Poisson statistics

Semiconductor Manufacturing Process: Poisson Statistics & Mathematical Modeling

1. Introduction: Why Poisson Statistics?

Semiconductor defects satisfy the classical Poisson conditions:

• Rare events — Defects are sparse relative to the total chip area
• Independence — Defect occurrences are approximately independent
• Homogeneity — Within local regions, defect rates are constant
• No simultaneity — At infinitesimal scales, simultaneous defects have zero probability

1.1 The Poisson Probability Mass Function

The probability of observing exactly k defects:

P(X = k) = (lambda^k e^-lambda)/(k!)

where the expected number of defects is:

lambda = D_0 · A

Parameter definitions:

• D_0 — Defect density (defects per unit area, typically defects/cm²)
• A — Chip area (cm²)
• lambda — Mean number of defects per chip

1.2 Key Statistical Properties

> Note: The equality of mean and variance (equidispersion) is a signature property of the Poisson distribution. Real semiconductor data often shows overdispersion (variance > mean), motivating compound models.

2. Fundamental Yield Equation

2.1 The Seeds Model (Simple Poisson)

A chip is functional if and only if it has zero killer defects. Under Poisson assumptions:

boxed(Y = P(X = 0) = e^-D_0 A)

Derivation:

P(X = 0) = (lambda^0 e^-lambda)/(0!) = e^-lambda = e^-D_0 A

2.2 Limitations of Simple Poisson

• Assumes uniform defect density across the wafer (unrealistic)
• Does not account for clustering of defects
• Consistently underestimates yield for large chips
• Ignores wafer-to-wafer and lot-to-lot variation

3. Compound Poisson Models

3.1 The Negative Binomial Approach

Model the defect density D_0 as a random variable with Gamma distribution:

D_0 ~ Gamma(alpha, (alpha)/(bar(D)))

Gamma probability density function:

f(D_0) = ((alpha/bar(D))^alpha)/(Gamma(alpha)) D_0^alpha-1 e^(-alpha D_0/bar(D))

where:

• bar(D) — Mean defect density
• alpha — Clustering parameter (shape parameter)

3.2 Resulting Yield Model

When defect density is Gamma-distributed, the defect count follows a Negative Binomial distribution, yielding:

boxed(Y = (1 + (D_0 A)/(alpha))^-alpha)

3.3 Physical Interpretation of Clustering Parameter alpha

3.4 Overdispersion

The variance-to-mean ratio for the Negative Binomial:

(Var(X))/(E[X]) = 1 + (bar(D)A)/(alpha) > 1

This overdispersion (ratio > 1) matches empirical observations in semiconductor manufacturing.

4. Classical Yield Models

4.1 Comparison Table

4.2 Murphy's Yield Model

Assumes triangular distribution of defect densities:

Y_Murphy = ((1 - e^-D_0 A)/(D_0 A))^2

Taylor expansion for small D_0 A:

Y_Murphy ≈ 1 - ((D_0 A)^2)/(12) + O((D_0 A)^4)

4.3 Limiting Behavior

As D_0 A → 0 (low defect density):

lim_D_0 A → 0 Y = 1 (all models)

As D_0 A → ∞ (high defect density):

lim_D_0 A → ∞ Y = 0 (all models)

5. Critical Area Analysis

5.1 Definition

Not all chip area is equally vulnerable. Critical area A_c is the region where a defect of size d causes circuit failure.

A_c(d) = ∫_layout 1[defect at (x,y) with size d causes failure] dx dy

5.2 Critical Area for Shorts

For two parallel conductors with:

• Length: L
• Spacing: S

A_c^short(d) =
begin(cases)
2L(d - S) & if d > S
0 & if d ≤ S
end(cases)

5.3 Critical Area for Opens

For a conductor with:

• Width: W
• Length: L

A_c^open(d) =
begin(cases)
L(d - W) & if d > W
0 & if d ≤ W
end(cases)

5.4 Total Critical Area

Integrate over the defect size distribution f(d):

A_c = ∫_0^∞ A_c(d) · f(d) dd

5.5 Defect Size Distribution

Typically modeled as power-law:

f(d) = C · d^-p for d ≥ d_min

Typical values:

• Exponent: p ≈ 2-4
• Normalization constant: C = (p-1) · d_min^p-1

Alternative: Log-normal distribution (common for particle contamination):

f(d) = (1)/(d sigma sqrt(2 pi)) exp(-((ln d - mu)^2)/(2 sigma^2))

6. Multi-Layer Yield Modeling

6.1 Modern IC Structure

Modern integrated circuits have 10-15+ metal layers. Each layer i has:

• Defect density: D_i
• Critical area: A_c,i
• Clustering parameter: alpha_i (for Negative Binomial)

6.2 Poisson Multi-Layer Yield

Y_total = ∏_i=1^n Y_i = ∏_i=1^n e^-D_i A_c,i

Simplified form:

boxed(Y_total = exp(-∑_i=1^n D_i A_c,i))

6.3 Negative Binomial Multi-Layer Yield

boxed(Y_total = ∏_i=1^n (1 + (D_i A_c,i)/(alpha_i))^-alpha_i)

6.4 Log-Yield Decomposition

Taking logarithms for analysis:

ln Y_total = -∑_i=1^n D_i A_c,i (Poisson)

ln Y_total = -∑_i=1^n alpha_i ln(1 + (D_i A_c,i)/(alpha_i)) (Negative Binomial)

7. Spatial Point Process Formulation

7.1 Inhomogeneous Poisson Process

Intensity function lambda(x, y) varies spatially across the wafer:

P(k defects in region R) = (Lambda(R)^k e^-Lambda(R))/(k!)

where the integrated intensity is:

Lambda(R) = iint_R lambda(x,y) dx dy

7.2 Cox Process (Doubly Stochastic)

The intensity lambda(x,y) is itself a random field:

lambda(x,y) = exp(mu + Z(x,y))

where:

• mu — Baseline log-intensity
• Z(x,y) — Gaussian random field with spatial correlation function rho(h)

Correlation structure:

Cov(Z(x_1, y_1), Z(x_2, y_2)) = sigma^2 rho(h)

where h = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)

7.3 Neyman Type A (Cluster Process)

Models defects occurring in clusters:

1. Cluster centers: Poisson process with intensity lambda_c
2. Defects per cluster: Poisson with mean mu
3. Defect positions: Scattered around cluster center (e.g., isotropic Gaussian)

Probability generating function:

G(s) = exp[lambda_c A (e^mu(s-1) - 1)]

Mean and variance:

E[N] = lambda_c A mu

Var(N) = lambda_c A mu (1 + mu)

8. Statistical Estimation Methods

8.1 Maximum Likelihood Estimation

8.1.1 Data Structure

Given:

• n chips with areas A_1, A_2, ldots, A_n
• Binary outcomes y_i in (0, 1) (pass/fail)

8.1.2 Likelihood Function

L(D_0, alpha) = ∏_i=1^n Y_i^y_i (1 - Y_i)^1-y_i

where Y_i = (1 + (D_0 A_i)/(alpha))^-alpha

8.1.3 Log-Likelihood

ell(D_0, alpha) = ∑_i=1^n [y_i ln Y_i + (1-y_i) ln(1-Y_i)]

8.1.4 Score Equations

(∂ ell)/(∂ D_0) = 0, (∂ ell)/(∂ alpha) = 0

> Note: Requires numerical optimization (Newton-Raphson, BFGS, or EM algorithm).

8.2 Bayesian Estimation

8.2.1 Prior Distribution

D_0 ~ Gamma(a, b)

pi(D_0) = (b^a)/(Gamma(a)) D_0^a-1 e^-b D_0

8.2.2 Posterior Distribution

Given defect count k on area A:

D_0 mid k ~ Gamma(a + k, b + A)

Posterior mean:

hat(D)_0 = (a + k)/(b + A)

Posterior variance:

Var(D_0 mid k) = (a + k)/((b + A)^2)

8.2.3 Sequential Updating

Bayesian framework enables sequential learning:

Prior_n xrightarrow(data k_n) Posterior_n = Prior_n+1

9. Statistical Process Control

9.1 c-Chart (Defect Counts)

For constant inspection area:

• Center line: bar(c) (average defect count)
• Upper Control Limit (UCL): bar(c) + 3sqrt(bar(c))
• Lower Control Limit (LCL): max(0, bar(c) - 3sqrt(bar(c)))

9.2 u-Chart (Defects per Unit Area)

For variable inspection area n_i:

u_i = (c_i)/(n_i)

• Center line: bar(u)
• Control limits: bar(u) ± 3sqrt((bar(u))/(n_i))

9.3 Overdispersion-Adjusted Charts

For clustered defects (Negative Binomial), inflate the variance:

UCL = bar(c) + 3sqrt(bar(c)(1 + (bar(c))/(alpha)))

LCL = max(0, bar(c) - 3sqrt(bar(c)(1 + (bar(c))/(alpha))))

9.4 CUSUM Chart

Cumulative sum for detecting small persistent shifts:

C_t^+ = max(0, C_t-1^+ + (x_t - mu_0 - K))

C_t^- = max(0, C_t-1^- - (x_t - mu_0 + K))

where:

• K — Slack value (typically 0.5 sigma)
• Signal when C_t^+ or C_t^- exceeds threshold H

10. EUV Lithography Stochastic Effects

10.1 Photon Shot Noise

At extreme ultraviolet wavelength (13.5 nm), photon shot noise becomes critical.

Number of photons absorbed in resist volume V:

N ~ Poisson(Phi · sigma · V)

where:

• Phi — Photon fluence (photons/area)
• sigma — Absorption cross-section
• V — Resist volume

10.2 Line Edge Roughness (LER)

Stochastic photon absorption causes spatial variation in resist exposure:

sigma_LER ∝ (1)/(sqrt(Phi · V))

Critical Design Rule:

LER_3 sigma < 0.1 × CD

where CD = Critical Dimension (feature size)

10.3 Stochastic Printing Failures

Probability of insufficient photons in a critical volume:

P(failure) = P(N < N_threshold) = ∑_k=0^N_threshold-1 (lambda^k e^-lambda)/(k!)

where lambda = Phi sigma V

11. Reliability and Latent Defects

11.1 Defect Classification

Not all defects cause immediate failure:

• Killer defects: Cause immediate functional failure
• Latent defects: May cause reliability failures over time

lambda_total = lambda_killer + lambda_latent

11.2 Yield vs. Reliability

Initial Yield:

Y = e^-lambda_killer · A

Reliability Function:

R(t) = e^-lambda_latent · A · H(t)

where H(t) is the cumulative hazard function for latent defect activation.

11.3 Weibull Activation Model

H(t) = ((t)/(eta))^beta

Parameters:

• eta — Scale parameter (characteristic life)
• beta — Shape parameter
• beta < 1: Decreasing failure rate (infant mortality)
• beta = 1: Constant failure rate (exponential)
• beta > 1: Increasing failure rate (wear-out)

12. Complete Mathematical Framework

12.1 Hierarchical Model Structure

12.2 Summary of Key Equations

12.3 Typical Parameter Values