - \sigma^2_{measurement} $$ $$ C_{pk,true} = C_{pk,observed} \times \sqrt{\frac{\sigma^2_{observed}}{\sigma^2_{observed} - \sigma^2_{measurement}}} $$ ## 9. Confidence Intervals for Capability Indices ### 9.1 Confidence Interval for $C_p$ $$ P\left(\hat{C}_p \sqrt{\frac{\chi^2_{n-1, \alpha/2}}{n-1}} \leq C_p \leq \hat{C}_p \sqrt{\frac{\chi^2_{n-1, 1-\alpha/2}}{n-1}}\right) = 1-\alpha $$ ### 9.2 Confidence Interval for $C_{pk}$ (Approximate) $$ \hat{C}_{pk} \pm z_{\alpha/2}\sqrt{\frac{1}{9n} + \frac{\hat{C}_{pk}^2}{2(n-1)}} $$ ### 9.3 Sample Size Requirements For desired relative precision $\epsilon$: $$ n \approx \frac{z_{\alpha/2}^2}{2\epsilon^2} + 1 $$ Practical guidelines: - 30 samples: Rough estimate - 50 samples: Reasonable precision - 100+ samples: Production qualification ### 9.4 Lower Confidence Bound Often used for acceptance decisions: $$ C_{pk,lower} = \hat{C}_{pk} - z_{\alpha}\sqrt{\frac{1}{9n} + \frac{\hat{C}_{pk}^2}{2(n-1)}} $$ ## 10. Dynamic Process Capability ### 10.1 Time-Varying Process Model Semiconductor processes drift due to: - Chamber conditioning/seasoning - Target erosion (PVD) - Consumable wear - Environmental drift **Drift Model:** $$ \mu(t) = \mu_0 + \delta t $$ **Periodic + Drift:** $$ \mu(t) = \mu_0 + \delta t + \sum_{k=1}^{K} A_k \sin(2\pi f_k t + \phi_k) $$ ### 10.2 EWMA-Based Monitoring **Mean Estimate:** $$ \hat{\mu}_t = \lambda x_t + (1-\lambda)\hat{\mu}_{t-1} $$ **Variance Estimate:** $$ \hat{\sigma}^2_t = \lambda(x_t - \hat{\mu}_{t-1})^2 + (1-\lambda)\hat{\sigma}^2_{t-1} $$ Where $0 < \lambda \leq 1$ is the smoothing constant. ### 10.3 Dynamic Capability Index $$ C_{pk}(t) = \min\left(\frac{USL - \hat{\mu}_t}{3\hat{\sigma}_t}, \frac{\hat{\mu}_t - LSL}{3\hat{\sigma}_t}\right) $$ ### 10.4 Control Chart Integration **EWMA Control Limits:** $$ UCL = \mu_0 + L\sigma\sqrt{\frac{\lambda}{2-\lambda}\left[1-(1-\lambda)^{2t}\right]} $$ $$ LCL = \mu_0 - L\sigma\sqrt{\frac{\lambda}{2-\lambda}\left[1-(1-\lambda)^{2t}\right]} $$ Where $L$ is the control limit factor (typically 2.7-3.0). ## 11. Run-to-Run Control Integration ### 11.1 Basic EWMA Controller $$ u_k = u_{k-1} + \frac{\eta}{\beta}(T - y_{k-1}) $$ Where: - $u_k$ = Recipe setting at run $k$ - $T$ = Target value - $\eta$ = Controller gain $(0 < \eta < 1)$ - $\beta$ = Process gain (sensitivity) ### 11.2 Double EWMA Controller For processes with drift: $$ \hat{a}_k = \lambda_1 y_k + (1-\lambda_1)(\hat{a}_{k-1} + \hat{b}_{k-1}) $$ $$ \hat{b}_k = \lambda_2(\hat{a}_k - \hat{a}_{k-1}) + (1-\lambda_2)\hat{b}_{k-1} $$ $$ u_k = \frac{T - \hat{a}_k - \hat{b}_k}{\beta} $$ ### 11.3 Achieved Capability Under Control **Variance of Controlled Output:** $$ \sigma^2_{controlled} = \frac{\sigma^2_\epsilon}{2\eta - \eta^2} $$ **Optimal Gain (Minimum Variance):** $$ \eta_{opt} = 1 \quad \text{(for i.i.d. disturbances)} $$ For autocorrelated disturbances, optimal gain depends on disturbance model. ### 11.4 Capability with APC $$ C_{pk,APC} = \min\left(\frac{USL - T}{3\sigma_{controlled}}, \frac{T - LSL}{3\sigma_{controlled}}\right) $$ ## 12. Advanced Topics ### 12.1 Bayesian Capability Analysis Useful for small sample sizes in development: **Posterior Distribution:** $$ P(C_{pk} | \text{data}) \propto L(\text{data} | C_{pk}) \cdot \pi(C_{pk}) $$ **With Non-informative Prior:** $$ C_{pk} | \text{data} \sim \text{Scaled-}t \text{ distribution} $$ **Credible Interval:** $$ P(C_{pk,L} < C_{pk} < C_{pk,U} | \text{data}) = 1 - \alpha $$ ### 12.2 Process Capability for Attributes **Equivalent Capability:** $$ C_{pk,attribute} = \frac{-\ln(p)}{3} $$ Where $p$ is the proportion defective. **For Defect Counts (Poisson):** $$ C_{pk,Poisson} = \frac{-\ln(1 - P(\text{acceptable}))}{3} $$ ### 12.3 Six Sigma and 1.5σ Shift **Short-term vs. Long-term:** $$ Z_{LT} = Z_{ST} - 1.5 $$ | Sigma Level | $Z_{ST}$ | $Z_{LT}$ | DPMO (Long-term) | |-------------|----------|----------|------------------| | 3σ | 3.0 | 1.5 | 66,807 | | 4σ | 4.0 | 2.5 | 6,210 | | 5σ | 5.0 | 3.5 | 233 | | 6σ | 6.0 | 4.5 | 3.4 | ### 12.4 Cpm and Cpkm (Taguchi Indices) **Cpm (accounts for deviation from target):** $$ C_{pm} = \frac{USL - LSL}{6\sqrt{\sigma^2 + (\mu - T)^2}} $$ $$ C_{pm} = \frac{USL - LSL}{6\tau} $$ Where $\tau = \sqrt{\sigma^2 + (\mu - T)^2}$ is the Taguchi loss function parameter. **Cpkm:** $$ C_{pkm} = \frac{C_{pk}}{\sqrt{1 + \left(\frac{\mu - T}{\sigma}\right)^2}} $$ ## 13. Practical Implementation Framework ### 13.1 Data Collection Strategy **Minimum Samples:** - Development: 30-50 wafers - Qualification: 100+ wafers - Monitoring: Per control chart rules **Rational Subgrouping:** - 5-25 wafers per lot - 9-49 measurement sites per wafer - Multiple lots across time windows ### 13.2 Capability Study Protocol 1. **Verify measurement system** - Complete Gauge R&R study - Requirement: P/T < 10%, ndc ≥ 5 2. **Collect data across variation sources** - Multiple lots - Multiple tools (if applicable) - Full wafer coverage 3. **Test for normality** - Shapiro-Wilk test - Anderson-Darling test - Visual: histogram, Q-Q plot 4. **Handle non-normality** - Transform (Box-Cox, Johnson) - Use percentile methods - Document approach 5. **Decompose variance components** - ANOVA or REML - Identify dominant sources 6. **Calculate indices with confidence intervals** - $C_p$, $C_{pk}$, $P_p$, $P_{pk}$ - Lower confidence bounds 7. **Assess spatial patterns** - Wafer maps - Radial plots - Systematic vs. random 8. **Document and establish monitoring** - Control charts - Trending - Review frequency ### 13.3 Decision Thresholds | $C_{pk}$ Range | Assessment | Required Action | |----------------|------------|-----------------| | < 1.0 | Not capable | Immediate improvement, 100% inspection | | 1.0 – 1.33 | Marginal | Improvement plan, enhanced monitoring | | 1.33 – 1.67 | Capable | Standard production controls | | > 1.67 | Highly capable | Reduced sampling possible | ## 14. Summary of Key Formulas ### Basic Indices $$ C_p = \frac{USL - LSL}{6\sigma} $$ $$ C_{pk} = \min\left(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma}\right) $$ ### Variance Decomposition $$ \sigma^2_{total} = \sigma^2_{within} + \sigma^2_{between} $$ ### Yield Relationship $$ Y = 2\Phi(3C_{pk}) - 1 $$ ### Confidence Interval $$ CI_{C_{pk}} = \hat{C}_{pk} \pm z_{\alpha/2}\sqrt{\frac{1}{9n} + \frac{\hat{C}_{pk}^2}{2(n-1)}} $$ ### Measurement System $$ \sigma^2_{observed} = \sigma^2_{actual} + \sigma^2_{measurement} $$
Cp adjusted for centering.
Measure of how well process stays within specification limits.
Cpm index penalizes deviation from target even within specifications.
Constrained Policy Optimization guarantees constraint satisfaction through trust region methods with constraint projections.
Conservative Q-Learning addresses distributional shift in offline RL by regularizing Q-functions to assign lower values to out-of-distribution actions.
Cradle-to-cradle design enables complete recycling or biodegradation eliminating waste concept.
Cradle-to-gate LCA analyzes impacts from resource extraction through manufacturing gate.
Cradle-to-grave LCA includes product use phase and end-of-life disposal.
Assess and correct retrieval quality.