capability
**Semiconductor Manufacturing Process SPC: Statistical Process Control Mathematics**
**1. Introduction**
**Why SPC Mathematics Matters in Semiconductor Fabs**
Semiconductor manufacturing operates at nanometer scales across hundreds of process steps, presenting unique challenges:
- **High Value**: A single wafer can be worth $10,000 to $100,000+
- **Tight Tolerances**: Process variations of a few nanometers cause yield collapse
- **Long Feedback Loops**: Days to weeks between process and measurement
- **Compounding Variation**: Multiple variance sources multiply through the process flow
The mathematics of SPC provides the framework to:
- Detect process shifts before they cause defects
- Quantify and decompose sources of variation
- Maintain processes within nanometer-scale tolerances
- Optimize yield through statistical understanding
**2. Fundamental Statistical Measures**
**2.1 Descriptive Statistics**
For a sample of $n$ measurements $x_1, x_2, \ldots, x_n$:
| Measure | Formula | Description |
|---------|---------|-------------|
| **Sample Mean** | $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$ | Central tendency |
| **Sample Variance** | $s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}$ | Spread (unbiased) |
| **Sample Std Dev** | $s = \sqrt{s^2}$ | Spread in original units |
| **Range** | $R = x_{max} - x_{min}$ | Total spread |
**2.2 The Normal (Gaussian) Distribution**
The mathematical backbone of classical SPC:
$$
f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)
$$
Where:
- $\mu$ = population mean
- $\sigma$ = population standard deviation
- $\sigma^2$ = population variance
**2.3 Critical Probability Intervals**
| Interval | Probability Contained | Application |
|----------|----------------------|-------------|
| $\pm 1\sigma$ | 68.27% | Typical variation |
| $\pm 2\sigma$ | 95.45% | Warning limits |
| $\pm 3\sigma$ | 99.73% | Control limits |
| $\pm 4\sigma$ | 99.9937% | Cpk = 1.33 |
| $\pm 5\sigma$ | 99.99994% | Cpk = 1.67 |
| $\pm 6\sigma$ | 99.9999998% | Six Sigma (3.4 DPMO) |
**2.4 Standard Normal Transformation**
Any normal variable can be standardized:
$$
Z = \frac{X - \mu}{\sigma}
$$
Where $Z \sim N(0, 1)$ (standard normal distribution).
**3. Control Chart Mathematics**
**3.1 Shewhart X̄-R Charts**
The workhorse of semiconductor SPC for monitoring subgroup data.
**X̄ Chart (Monitoring Process Mean)**
$$
\begin{aligned}
CL &= \bar{\bar{X}} \quad \text{(grand mean of subgroup means)} \\
UCL &= \bar{\bar{X}} + A_2 \bar{R} \\
LCL &= \bar{\bar{X}} - A_2 \bar{R}
\end{aligned}
$$
**Theoretical basis:**
$$
UCL / LCL = \mu \pm \frac{3\sigma}{\sqrt{n}}
$$
**R Chart (Monitoring Process Spread)**
$$
\begin{aligned}
CL &= \bar{R} \\
UCL &= D_4 \bar{R} \\
LCL &= D_3 \bar{R}
\end{aligned}
$$
**Control Chart Constants**
| $n$ | $A_2$ | $D_3$ | $D_4$ | $d_2$ |
|-----|-------|-------|-------|-------|
| 2 | 1.880 | 0 | 3.267 | 1.128 |
| 3 | 1.023 | 0 | 2.574 | 1.693 |
| 4 | 0.729 | 0 | 2.282 | 2.059 |
| 5 | 0.577 | 0 | 2.114 | 2.326 |
| 6 | 0.483 | 0 | 2.004 | 2.534 |
| 7 | 0.419 | 0.076 | 1.924 | 2.704 |
| 8 | 0.373 | 0.136 | 1.864 | 2.847 |
| 9 | 0.337 | 0.184 | 1.816 | 2.970 |
| 10 | 0.308 | 0.223 | 1.777 | 3.078 |
**3.2 Individuals-Moving Range (I-MR) Charts**
Common in semiconductor when rational subgrouping isn't practical (e.g., one measurement per wafer lot).
**Individuals Chart**
$$
\begin{aligned}
CL &= \bar{X} \\
UCL &= \bar{X} + 3 \cdot \frac{\overline{MR}}{d_2} \\
LCL &= \bar{X} - 3 \cdot \frac{\overline{MR}}{d_2}
\end{aligned}
$$
Where $d_2 = 1.128$ for moving range of span 2.
**Moving Range Chart**
$$
\begin{aligned}
CL &= \overline{MR} \\
UCL &= D_4 \cdot \overline{MR} = 3.267 \cdot \overline{MR} \\
LCL &= D_3 \cdot \overline{MR} = 0
\end{aligned}
$$
**3.3 EWMA Charts (Exponentially Weighted Moving Average)**
More sensitive to small, persistent shifts than Shewhart charts.
**EWMA Statistic**
$$
EWMA_t = \lambda x_t + (1-\lambda) EWMA_{t-1}
$$
Where:
- $\lambda$ = smoothing parameter ($0 < \lambda \leq 1$, typically 0.05–0.25)
- $EWMA_0 = \mu_0$ (target mean)
**Control Limits (Time-Varying)**
$$
UCL/LCL = \mu_0 \pm L\sigma\sqrt{\frac{\lambda}{2-\lambda}\left[1-(1-\lambda)^{2t}\right]}
$$
**Asymptotic Control Limits**
As $t \to \infty$:
$$
UCL/LCL = \mu_0 \pm L\sigma\sqrt{\frac{\lambda}{2-\lambda}}
$$
**Typical parameters:**
- $\lambda = 0.10$ to $0.20$
- $L = 2.5$ to $3.0$
**EWMA Variance**
$$
Var(EWMA_t) = \sigma^2 \cdot \frac{\lambda}{2-\lambda} \left[1 - (1-\lambda)^{2t}\right]
$$
**3.4 CUSUM Charts (Cumulative Sum)**
Accumulates deviations from target—excellent for detecting sustained shifts.
**Tabular CUSUM**
**Upper CUSUM (detecting upward shifts):**
$$
C_t^+ = \max\left[0, x_t - (\mu_0 + K) + C_{t-1}^+\right]
$$
**Lower CUSUM (detecting downward shifts):**
$$
C_t^- = \max\left[0, (\mu_0 - K) - x_t + C_{t-1}^-\right]
$$
**Signal condition:**
$$
C_t^+ > H \quad \text{or} \quad C_t^- > H
$$
Where:
- $K$ = allowable slack (reference value), typically $0.5\sigma$
- $H$ = decision interval, typically $4\sigma$ to $5\sigma$
- $C_0^+ = C_0^- = 0$
**Standardized Form**
For standardized observations $z_t = (x_t - \mu_0)/\sigma$:
$$
\begin{aligned}
S_t^+ &= \max(0, z_t - k + S_{t-1}^+) \\
S_t^- &= \max(0, -z_t - k + S_{t-1}^-)
\end{aligned}
$$
With $k = 0.5$ (half the shift to detect) and $h = 4$ or $5$.
**4. Process Capability Indices**
**4.1 Cp (Potential Capability)**
Measures the ratio of specification width to process spread:
$$
C_p = \frac{USL - LSL}{6\sigma}
$$
Where:
- $USL$ = Upper Specification Limit
- $LSL$ = Lower Specification Limit
- $\sigma$ = process standard deviation
**Interpretation:**
- $C_p$ does **not** account for centering
- Represents potential capability if process were perfectly centered
**4.2 Cpk (Actual Capability)**
Accounts for off-center processes:
$$
C_{pk} = \min\left[\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma}\right]
$$
**Alternative formulation:**
$$
C_{pk} = C_p(1 - k)
$$
Where $k = \frac{|T - \mu|}{(USL - LSL)/2}$ and $T$ is the target (specification midpoint).
**Key property:** $C_{pk} \leq C_p$ always.
**4.3 Cpm (Taguchi Capability Index)**
Penalizes deviation from target $T$:
$$
C_{pm} = \frac{USL - LSL}{6\sqrt{\sigma^2 + (\mu - T)^2}}
$$
Or equivalently:
$$
C_{pm} = \frac{C_p}{\sqrt{1 + \left(\frac{\mu - T}{\sigma}\right)^2}}
$$
**4.4 Pp and Ppk (Performance Indices)**
Same formulas but use **overall** standard deviation (including between-subgroup variation):
$$
P_p = \frac{USL - LSL}{6s_{overall}}
$$
$$
P_{pk} = \min\left[\frac{USL - \bar{x}}{3s_{overall}}, \frac{\bar{x} - LSL}{3s_{overall}}\right]
$$
**Relationship:**
- $C_p, C_{pk}$: Use within-subgroup $\sigma$ (short-term capability)
- $P_p, P_{pk}$: Use overall $s$ (long-term performance)
**4.5 Relating Cpk to Defect Rates**
| $C_{pk}$ | $\sigma$-level | DPMO | Yield |
|----------|----------------|------|-------|
| 0.33 | 1σ | 317,311 | 68.27% |
| 0.67 | 2σ | 45,500 | 95.45% |
| 1.00 | 3σ | 2,700 | 99.73% |
| 1.33 | 4σ | 63 | 99.9937% |
| 1.67 | 5σ | 0.57 | 99.99994% |
| 2.00 | 6σ | 0.002 | 99.9999998% |
> **Note:** With 1.5σ shift allowance (industry standard), 6σ = 3.4 DPMO.
**4.6 Confidence Intervals for Cpk**
$$
\hat{C}_{pk} \pm z_{\alpha/2} \sqrt{\frac{1}{9n} + \frac{C_{pk}^2}{2(n-1)}}
$$
For reliable capability estimates, need $n \geq 30$, preferably $n \geq 50$.
**5. Variance Components Analysis**
**5.1 Typical Variance Hierarchy in Semiconductor**
$$
\sigma^2_{total} = \sigma^2_{lot} + \sigma^2_{wafer(lot)} + \sigma^2_{site(wafer)} + \sigma^2_{measurement}
$$
Each component represents:
- **Lot-to-lot ($\sigma^2_{lot}$)**: Variation between production lots
- **Wafer-to-wafer ($\sigma^2_{wafer}$)**: Variation between wafers within a lot
- **Within-wafer ($\sigma^2_{site}$)**: Variation across measurement sites on a wafer
- **Measurement ($\sigma^2_{meas}$)**: Gauge/metrology variation
**5.2 One-Way ANOVA**
**Sum of Squares Decomposition**
$$
SS_T = SS_B + SS_W
$$
**Total Sum of Squares:**
$$
SS_T = \sum_{i=1}^{k}\sum_{j=1}^{n}(x_{ij} - \bar{x}_{..})^2
$$
**Between-Groups Sum of Squares:**
$$
SS_B = n\sum_{i=1}^{k}(\bar{x}_{i.} - \bar{x}_{..})^2
$$
**Within-Groups Sum of Squares:**
$$
SS_W = \sum_{i=1}^{k}\sum_{j=1}^{n}(x_{ij} - \bar{x}_{i.})^2
$$
**Mean Squares**
$$
\begin{aligned}
MS_B &= \frac{SS_B}{k-1} \\
MS_W &= \frac{SS_W}{N-k}
\end{aligned}
$$
**F-Statistic**
$$
F = \frac{MS_B}{MS_W} \sim F_{k-1, N-k}
$$
**5.3 Variance Component Estimation**
From mean squares:
$$
\begin{aligned}
\hat{\sigma}^2_{within} &= MS_W \\
\hat{\sigma}^2_{between} &= \frac{MS_B - MS_W}{n}
\end{aligned}
$$
If $MS_B < MS_W$, set $\hat{\sigma}^2_{between} = 0$.
**5.4 Nested (Hierarchical) ANOVA**
For semiconductor's nested structure (sites within wafers within lots):
$$
x_{ijk} = \mu + \alpha_i + \beta_{j(i)} + \varepsilon_{k(ij)}
$$
Where:
- $\alpha_i$ = lot effect (random)
- $\beta_{j(i)}$ = wafer effect nested within lot (random)
- $\varepsilon_{k(ij)}$ = site/measurement error
**6. Measurement System Analysis (Gauge R&R)**
**6.1 Variance Decomposition**
$$
\sigma^2_{total} = \sigma^2_{part} + \sigma^2_{gauge}
$$
$$
\sigma^2_{gauge} = \sigma^2_{repeatability} + \sigma^2_{reproducibility}
$$
Where:
- **Repeatability (Equipment Variation):** Same operator, same equipment, multiple measurements
- **Reproducibility (Appraiser Variation):** Different operators or equipment
**6.2 ANOVA Method for Gauge R&R**
**Two-Factor Crossed Design**
| Source | SS | df | MS | EMS |
|--------|----|----|----|----|
| Part (P) | $SS_P$ | $p-1$ | $MS_P$ | $\sigma^2_E + r\sigma^2_{OP} + or\sigma^2_P$ |
| Operator (O) | $SS_O$ | $o-1$ | $MS_O$ | $\sigma^2_E + r\sigma^2_{OP} + pr\sigma^2_O$ |
| P×O | $SS_{PO}$ | $(p-1)(o-1)$ | $MS_{PO}$ | $\sigma^2_E + r\sigma^2_{OP}$ |
| Error (E) | $SS_E$ | $po(r-1)$ | $MS_E$ | $\sigma^2_E$ |
**Variance Component Estimates**
$$
\begin{aligned}
\hat{\sigma}^2_{repeatability} &= MS_E \\
\hat{\sigma}^2_{operator} &= \frac{MS_O - MS_{PO}}{pr} \\
\hat{\sigma}^2_{interaction} &= \frac{MS_{PO} - MS_E}{r} \\
\hat{\sigma}^2_{reproducibility} &= \hat{\sigma}^2_{operator} + \hat{\sigma}^2_{interaction} \\
\hat{\sigma}^2_{part} &= \frac{MS_P - MS_{PO}}{or}
\end{aligned}
$$
**6.3 Key Metrics**
**Percentage of Total Variation**
$$
\%GRR = 100 \times \frac{\sigma_{gauge}}{\sigma_{total}}
$$
Or using study variation (5.15σ for 99%):
$$
\%GRR = 100 \times \frac{5.15 \cdot \sigma_{gauge}}{5.15 \cdot \sigma_{total}}
$$
**Precision-to-Tolerance Ratio (P/T)**
$$
P/T = \frac{k \cdot \sigma_{gauge}}{USL - LSL}
$$
Where $k = 5.15$ (99%) or $k = 6$ (99.73%).
**Number of Distinct Categories (ndc)**
$$
ndc = 1.41 \cdot \frac{\sigma_{part}}{\sigma_{gauge}}
$$
**6.4 Acceptance Criteria**
| %GRR | Assessment | Action |
|------|------------|--------|
| < 10% | Excellent | Acceptable for all applications |
| 10–30% | Acceptable | May be acceptable depending on application |
| > 30% | Unacceptable | Measurement system needs improvement |
| ndc | Assessment |
|-----|------------|
| ≥ 5 | Acceptable |
| < 5 | Measurement system cannot distinguish parts |
**7. Run Rules (Western Electric / Nelson Rules)**
**7.1 Standard Run Rules**
Pattern detection beyond simple control limits:
| Rule | Pattern | Interpretation |
|------|---------|----------------|
| **1** | 1 point beyond ±3σ | Large shift or outlier |
| **2** | 9 consecutive points on same side of CL | Small sustained shift |
| **3** | 6 consecutive points steadily increasing or decreasing | Trend/drift |
| **4** | 14 consecutive points alternating up and down | Systematic oscillation (over-adjustment) |
| **5** | 2 of 3 consecutive points beyond ±2σ (same side) | Shift warning |
| **6** | 4 of 5 consecutive points beyond ±1σ (same side) | Shift warning |
| **7** | 15 consecutive points within ±1σ | Stratification (mixture) |
| **8** | 8 consecutive points beyond ±1σ (either side) | Mixture of populations |
**7.2 Zone Definitions**
Control charts are divided into zones:
- **Zone A:** Between 2σ and 3σ from center line
- **Zone B:** Between 1σ and 2σ from center line
- **Zone C:** Within 1σ of center line
**7.3 False Alarm Probabilities**
| Rule | Probability (per test) |
|------|----------------------|
| Rule 1 (±3σ) | 0.0027 |
| Rule 2 (9 same side) | 0.0039 |
| Rule 3 (6 trending) | 0.0028 |
| Rule 5 (2 of 3 in Zone A) | 0.0044 |
**Combined false alarm rate** increases when multiple rules are applied.
**8. Average Run Length (ARL)**
**8.1 Definitions**
- **ARL₀ (In-Control ARL):** Average number of samples until false alarm when process is in control (want high)
- **ARL₁ (Out-of-Control ARL):** Average number of samples to detect a shift (want low)
**8.2 Shewhart Chart ARL**
For 3σ limits:
$$
ARL_0 = \frac{1}{\alpha} = \frac{1}{0.0027} \approx 370
$$
For detecting a shift of $\delta$ standard deviations:
$$
ARL_1 = \frac{1}{P(\text{signal} | \text{shift})}
$$
$$
P(\text{signal}) = 1 - \Phi(3-\delta) + \Phi(-3-\delta)
$$
**8.3 Comparison of Chart Performance**
| Shift ($\delta\sigma$) | Shewhart ARL₁ | EWMA ARL₁ ($\lambda$=0.1) | CUSUM ARL₁ |
|------------------------|---------------|---------------------------|------------|
| 0.25 | 281 | 66 | 38 |
| 0.50 | 155 | 26 | 17 |
| 0.75 | 81 | 15 | 10 |
| 1.00 | 44 | 10 | 8 |
| 1.50 | 15 | 5 | 5 |
| 2.00 | 6 | 4 | 4 |
| 3.00 | 2 | 2 | 2 |
**Key insight:** EWMA and CUSUM are far superior for detecting small shifts ($\delta < 1.5\sigma$).
**8.4 ARL Formulas for CUSUM**
Approximate ARL for CUSUM detecting shift of size $\delta$:
$$
ARL_1 \approx \frac{e^{-2\Delta b} + 2\Delta b - 1}{2\Delta^2}
$$
Where:
- $\Delta = \delta - k$ (excess over reference value)
- $b = h + 1.166$ (adjusted decision interval)
**9. Multivariate SPC**
**9.1 Why Multivariate?**
Semiconductor processes involve many correlated parameters. Univariate charts on correlated variables:
- Increase false alarm rates
- Miss shifts in correlated directions
- Fail to detect process changes that affect multiple parameters simultaneously
**9.2 Hotelling's T² Statistic**
For $p$ variables measured on a sample of size $n$:
$$
T^2 = n(\bar{\mathbf{x}} - \boldsymbol{\mu}_0)' \mathbf{S}^{-1} (\bar{\mathbf{x}} - \boldsymbol{\mu}_0)
$$
Where:
- $\bar{\mathbf{x}}$ = sample mean vector ($p \times 1$)
- $\boldsymbol{\mu}_0$ = target mean vector ($p \times 1$)
- $\mathbf{S}$ = sample covariance matrix ($p \times p$)
**9.3 Control Limit for T²**
**Phase I (establishing control):**
$$
UCL = \frac{(m-1)(m+1)p}{m(m-p)} F_{\alpha, p, m-p}
$$
**Phase II (monitoring):**
$$
UCL = \frac{p(m+1)(m-1)}{m(m-p)} F_{\alpha, p, m-p}
$$
Where $m$ = number of historical samples.
For large $m$:
$$
UCL \approx \chi^2_{\alpha, p}
$$
**9.4 Multivariate EWMA (MEWMA)**
$$
\mathbf{Z}_t = \Lambda\mathbf{X}_t + (\mathbf{I} - \Lambda)\mathbf{Z}_{t-1}
$$
Where $\Lambda = \text{diag}(\lambda_1, \lambda_2, \ldots, \lambda_p)$.
**Statistic:**
$$
T^2_t = \mathbf{Z}_t' \boldsymbol{\Sigma}_{\mathbf{Z}_t}^{-1} \mathbf{Z}_t
$$
**Covariance of MEWMA:**
$$
\boldsymbol{\Sigma}_{\mathbf{Z}_t} = \frac{\lambda}{2-\lambda}\left[1 - (1-\lambda)^{2t}\right]\boldsymbol{\Sigma}
$$
**9.5 Principal Component Analysis (PCA) for SPC**
Decompose correlated variables into uncorrelated principal components:
$$
\mathbf{X} = \mathbf{T}\mathbf{P}' + \mathbf{E}
$$
Where:
- $\mathbf{T}$ = scores matrix
- $\mathbf{P}$ = loadings matrix
- $\mathbf{E}$ = residuals
**Hotelling's T² in PC space:**
$$
T^2 = \sum_{i=1}^{k} \frac{t_i^2}{\lambda_i}
$$
**Squared Prediction Error (SPE):**
$$
SPE = \mathbf{e}'\mathbf{e} = \sum_{i=k+1}^{p} t_i^2
$$
**10. Autocorrelation Handling**
**10.1 The Problem**
Semiconductor tool data often exhibits serial correlation, violating the independence assumption of standard SPC.
**Consequences of ignoring autocorrelation:**
- Actual ARL₀ << 370 (excessive false alarms)
- Control limits are too tight
- Patterns in data are misinterpreted
**10.2 Autocorrelation Function (ACF)**
**Population autocorrelation at lag $k$:**
$$
\rho_k = \frac{Cov(X_t, X_{t+k})}{Var(X_t)} = \frac{\gamma_k}{\gamma_0}
$$
**Sample autocorrelation:**
$$
r_k = \frac{\sum_{t=1}^{n-k}(x_t - \bar{x})(x_{t+k} - \bar{x})}{\sum_{t=1}^{n}(x_t - \bar{x})^2}
$$
**10.3 AR(1) Process**
The simplest autocorrelated model:
$$
X_t = \phi X_{t-1} + \varepsilon_t
$$
Where:
- $\phi$ = autoregressive parameter ($|\phi| < 1$ for stationarity)
- $\varepsilon_t \sim N(0, \sigma^2_\varepsilon)$ (white noise)
**Properties:**
$$
\begin{aligned}
Var(X_t) &= \frac{\sigma^2_\varepsilon}{1 - \phi^2} \\
\rho_k &= \phi^k
\end{aligned}
$$
**10.4 Solutions for Autocorrelated Data**
1. **Residual Charts:**
- Fit time series model (AR, ARMA, etc.)
- Apply SPC to residuals $\hat{\varepsilon}_t = X_t - \hat{X}_t$
2. **Modified Control Limits:**
$$
UCL/LCL = \mu \pm 3\sigma_X \sqrt{\frac{1 + \phi}{1 - \phi}}
$$
3. **EWMA with Adjusted Parameters:**
- Use $\lambda = 1 - \phi$ for optimal smoothing
4. **Special Cause Charts:**
- Designed specifically for autocorrelated processes
**11. Run-to-Run (R2R) Process Control**
**11.1 Basic Concept**
Active feedback control layered on SPC—adjust recipe parameters based on measured outputs.
**11.2 EWMA Controller**
**Prediction:**
$$
\hat{y}_{t+1} = \lambda y_t + (1-\lambda)\hat{y}_t
$$
**Recipe Adjustment:**
$$
u_{t+1} = u_t - G(\hat{y}_t - y_{target})
$$
Where:
- $G$ = controller gain
- $u$ = recipe parameter (e.g., etch time, dose)
- $y$ = output measurement (e.g., CD, thickness)
**11.3 Double EWMA (for Drifting Processes)**
Track both level and slope:
**Level estimate:**
$$
L_t = \lambda y_t + (1-\lambda)(L_{t-1} + T_{t-1})
$$
**Trend estimate:**
$$
T_t = \gamma(L_t - L_{t-1}) + (1-\gamma)T_{t-1}
$$
**Forecast:**
$$
\hat{y}_{t+1} = L_t + T_t
$$
**11.4 Process Model Integration**
For process with known gain $\beta$:
$$
y_t = \alpha + \beta u_t + \varepsilon_t
$$
**Optimal control:**
$$
u_{t+1} = \frac{y_{target} - \hat{\alpha}_{t+1}}{\beta}
$$
**12. Yield Modeling Mathematics**
**12.1 Defect Density**
$$
D_0 = \frac{\text{Number of defects}}{\text{Area (cm}^2\text{)}}
$$
**12.2 Poisson Model (Random Defects)**
Assumes defects are randomly distributed:
$$
Y = e^{-D_0 A}
$$
Where:
- $D_0$ = defect density (defects/cm²)
- $A$ = die area (cm²)
**Probability of $k$ defects on a die:**
$$
P(k) = \frac{(D_0 A)^k e^{-D_0 A}}{k!}
$$
**12.3 Murphy's Model (Distributed Defects)**
Accounts for defect density variation across wafer:
$$
Y = \left[\frac{1 - e^{-D_0 A}}{D_0 A}\right]^2
$$
**12.4 Negative Binomial Model (Clustered Defects)**
More realistic for semiconductor:
$$
Y = \left(1 + \frac{D_0 A}{\alpha}\right)^{-\alpha}
$$
Where $\alpha$ = clustering parameter:
- $\alpha \to \infty$: Approaches Poisson (random)
- $\alpha$ small: Highly clustered
**12.5 Seeds Model**
$$
Y = e^{-D_0 A_s}
$$
Where $A_s$ = sensitive area (fraction of die area susceptible to defects).
**12.6 Yield Loss Calculations**
**Defect-Limited Yield:**
$$
Y_D = e^{-D_0 A}
$$
**Parametric Yield:**
$$
Y_P = \prod_{i} P(\text{parameter}_i \text{ in spec})
$$
**Total Yield:**
$$
Y_{total} = Y_D \times Y_P
$$
**13. Spatial Statistics for Wafer Maps**
**13.1 Radial Uniformity**
$$
\sigma_{radial} = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i - f(r_i))^2}
$$
Where $f(r_i)$ is the fitted radial profile at radius $r_i$.
**13.2 Wafer-Level Variation Components**
$$
\sigma^2_{total} = \sigma^2_{W2W} + \sigma^2_{WIW}
$$
Within-wafer variation often decomposed:
$$
\sigma^2_{WIW} = \sigma^2_{systematic} + \sigma^2_{random}
$$
Where:
- **Systematic WIW:** Modeled and corrected (radial, azimuthal patterns)
- **Random WIW:** Inherent noise
**13.3 Spatial Correlation Function**
For locations $\mathbf{s}_i$ and $\mathbf{s}_j$:
$$
C(h) = Cov(X(\mathbf{s}_i), X(\mathbf{s}_j))
$$
Where $h = \|\mathbf{s}_i - \mathbf{s}_j\|$ (distance between points).
**Variogram:**
$$
\gamma(h) = \frac{1}{2}Var[X(\mathbf{s}_i) - X(\mathbf{s}_j)]
$$
**13.4 Common Wafer Signatures**
Mathematical models for common spatial patterns:
**Radial (bowl/dome):**
$$
f(r) = a_0 + a_1 r + a_2 r^2
$$
**Azimuthal:**
$$
f(\theta) = b_0 + b_1 \cos(\theta) + b_2 \sin(\theta)
$$
**Combined:**
$$
f(r, \theta) = \sum_{n,m} a_{nm} Z_n^m(r, \theta)
$$
Where $Z_n^m$ are Zernike polynomials.
**14. Practical Implementation Considerations**
**14.1 Sample Size Effects**
Uncertainty in estimated standard deviation:
$$
SE(\hat{\sigma}) \approx \frac{\sigma}{\sqrt{2(n-1)}}
$$
**For reliable capability estimates:**
- Minimum: $n \geq 30$
- Preferred: $n \geq 50$
- For critical processes: $n \geq 100$
**14.2 Confidence Interval for σ**
$$
\sqrt{\frac{(n-1)s^2}{\chi^2_{\alpha/2, n-1}}} \leq \sigma \leq \sqrt{\frac{(n-1)s^2}{\chi^2_{1-\alpha/2, n-1}}}
$$
**14.3 Rational Subgrouping**
**Principles:**
- Subgroups should capture short-term (within) variation
- Between-subgroup variation captures long-term drift
- Subgroup size $n$ typically 3–5 for continuous data
**In semiconductor:**
- Subgroup = wafers from same lot, run, or time window
- Site-to-site variation often treated as within-subgroup
**14.4 Control Limit Estimation**
**Using Range Method:**
$$
\hat{\sigma} = \frac{\bar{R}}{d_2}
$$
**Using Sample Standard Deviation:**
$$
\hat{\sigma} = \frac{\bar{s}}{c_4}
$$
Where $c_4$ is the unbiasing constant for standard deviation.
**14.5 Short-Run SPC**
For limited data (new process, low volume):
**Z-MR charts using target:**
$$
Z_i = \frac{x_i - T}{\sigma_0}
$$
**Q-charts (self-starting):**
$$
Q_i = \Phi^{-1}\left(F_{i-1}\left(\frac{x_i - \bar{x}_{i-1}}{s_{i-1}\sqrt{1 + 1/(i-1)}}\right)\right)
$$
**15. Key Mathematical Relationships**
**Quick Reference Table**
| Concept | Core Mathematics |
|---------|------------------|
| **Control Limits** | $\mu \pm \frac{3\sigma}{\sqrt{n}}$ |
| **Cp** | $\frac{USL - LSL}{6\sigma}$ |
| **Cpk** | $\min\left[\frac{USL-\mu}{3\sigma}, \frac{\mu-LSL}{3\sigma}\right]$ |
| **EWMA** | $\lambda x_t + (1-\lambda)EWMA_{t-1}$ |
| **CUSUM** | $\max[0, x_t - (\mu_0 + K) + C_{t-1}]$ |
| **Hotelling's T²** | $n(\bar{\mathbf{x}}-\boldsymbol{\mu})'S^{-1}(\bar{\mathbf{x}}-\boldsymbol{\mu})$ |
| **Gauge R&R** | $\sigma^2_{total} = \sigma^2_{part} + \sigma^2_{gauge}$ |
| **Yield (Poisson)** | $Y = e^{-D_0 A}$ |
| **ARL₀ (3σ)** | $\frac{1}{0.0027} \approx 370$ |
| **AR(1) Variance** | $\frac{\sigma^2_\varepsilon}{1-\phi^2}$ |
**Decision Guide: Which Chart to Use?**
| Situation | Recommended Chart |
|-----------|------------------|
| Standard monitoring, subgroups | X̄-R or X̄-S |
| Individual measurements | I-MR |
| Detect small shifts ($< 1.5\sigma$) | EWMA or CUSUM |
| Multiple correlated parameters | Hotelling's T² or MEWMA |
| Autocorrelated data | Residual charts or modified EWMA |
| Short production runs | Q-charts or Z-MR |
**Critical Success Factors**
1. **Validate measurement system first** (Gauge R&R < 10%)
2. **Ensure rational subgrouping** captures meaningful variation
3. **Check for autocorrelation** before applying standard charts
4. **Use appropriate capability indices** (Cpk vs Ppk)
5. **Decompose variance** to target improvement efforts
6. **Match chart sensitivity** to required detection speed
**Control Chart Constant Tables**
**Constants for X̄ and R Charts**
| $n$ | $A_2$ | $A_3$ | $d_2$ | $d_3$ | $D_3$ | $D_4$ | $c_4$ | $B_3$ | $B_4$ |
|-----|-------|-------|-------|-------|-------|-------|-------|-------|-------|
| 2 | 1.880 | 2.659 | 1.128 | 0.853 | 0 | 3.267 | 0.7979 | 0 | 3.267 |
| 3 | 1.023 | 1.954 | 1.693 | 0.888 | 0 | 2.574 | 0.8862 | 0 | 2.568 |
| 4 | 0.729 | 1.628 | 2.059 | 0.880 | 0 | 2.282 | 0.9213 | 0 | 2.266 |
| 5 | 0.577 | 1.427 | 2.326 | 0.864 | 0 | 2.114 | 0.9400 | 0 | 2.089 |
| 6 | 0.483 | 1.287 | 2.534 | 0.848 | 0 | 2.004 | 0.9515 | 0.030 | 1.970 |
| 7 | 0.419 | 1.182 | 2.704 | 0.833 | 0.076 | 1.924 | 0.9594 | 0.118 | 1.882 |
| 8 | 0.373 | 1.099 | 2.847 | 0.820 | 0.136 | 1.864 | 0.9650 | 0.185 | 1.815 |
| 9 | 0.337 | 1.032 | 2.970 | 0.808 | 0.184 | 1.816 | 0.9693 | 0.239 | 1.761 |
| 10 | 0.308 | 0.975 | 3.078 | 0.797 | 0.223 | 1.777 | 0.9727 | 0.284 | 1.716 |
**Standard Normal Distribution Critical Values**
| Confidence | $z_{\alpha/2}$ |
|------------|----------------|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
| 99.73% | 3.000 |