Capability Analysis

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:

$$ C_{pk} = \min\left(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma} ight) $$

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)

$$ C_{pkm} = \frac{C_{pk}}{\sqrt{1 + \left(\frac{\mu - T}{\sigma} ight)^2}} $$

1.6 Industry Targets for Semiconductor Etch

2. Etch-Specific Uniformity Mathematics

2.1 Within-Wafer Uniformity (WIW)

• Range-based method:

$$ \%U_{WIW} = \frac{X_{max} - X_{min}}{2 \cdot \bar{X}} imes 100\% $$

• Standard deviation-based method (preferred):

$$ \%U_{1\sigma} = \frac{s}{\bar{X}} imes 100\% $$

• Typical target: $<1\%$ $(1\sigma)$ uniformity for etch rate

2.2 Wafer-to-Wafer Uniformity (WtW)

$$ \%U_{WtW} = \frac{s_{ ext{wafer means}}}{\bar{X}_{ ext{overall}}} imes 100\% $$

2.3 Total Variance Decomposition

Via nested ANOVA:

$$ \sigma^2_{ ext{total}} = \sigma^2_{WIW} + \sigma^2_{WtW} + \sigma^2_{LtL} + \sigma^2_{TtT} $$

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

$$ CD_{ ext{uniformity}} = \frac{CD_{max} - CD_{min}}{CD_{target}} imes 100\% $$

3.2 Etch Bias

$$ ext{Etch Bias} = CD_{ ext{after etch}} - CD_{ ext{after litho}} $$

For anisotropic etch with undercut angle $ heta$:

$$ \Delta CD = 2 \cdot d \cdot an( heta) $$

Where:
- $d$ = etch depth
- $ heta$ = undercut angle
- For ideal anisotropic etch: $ heta = 0 \Rightarrow \Delta CD = 0$

3.3 Iso-Dense Bias (IDB)

$$ IDB = CD_{ ext{isolated}} - CD_{ ext{dense}} $$

Capability for IDB:

$$ C_{pk,IDB} = \min\left(\frac{IDB_{USL} - \overline{IDB}}{3s_{IDB}}, \frac{\overline{IDB} - IDB_{LSL}}{3s_{IDB}} ight) $$

3.4 Line Edge Roughness (LER) / Line Width Roughness (LWR)

• LER Definition:

$$ LER = 3\sigma_{ ext{edge position}} $$

• LWR Definition:

$$ LWR = 3\sigma_{ ext{line width}} $$

• One-sided capability (upper limit only):

$$ C_{pk,LER} = \frac{USL_{LER} - \overline{LER}}{3s_{LER}} $$

4. Selectivity Mathematics

4.1 Basic Selectivity Definition

$$ ext{Selectivity} = \frac{ER_{ ext{target material}}}{ER_{ ext{mask or stop layer}}} $$

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

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):

$$ \hat{\sigma}^2_{ ext{total}} = \hat{\sigma}^2_W + \hat{\sigma}^2_L + \hat{\sigma}^2_T + \hat{\sigma}^2_S + \hat{\sigma}^2_\epsilon $$

5.3 Percent Contribution

$$ \% ext{Contribution}_i = \frac{\hat{\sigma}^2_i}{\hat{\sigma}^2_{ ext{total}}} imes 100\% $$

6. Response Surface Modeling for Etch

6.1 Second-Order Polynomial Model

$$ ER = \beta_0 + \sum_{i}\beta_i x_i + \sum_{i}\beta_{ii}x_i^2 + \sum_{i<j}\beta_{ij}x_i x_j + \epsilon $$

Where $x_i$ represents process parameters:
- $P$ = RF Power
- $p$ = Chamber pressure
- $F$ = Gas flow rate
- $T$ = Temperature

6.2 Process Window Definition

$$ \mathcal{W} = \bigcap_{i=1}^{n} \{(P, p, F, T) : LSL_i \leq Y_i \leq USL_i\} $$

6.3 Desirability Function

Overall desirability:

$$ D = \left(\prod_{i=1}^{n} d_i^{w_i} ight)^{1/\sum w_i} $$

Individual desirability functions:

• Target is best:

$$ d = \exp\left(-\left|\frac{y-T}{s} ight|^r ight) $$

• Larger is better:

$$ d = \left(\frac{y - L}{T - L} ight)^r \quad ext{for } L < y < T $$

• Smaller is better:

$$ d = \left(\frac{U - y}{U - T} ight)^r \quad ext{for } T < y < U $$

7. Loading Effect Models

7.1 Macro-Loading

As exposed area $A$ increases, etch rate decreases:

$$ ER(A) = ER_0 \cdot \frac{1}{1 + kA} $$

7.2 Micro-Loading (ARDE)

Aspect Ratio Dependent Etching:

$$ \frac{ER_{ ext{trench}}}{ER_{ ext{open}}} = f(AR) = f\left(\frac{ ext{depth}}{ ext{width}} ight) $$

Knudsen diffusion model:

$$ ER \propto \frac{1}{1 + \alpha \cdot AR} $$

7.3 RIE Lag Correction

For high aspect ratio features $(AR > 20:1)$:

$$ ER_{ ext{corrected}} = ER_{ ext{open}} \cdot \exp\left(-\beta \cdot AR^{\gamma} ight) $$

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):

8.3 EWMA (Exponentially Weighted Moving Average)

Recursive formula:

$$ EWMA_t = \lambda x_t + (1-\lambda)EWMA_{t-1} $$

Control limits:

$$ UCL = \mu_0 + L\sigma\sqrt{\frac{\lambda}{2-\lambda}\left[1-(1-\lambda)^{2t} ight]} $$

$$ LCL = \mu_0 - L\sigma\sqrt{\frac{\lambda}{2-\lambda}\left[1-(1-\lambda)^{2t} ight]} $$

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

$$ d = \frac{N \lambda}{2n \cos heta} $$

Where:
- $N$ = Number of interference fringes counted
- $\lambda$ = Wavelength of light
- $n$ = Refractive index of material
- $ heta$ = Angle of incidence

9.2 Optical Emission Spectroscopy (OES)

Endpoint trigger condition:

$$ \left|\frac{dI(\lambda, t)}{dt} ight| > ext{threshold} $$

Normalized derivative:

$$ \frac{d}{dt}\left[\frac{I(\lambda, t)}{I_{ref}} ight] > ext{threshold} $$

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:

$$ \sigma^2_{ ext{observed}} = \sigma^2_{ ext{part}} + \sigma^2_{ ext{measurement}} $$

Measurement variance:

$$ \sigma^2_{ ext{measurement}} = \sigma^2_{ ext{repeatability}} + \sigma^2_{ ext{reproducibility}} $$

10.2 Percent GRR Calculations

To total variation:

$$ \%GRR_{ ext{TV}} = \frac{\sigma_{ ext{GRR}}}{\sigma_{ ext{total}}} imes 100\% $$

To tolerance:

$$ \%GRR_{ ext{Tol}} = \frac{6\sigma_{ ext{GRR}}}{USL - LSL} imes 100\% $$

10.3 GRR Assessment Criteria

10.4 Number of Distinct Categories (ndc)

$$ ndc = 1.41 \cdot \frac{\sigma_{ ext{part}}}{\sigma_{ ext{GRR}}} $$

Requirement: $ndc \geq 5$

11. Confidence Intervals for Capability

11.1 Confidence Interval for Cp

Chi-square based:

$$ P\left(\hat{C}_p \sqrt{\frac{\chi^2_{n-1, 1-\alpha/2}}{n-1}} \leq C_p \leq \hat{C}_p \sqrt{\frac{\chi^2_{n-1, \alpha/2}}{n-1}} ight) = 1-\alpha $$

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

$$ y^{(\lambda)} = \begin{cases} \dfrac{y^\lambda - 1}{\lambda} & ext{if } \lambda eq 0 \\[10pt] \ln(y) & ext{if } \lambda = 0 \end{cases} $$

Common transformations:
- $\lambda = 0.5$: Square root
- $\lambda = 0$: Natural log
- $\lambda = -1$: Inverse

12.2 Percentile-Based Capability

$$ C_p = \frac{USL - LSL}{X_{99.865\%} - X_{0.135\%}} $$

$$ C_{pk} = \min\left(\frac{USL - X_{50\%}}{X_{99.865\%} - X_{50\%}}, \frac{X_{50\%} - LSL}{X_{50\%} - X_{0.135\%}} ight) $$

12.3 Johnson Transformation System

Three distribution families:

• $S_B$ (bounded):

$$ z = \gamma + \delta \ln\left(\frac{x - \xi}{\lambda + \xi - x} ight) $$

• $S_L$ (lognormal):

$$ z = \gamma + \delta \ln(x - \xi) $$

• $S_U$ (unbounded):

$$ z = \gamma + \delta \sinh^{-1}\left(\frac{x - \xi}{\lambda} ight) $$

13. Multivariate Capability

13.1 Multivariate Capability Index (MCp)

$$ MC_p = \frac{ ext{Vol}( ext{specification region})}{ ext{Vol}( ext{process region})} $$

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

14.2 Data Collection

• Wafers: 25 wafers
• Sites per wafer: 49 sites
• Total measurements: $25 imes 49 = 1,225$

14.3 Results Summary

14.4 Cpk Calculations

CD Cpk:

$$ C_{pk,CD} = \min\left(\frac{48-44.8}{3 imes 0.9}, \frac{44.8-42}{3 imes 0.9} ight) = \min(1.19, 1.04) = 1.04 $$

Depth Cpk:

$$ C_{pk,Depth} = \min\left(\frac{210-199}{3 imes 2.5}, \frac{199-190}{3 imes 2.5} ight) = \min(1.47, 1.20) = 1.20 $$

LWR Cpk (one-sided):

$$ C_{pk,LWR} = \frac{4 - 3.2}{3 imes 0.4} = \frac{0.8}{1.2} = 0.67 $$

14.5 Variance Decomposition for CD

Conclusions:
- Chamber uniformity issue (WIW dominant)
- Consider improving CD-SEM recipe to reduce measurement variance

Key Mathematical Tools

Quick Reference: Essential Formulas