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Semiconductor Etch Process Capability Mathematics

Keywords: 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<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}\right)^{1/\sum w_i}

Individual desirability functions:

Target is best:

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

Larger is better:

d = \left(\frac{y - L}{T - L}\right)^r \quad \text{for } L < y < T

Smaller is better:

d = \left(\frac{U - y}{U - T}\right)^r \quad \text{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_{\text{trench}}}{ER_{\text{open}}} = f(AR) = f\left(\frac{\text{depth}}{\text{width}}\right)

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_{\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	$A_2$	$D_4$
2	1.880	3.267
3	1.023	2.575
4	0.729	2.282
5	0.577	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 eq 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}                            │
└─────────────────────────────────────────────────────────────┘

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