c chart, spc
Count of defects.
c-sam, c-sam, failure analysis advanced
C-mode Scanning Acoustic Microscopy detects delamination and voids through ultrasonic reflections at material interfaces.
c-sam,failure analysis
C-mode acoustic microscopy.
c-v curve,metrology
Capacitance-voltage characteristic.
c-v profiling, c-v, yield enhancement
Capacitance-voltage profiling determines dopant concentration depth profiles in semiconductors.
c&w attack, c&w, ai safety
Optimization-based attack finding minimal perturbations.
c&w attack, c&w, interpretability
Carlini & Wagner attack optimizes adversarial perturbations using specialized loss functions.
c2pa (coalition for content provenance and authenticity),c2pa,coalition for content provenance and authenticity,standards
Standard for content authenticity metadata.
c3d, c3d, video understanding
Early 3D CNN for video.
c4, c4, packaging
IBM flip-chip technology.
c51, c51, reinforcement learning
Categorical distributional RL.
c51, c51, reinforcement learning advanced
Categorical DQN represents return distribution over discrete atoms improving performance through distributional RL.
cache eviction, llm optimization
Cache eviction policies determine which entries to remove when storage fills.
cache hit rate, llm optimization
Cache hit rate measures percentage of requests served from cache rather than recomputation.
cache hit rate,optimization
Percentage of requests served from cache.
cache invalidation,optimization
Remove stale entries from cache.
cache warming, llm optimization
Cache warming preloads frequently needed items improving initial response times.
caching in retrieval, rag
Store frequent queries.
caching strategies,optimization
Store results to avoid recomputation.
caching,cache strategy,redis
Cache frequent LLM responses to reduce cost and latency. Use semantic similarity for cache lookup. Redis/Memcached for storage.
cad model generation,engineering
Create 3D CAD models.
cait, computer vision
Separate class token attention.
calculator use, tool use
Perform exact arithmetic.
calibrated rec, recommendation systems
Calibrated recommendations ensure recommendation distributions match user preference distributions.
calibrated recommendations,recommender systems
Match user's preference distribution.
calibration (tcad),calibration,tcad,simulation
Fit simulation parameters to match real data.
calibration certificate,quality
Document proving measurement accuracy.
calibration curve, metrology
Relationship between signal and concentration.
calibration prompting, prompting techniques
Calibration prompting adjusts output probabilities to reduce label bias.
calibration verification, quality
Check calibration remains valid.
calibration, ai safety
Calibration ensures predicted probabilities match actual outcome frequencies.
calibration,metrology
Adjust tool measurements to match known standards.
calibration,probability,confidence
Calibration: predicted probabilities match actual frequencies. Important for decision-making.
caliper,metrology
Measuring tool for length thickness.
canary circuits, design
Sentinels indicating degradation.
canary deployment,mlops
Gradually roll out new model to small percentage of traffic.
candle,rust,ml
Candle is Rust ML framework by Hugging Face. Fast, minimal dependencies.
canny edge control, generative models
Condition on edge detection.
canonical correlation analysis for networks, explainable ai
Find correlated components.
canonical correlation analysis, cca, multi-view learning
Find correlated components across views.
cantilever probe, advanced test & probe
Cantilever probes use spring-loaded needles in probe cards that deflect upon contact providing reliable electrical connections during wafer testing.
cap wafer bonding, packaging
Bond cap for protection.
capa, capa, quality & reliability
Corrective and Preventive Action systems manage nonconformance investigation and resolution.
capability analysis, spc, statistical process control, etch capability, process metrics, defect analysis, yield optimization
# Semiconductor Etch Process Capability Mathematics
## 1. Fundamental Capability Indices
### 1.1 Basic Statistical Measures
- **Sample Mean ($\bar{x}$):**
$$
\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i
$$
- **Sample Standard Deviation ($s$):**
$$
s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2}
$$
### 1.2 Process Capability (Cp)
The **potential capability** measures the process spread relative to specification width:
$$
C_p = \frac{USL - LSL}{6\sigma}
$$
Where:
- $USL$ = Upper Specification Limit
- $LSL$ = Lower Specification Limit
- $\sigma$ = Process standard deviation
**Interpretation:**
- $C_p = 1.0$ means the process $\pm 3\sigma$ exactly fills the spec window
- Higher $C_p$ indicates greater potential capability
### 1.3 Process Capability Index (Cpk)
The **actual capability** accounts for process centering:
$$
C_{pk} = \min\left(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma}\right)
$$
**Key relationship:**
- $C_{pk} \leq C_p$ (always)
- $C_{pk} = C_p$ only when process is perfectly centered
### 1.4 Taguchi Capability Index (Cpm)
Penalizes deviation from target $T$, not merely being within spec:
$$
C_{pm} = \frac{USL - LSL}{6\sqrt{\sigma^2 + (\mu - T)^2}}
$$
### 1.5 Combined Index (Cpkm)
$$
C_{pkm} = \frac{C_{pk}}{\sqrt{1 + \left(\frac{\mu - T}{\sigma}\right)^2}}
$$
### 1.6 Industry Targets for Semiconductor Etch
| Cpk Value | Sigma Level | Defect Rate | Typical Application |
|:---------:|:-----------:|:-----------:|:-------------------:|
| 1.00 | 3σ | 2,700 ppm | Minimum acceptable |
| 1.33 | 4σ | 63 ppm | Standard processes |
| 1.67 | 5σ | 0.57 ppm | Critical dimensions |
| 2.00 | 6σ | 0.002 ppm | Advanced nodes |
## 2. Etch-Specific Uniformity Mathematics
### 2.1 Within-Wafer Uniformity (WIW)
- **Range-based method:**
$$
\%U_{WIW} = \frac{X_{max} - X_{min}}{2 \cdot \bar{X}} \times 100\%
$$
- **Standard deviation-based method (preferred):**
$$
\%U_{1\sigma} = \frac{s}{\bar{X}} \times 100\%
$$
- **Typical target:** $<1\%$ $(1\sigma)$ uniformity for etch rate
### 2.2 Wafer-to-Wafer Uniformity (WtW)
$$
\%U_{WtW} = \frac{s_{\text{wafer means}}}{\bar{X}_{\text{overall}}} \times 100\%
$$
### 2.3 Total Variance Decomposition
Via nested ANOVA:
$$
\sigma^2_{\text{total}} = \sigma^2_{WIW} + \sigma^2_{WtW} + \sigma^2_{LtL} + \sigma^2_{TtT}
$$
Where:
- $\sigma^2_{WIW}$ = Within-Wafer variance
- $\sigma^2_{WtW}$ = Wafer-to-Wafer variance
- $\sigma^2_{LtL}$ = Lot-to-Lot variance
- $\sigma^2_{TtT}$ = Tool-to-Tool (chamber-to-chamber) variance
## 3. Critical Dimension (CD) Control
### 3.1 CD Uniformity
$$
CD_{\text{uniformity}} = \frac{CD_{max} - CD_{min}}{CD_{target}} \times 100\%
$$
### 3.2 Etch Bias
$$
\text{Etch Bias} = CD_{\text{after etch}} - CD_{\text{after litho}}
$$
For anisotropic etch with undercut angle $\theta$:
$$
\Delta CD = 2 \cdot d \cdot \tan(\theta)
$$
Where:
- $d$ = etch depth
- $\theta$ = undercut angle
- For ideal anisotropic etch: $\theta = 0 \Rightarrow \Delta CD = 0$
### 3.3 Iso-Dense Bias (IDB)
$$
IDB = CD_{\text{isolated}} - CD_{\text{dense}}
$$
**Capability for IDB:**
$$
C_{pk,IDB} = \min\left(\frac{IDB_{USL} - \overline{IDB}}{3s_{IDB}}, \frac{\overline{IDB} - IDB_{LSL}}{3s_{IDB}}\right)
$$
### 3.4 Line Edge Roughness (LER) / Line Width Roughness (LWR)
- **LER Definition:**
$$
LER = 3\sigma_{\text{edge position}}
$$
- **LWR Definition:**
$$
LWR = 3\sigma_{\text{line width}}
$$
- **One-sided capability (upper limit only):**
$$
C_{pk,LER} = \frac{USL_{LER} - \overline{LER}}{3s_{LER}}
$$
## 4. Selectivity Mathematics
### 4.1 Basic Selectivity Definition
$$
\text{Selectivity} = \frac{ER_{\text{target material}}}{ER_{\text{mask or stop layer}}}
$$
### 4.2 Selectivity Capability (One-Sided)
$$
C_{pk,sel} = \frac{\overline{Sel} - LSL_{Sel}}{3s_{Sel}}
$$
**Note:** Higher selectivity is always better, so this is typically a one-sided specification.
### 4.3 Common Selectivity Requirements
| Etch Type | Material System | Typical Selectivity |
|:----------|:----------------|:-------------------:|
| SAC Etch | Oxide:Nitride | >30:1 |
| Gate Etch | Poly-Si:Oxide | >50:1 |
| Metal Etch | Al:Resist | >5:1 |
| Via Etch | Oxide:TiN | >20:1 |
## 5. Variance Component Analysis
### 5.1 Mixed-Effects Model
$$
X_{ijkl} = \mu + W_i + L_j + T_k + S_{l(ijk)} + \epsilon_{ijkl}
$$
Where:
- $\mu$ = Grand mean
- $W_i$ = Wafer random effect
- $L_j$ = Lot random effect
- $T_k$ = Tool/chamber random effect
- $S_{l(ijk)}$ = Site (within-wafer) effect
- $\epsilon_{ijkl}$ = Residual measurement error
### 5.2 Variance Component Estimation
Via REML (Restricted Maximum Likelihood):
$$
\hat{\sigma}^2_{\text{total}} = \hat{\sigma}^2_W + \hat{\sigma}^2_L + \hat{\sigma}^2_T + \hat{\sigma}^2_S + \hat{\sigma}^2_\epsilon
$$
### 5.3 Percent Contribution
$$
\%\text{Contribution}_i = \frac{\hat{\sigma}^2_i}{\hat{\sigma}^2_{\text{total}}} \times 100\%
$$
## 6. Response Surface Modeling for Etch
### 6.1 Second-Order Polynomial Model
$$
ER = \beta_0 + \sum_{i}\beta_i x_i + \sum_{i}\beta_{ii}x_i^2 + \sum_{i 20:1)$:
$$
ER_{\text{corrected}} = ER_{\text{open}} \cdot \exp\left(-\beta \cdot AR^{\gamma}\right)
$$
## 8. Statistical Process Control Mathematics
### 8.1 X-bar Chart Control Limits
$$
UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}
$$
$$
LCL_{\bar{x}} = \bar{\bar{x}} - A_2 \bar{R}
$$
### 8.2 R Chart Control Limits
$$
UCL_R = D_4 \bar{R}
$$
$$
LCL_R = D_3 \bar{R}
$$
**Control chart constants (selected values):**
| n | $A_2$ | $D_3$ | $D_4$ |
|:-:|:-----:|:-----:|:-----:|
| 2 | 1.880 | 0 | 3.267 |
| 3 | 1.023 | 0 | 2.575 |
| 4 | 0.729 | 0 | 2.282 |
| 5 | 0.577 | 0 | 2.115 |
### 8.3 EWMA (Exponentially Weighted Moving Average)
**Recursive formula:**
$$
EWMA_t = \lambda x_t + (1-\lambda)EWMA_{t-1}
$$
**Control limits:**
$$
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]}
$$
**Typical parameters:**
- $\lambda = 0.2$
- $L = 3$
### 8.4 CUSUM (Cumulative Sum)
**Upper CUSUM:**
$$
C^+_t = \max[0, x_t - (\mu_0 + K) + C^+_{t-1}]
$$
**Lower CUSUM:**
$$
C^-_t = \max[0, (\mu_0 - K) - x_t + C^-_{t-1}]
$$
Where:
- $K = \frac{\delta \sigma}{2}$ (reference value)
- $H = h\sigma$ (decision interval)
## 9. Endpoint Detection Mathematics
### 9.1 Interferometric Endpoint
$$
d = \frac{N \lambda}{2n \cos\theta}
$$
Where:
- $N$ = Number of interference fringes counted
- $\lambda$ = Wavelength of light
- $n$ = Refractive index of material
- $\theta$ = Angle of incidence
### 9.2 Optical Emission Spectroscopy (OES)
**Endpoint trigger condition:**
$$
\left|\frac{dI(\lambda, t)}{dt}\right| > \text{threshold}
$$
**Normalized derivative:**
$$
\frac{d}{dt}\left[\frac{I(\lambda, t)}{I_{ref}}\right] > \text{threshold}
$$
### 9.3 Multi-Wavelength PCA Endpoint
Principal component score:
$$
PC_1(t) = \sum_{i=1}^{p} w_i \cdot I_i(t)
$$
Where $w_i$ are PCA loadings for wavelength $i$.
## 10. Measurement System Analysis (Gauge R&R)
### 10.1 Variance Decomposition
**Total observed variance:**
$$
\sigma^2_{\text{observed}} = \sigma^2_{\text{part}} + \sigma^2_{\text{measurement}}
$$
**Measurement variance:**
$$
\sigma^2_{\text{measurement}} = \sigma^2_{\text{repeatability}} + \sigma^2_{\text{reproducibility}}
$$
### 10.2 Percent GRR Calculations
**To total variation:**
$$
\%GRR_{\text{TV}} = \frac{\sigma_{\text{GRR}}}{\sigma_{\text{total}}} \times 100\%
$$
**To tolerance:**
$$
\%GRR_{\text{Tol}} = \frac{6\sigma_{\text{GRR}}}{USL - LSL} \times 100\%
$$
### 10.3 GRR Assessment Criteria
| %GRR | Assessment | Action |
|:----:|:----------:|:------:|
| <10% | Excellent | Acceptable |
| 10-30% | Marginal | May be acceptable |
| >30% | Unacceptable | Improve measurement system |
### 10.4 Number of Distinct Categories (ndc)
$$
ndc = 1.41 \cdot \frac{\sigma_{\text{part}}}{\sigma_{\text{GRR}}}
$$
**Requirement:** $ndc \geq 5$
## 11. Confidence Intervals for Capability
### 11.1 Confidence Interval for Cp
**Chi-square based:**
$$
P\left(\hat{C}_p \sqrt{\frac{\chi^2_{n-1, 1-\alpha/2}}{n-1}} \leq C_p \leq \hat{C}_p \sqrt{\frac{\chi^2_{n-1, \alpha/2}}{n-1}}\right) = 1-\alpha
$$
**Approximate form:**
$$
\hat{C}_p \pm z_{\alpha/2}\sqrt{\frac{C_p^2}{2(n-1)}}
$$
### 11.2 Lower Confidence Bound for Cpk
$$
LCL_{C_{pk}} = \hat{C}_{pk} - z_{\alpha}\sqrt{\frac{1}{9n\hat{C}_{pk}^2} + \frac{1}{2(n-1)}}
$$
### 11.3 Sample Size Guidelines
**Rule of thumb for Cpk studies:**
- Minimum: $n \geq 50$ data points
- Recommended: $n \geq 100$ data points
- For high confidence: $n \geq 200$ data points
## 12. Non-Normal Data Handling
### 12.1 Box-Cox Transformation
$$
y^{(\lambda)} = \begin{cases}
\dfrac{y^\lambda - 1}{\lambda} & \text{if } \lambda \neq 0 \\[10pt]
\ln(y) & \text{if } \lambda = 0
\end{cases}
$$
**Common transformations:**
- $\lambda = 0.5$: Square root
- $\lambda = 0$: Natural log
- $\lambda = -1$: Inverse
### 12.2 Percentile-Based Capability
$$
C_p = \frac{USL - LSL}{X_{99.865\%} - X_{0.135\%}}
$$
$$
C_{pk} = \min\left(\frac{USL - X_{50\%}}{X_{99.865\%} - X_{50\%}}, \frac{X_{50\%} - LSL}{X_{50\%} - X_{0.135\%}}\right)
$$
### 12.3 Johnson Transformation System
**Three distribution families:**
- **$S_B$ (bounded):**
$$
z = \gamma + \delta \ln\left(\frac{x - \xi}{\lambda + \xi - x}\right)
$$
- **$S_L$ (lognormal):**
$$
z = \gamma + \delta \ln(x - \xi)
$$
- **$S_U$ (unbounded):**
$$
z = \gamma + \delta \sinh^{-1}\left(\frac{x - \xi}{\lambda}\right)
$$
## 13. Multivariate Capability
### 13.1 Multivariate Capability Index (MCp)
$$
MC_p = \frac{\text{Vol}(\text{specification region})}{\text{Vol}(\text{process region})}
$$
### 13.2 Principal Component Approach
For correlated outputs, transform to uncorrelated PCs:
$$
\mathbf{z} = \mathbf{P}^T(\mathbf{x} - \boldsymbol{\mu})
$$
Where $\mathbf{P}$ is the matrix of eigenvectors.
**Capability on each PC:**
$$
C_{pk,i} = \frac{\min(|USL_{z_i}|, |LSL_{z_i}|)}{3\sqrt{\lambda_i}}
$$
Where $\lambda_i$ is the eigenvalue (variance) of PC $i$.
### 13.3 Hotelling's T² Statistic
$$
T^2 = n(\bar{\mathbf{x}} - \boldsymbol{\mu}_0)^T \mathbf{S}^{-1} (\bar{\mathbf{x}} - \boldsymbol{\mu}_0)
$$
**Control limit:**
$$
UCL = \frac{p(n-1)(n+1)}{n(n-p)} F_{\alpha, p, n-p}
$$
## 14. Practical Example: Gate Etch Capability Study
### 14.1 Process Specifications
| Parameter | Target | LSL | USL | Unit |
|:----------|:------:|:---:|:---:|:----:|
| CD | 45 | 42 | 48 | nm |
| Etch Depth | 200 | 190 | 210 | nm |
| Selectivity | >20:1 | 20 | - | ratio |
| LWR | <4 | - | 4 | nm |
### 14.2 Data Collection
- **Wafers:** 25 wafers
- **Sites per wafer:** 49 sites
- **Total measurements:** $25 \times 49 = 1,225$
### 14.3 Results Summary
| Parameter | Mean | σ | Cpk | Status |
|:----------|:----:|:-:|:---:|:------:|
| CD | 44.8 nm | 0.9 nm | 1.03 | ❌ Below target |
| Depth | 199 nm | 2.5 nm | 1.33 | ✓ Acceptable |
| LWR | 3.2 nm | 0.4 nm | 0.67 | ❌ Major issue |
### 14.4 Cpk Calculations
**CD Cpk:**
$$
C_{pk,CD} = \min\left(\frac{48-44.8}{3 \times 0.9}, \frac{44.8-42}{3 \times 0.9}\right) = \min(1.19, 1.04) = 1.04
$$
**Depth Cpk:**
$$
C_{pk,Depth} = \min\left(\frac{210-199}{3 \times 2.5}, \frac{199-190}{3 \times 2.5}\right) = \min(1.47, 1.20) = 1.20
$$
**LWR Cpk (one-sided):**
$$
C_{pk,LWR} = \frac{4 - 3.2}{3 \times 0.4} = \frac{0.8}{1.2} = 0.67
$$
### 14.5 Variance Decomposition for CD
| Source | Variance (nm²) | % Contribution |
|:-------|:--------------:|:--------------:|
| Within-Wafer | 0.53 | 65% |
| Wafer-to-Wafer | 0.16 | 20% |
| Measurement | 0.12 | 15% |
| **Total** | **0.81** | **100%** |
**Conclusions:**
- Chamber uniformity issue (WIW dominant)
- Consider improving CD-SEM recipe to reduce measurement variance
## Key Mathematical Tools
| Application | Key Mathematics |
|:------------|:----------------|
| Basic capability | $C_p$, $C_{pk}$, $C_{pm}$ |
| Uniformity | $1\sigma\%$, range-based $\%$ |
| Variance sourcing | Nested ANOVA, variance components |
| Process optimization | RSM, desirability functions |
| Drift detection | EWMA, CUSUM charts |
| Measurement quality | Gauge R&R, $\%GRR$, $ndc$ |
| Non-normal data | Box-Cox, percentile methods |
| Loading effects | ARDE models, Knudsen transport |
| Multi-response | Multivariate $C_p$, Hotelling's $T^2$ |
## Quick Reference: Essential Formulas
```
-
┌─────────────────────────────────────────────────────────────┐
│ Cp = (USL - LSL) / 6σ │
│ Cpk = min[(USL - μ)/3σ, (μ - LSL)/3σ] │
│ %U = (s / x̄) × 100% │
│ GRR = √(σ²_repeatability + σ²_reproducibility) │
│ EWMA_t = λx_t + (1-λ)EWMA_{t-1} │
└─────────────────────────────────────────────────────────────┘
```
capability control,limit capability,scope
Capability control limits what models can do. Prevent misuse while enabling good uses.
capability elicitation, ai safety
Capability elicitation discovers full extent of model abilities through systematic testing.
capability for attribute data, quality
Capability with pass/fail data.
capability plateau, theory
Scaling without improvement.
capability study duration, quality & reliability
Capability study duration must be sufficient to capture representative variation.
capability testing, testing
Test specific linguistic capabilities.