principal component control charts, spc
Use PCA for dimensionality reduction.
9,967 technical terms and definitions
Use PCA for dimensionality reduction.
Find existing inventions.
Prioritization matrices weight criteria ranking options by importance.
Sample important transitions more.
Priority queues serve high-importance requests before lower-priority ones.
Schedule jobs by priority.
Privacy budget quantifies cumulative privacy loss across queries or training iterations.
Cumulative privacy loss parameter in differential privacy.
Combine federated learning with privacy.
Training and inference techniques that protect sensitive data (federated learning differential privacy).
Privacy-preserving recommendations protect user data through techniques like differential privacy and secure aggregation.
Techniques to protect data privacy during training.
For sensitive data, run models on-prem or in a VPC. No logs to third-party clouds, strict access control, encryption, and auditing of all requests.
Use proprietary data for pre-training.
Use extra info during training only.
Predict full probability distributions.
Express probabilistic models as programs.
Deterministic ODE with same marginals as SDE.
Probe alignment ensures probe tips accurately contact designated pads using vision systems and fine positioning.
Probe card cleaning removes oxide buildup and contamination from tips using techniques like scrubbing or plasma.
Probe card life is measured in touchdowns before cleaning or replacement is required due to wear or contamination.
Probe card planarity ensures all probe tips contact the wafer simultaneously despite variations requiring precise mechanical adjustment and fabrication.
Probe card repair replaces damaged needles cleans contamination and realigns tips to restore measurement accuracy.
Probe marks are indentations or scratches left on device pads after electrical testing indicating contact quality and alignment.
Probe scrub is the lateral movement of probe tips on contact pads during touchdown to penetrate oxide and establish low-resistance contact.
Probe tip geometry affects contact area resistance and pad damage with designs like pyramid cantilever and cobra.
Percentage passing wafer probe test.
Mechanistic interpretability reverse-engineers model internals: circuits, features, representations.
Probing classifiers test whether representations encode specific linguistic or semantic properties.
Train classifiers on representations.
Train classifiers on internal representations to see what information is encoded.
Probing trains classifiers on hidden states. Reveals what layers encode. Understanding model internals.
Problem escalation ensures unresolved issues reach appropriate decision makers.
Problem notification systems alert appropriate personnel of issues requiring attention.
AI-assisted procedural content creation.
Process audits examine activities for adherence to procedures.
Process capability indices compare process spread to specification width.
Compare capability to requirements.
Assess process ability to meet specs.
Process capability studies quantify ability of stable processes to meet specifications.
Separate assessments.
# 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. 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 $$ \sigma^2_{observed} = \sigma^2_{actual} + \sigma^2_{measurement} $$
Manage modifications to processes.
Adjust voltage/bias to compensate for process variation.
Process control loops continuously adjust parameters maintaining targets despite disturbances.
Process control monitors are specialized structures measuring process parameters for SPC.
Chilled water loop for cooling process tools and chambers.
Process cooling maintains precise temperatures for chemical reactions and depositions.
Quality issues requiring rework.
Files and models needed to design for specific foundry process.