Weibull distribution

Weibull Distribution Mathematics in Semiconductor Manufacturing

A comprehensive guide to the mathematical foundations and applications of Weibull distribution in semiconductor reliability engineering.

1. Fundamental Weibull Mathematics

1.1 The Core Equations

Two-parameter Weibull Probability Density Function (PDF):

f(t) = (beta)/(eta) ((t)/(eta))^beta-1 exp[-((t)/(eta))^beta]

Cumulative Distribution Function (CDF) — probability of failure by time t:

F(t) = 1 - exp[-((t)/(eta))^beta]

Reliability (Survival) Function:

R(t) = exp[-((t)/(eta))^beta]

Parameter Definitions:

• t ≥ 0 — random variable (typically time or stress cycles)
• beta > 0 — shape parameter (Weibull slope/modulus)
• eta > 0 — scale parameter (characteristic life, where F(eta) = 0.632)

1.2 Three-Parameter Weibull

Adding a location parameter gamma (threshold/minimum life):

F(t) = 1 - exp[-((t-gamma)/(eta))^beta], t ≥ gamma

1.3 The Hazard Function (Instantaneous Failure Rate)

h(t) = (f(t))/(R(t)) = (beta)/(eta) ((t)/(eta))^beta-1

Physical Interpretation of Shape Parameter beta:

This directly models the semiconductor bathtub curve.

2. Semiconductor-Specific Applications

2.1 Time-Dependent Dielectric Breakdown (TDDB)

Gate oxide breakdown follows Weibull statistics. The area scaling law derives from weakest-link theory:

eta_2 = eta_1 ((A_1)/(A_2))^1/beta

Where:

• A_1 — reference test area
• A_2 — target device area
• eta_1 — characteristic life at area A_1
• eta_2 — predicted characteristic life at area A_2

Typical beta values for oxide breakdown:

• Intrinsic breakdown: beta ≈ 10–30 (tight distribution)
• Extrinsic/defect-related: beta ≈ 1–5 (broader distribution)

2.2 Electromigration

Metal interconnect failure combines Black's equation with Weibull statistics:

MTF = A · j^-n · exp((E_a)/(k_B T))

Where:

• MTF — median time to failure
• j — current density (A/cm^2)
• n — current density exponent (typically 1–2)
• E_a — activation energy (eV)
• k_B — Boltzmann constant (8.617 × 10^-5 eV/K)
• T — absolute temperature (K)

Typical beta values: 2–4 (wear-out behavior)

2.3 Hot Carrier Injection (HCI)

Degradation follows power-law kinetics:

Delta V_th = A · t^n

Where:

• Delta V_th — threshold voltage shift
• t — stress time
• n — time exponent (typically 0.3–0.5)

2.4 Negative Bias Temperature Instability (NBTI)

For PMOS transistors:

Delta V_th = A · t^n · exp(-(E_a)/(k_B T))

3. Statistical Analysis Methods

3.1 Weibull Probability Plotting

Linearization transformation — take double logarithm of CDF:

ln[-ln(1-F(t))] = beta ln(t) - beta ln(eta)

Plotting ln[-ln(1-F)] vs ln(t):

• Slope = beta
• Intercept at F = 0.632 gives t = eta

Bernard's Median Rank Approximation for ranking data:

hat(F)(t_(r)) ≈ (r - 0.3)/(n + 0.4)

Where:

• r — rank of the r-th ordered failure
• n — total sample size

3.2 Maximum Likelihood Estimation (MLE)

Log-likelihood function for n samples with r failures and (n-r) censored units:

L(beta, eta) = ∑_i=1^r [ln beta - beta ln eta + (beta-1)ln t_i - ((t_i)/(eta))^beta] - ∑_j=1^n-r((t_j)/(eta))^beta

MLE Estimator for eta:

hat(eta) = [(1)/(r)∑_i=1^n t_i^(hat(beta))]^(1/hat(beta))

MLE Equation for beta (solve numerically):

(1)/(hat(beta)) + (∑_i=1^n t_i^(hat(beta)) ln t_i)/(∑_i=1^n t_i^(hat(beta))) - (1)/(r)∑_i=1^r ln t_i = 0

4. Accelerated Life Testing Mathematics

4.1 Acceleration Factors

Arrhenius Model (Thermal Acceleration):

AF = exp[(E_a)/(k_B)((1)/(T_use) - (1)/(T_stress))]

Exponential Voltage Acceleration:

AF = exp[gamma(V_stress - V_use)]

Power-Law Voltage Acceleration:

AF = ((V_stress)/(V_use))^n

Life Extrapolation:

eta_use = AF × eta_stress

4.2 Combined Stress Models (Eyring)

AF = A · exp((E_a)/(k_B T)) · V^n · (RH)^m

Where:

• RH — relative humidity
• m — humidity exponent
• Additional stress factors can be included

5. Competing Failure Modes

5.1 Series (Competing Risks) Model

Device fails when the first mechanism fails:

R(t) = ∏_i=1^k exp[-((t)/(eta_i))^beta_i] = exp[-∑_i=1^k((t)/(eta_i))^beta_i]

Combined CDF:

F(t) = 1 - exp[-∑_i=1^k((t)/(eta_i))^beta_i]

5.2 Mixture Model

Different subpopulations with different failure characteristics:

F(t) = ∑_i=1^k p_i · F_i(t)

Where:

• p_i — proportion in subpopulation i
• ∑_i=1^k p_i = 1
• F_i(t) — CDF for subpopulation i

PDF for mixture:

f(t) = ∑_i=1^k p_i · f_i(t)

6. Key Derived Quantities

6.1 Moments of the Weibull Distribution

k-th Raw Moment:

E[T^k] = eta^k · Gamma(1 + (k)/(beta))

Mean (MTTF — Mean Time To Failure):

mu = eta · Gamma(1 + (1)/(beta))

Variance:

sigma^2 = eta^2 [Gamma(1 + (2)/(beta)) - Gamma^2(1 + (1)/(beta))]

Standard Deviation:

sigma = eta sqrt(Gamma(1 + (2)/(beta)) - Gamma^2(1 + (1)/(beta)))

6.2 Percentile Lives (BX Life)

Time by which X% have failed:

t_X = eta · [ln((1)/(1-X/100))]^1/beta

Common Percentile Lives:

6.3 Characteristic Life Significance

At t = eta:

F(eta) = 1 - exp(-1) = 1 - 0.368 = 0.632

This means 63.2% of units have failed by the characteristic life, regardless of beta.

7. Confidence Bounds

7.1 Fisher Information Matrix Approach

Information Matrix:

I(beta, eta) = -E[(∂^2 L)/(∂ theta_i ∂ theta_j)]

Asymptotic Variance-Covariance Matrix:

Var(hat(theta)) ≈ I^-1(hat(theta))

Fisher Matrix Elements:

I_beta beta = (r)/(beta^2)[1 + (pi^2)/(6)]

I_eta eta = (r beta^2)/(eta^2)

I_beta eta = (r)/(eta)(1 - gamma_E)

Where gamma_E ≈ 0.5772 is the Euler-Mascheroni constant.

7.2 Likelihood Ratio Bounds (Preferred for Small Samples)

-2[L(theta_0) - L(hat(theta))] ≤ chi^2_alpha, df

Approximate (1-alpha) Confidence Interval:

(theta : -2[L(theta) - L(hat(theta))] ≤ chi^2_alpha, p)

8. Order Statistics

8.1 Expected Value of Order Statistics

For n samples, the expected value of the r-th order statistic:

E[t_(r)] = eta · Gamma(1 + (1)/(beta)) · ∑_j=0^r-1 ((-1)^j binom(r-1)(j))/((n-r+1+j)^1+1/beta)

8.2 Plotting Positions

Bernard's Approximation (recommended):

hat(F)_i = (i - 0.3)/(n + 0.4)

Hazen's Approximation:

hat(F)_i = (i - 0.5)/(n)

Mean Rank:

hat(F)_i = (i)/(n + 1)

9. Practical Example: Gate Oxide Qualification

9.1 Test Setup

• Sample size: 50 oxide capacitors
• Stress conditions: 125°C, 1.2× nominal voltage
• Test duration: 1000 hours
• Failures: 8 units at times: 156, 289, 412, 523, 678, 734, 891, 967 hours
• Censored: 42 units still running at 1000h

9.2 Analysis Steps

Step 1: Calculate Median Ranks

Step 2: MLE Results

hat(beta) ≈ 2.1, hat(eta) ≈ 1850 hours (at stress)

Step 3: Calculate Acceleration Factor

Given: E_a = 0.7 eV, voltage exponent n = 40

AF_thermal = exp[(0.7)/(8.617 × 10^-5)((1)/(298) - (1)/(398))] ≈ 85

AF_voltage = (1.2)^40 ≈ 1.8

AF_total ≈ 85 × 1.8 ≈ 150

Step 4: Extrapolate to Use Conditions

eta_use = 1850 × 150 = 277,500 hours

Step 5: Calculate B0.1 Life

t_0.1 = 277,500 × (0.001001)^1/2.1 ≈ 3,200 hours

10. Key Equations

10.1 Quick Reference Table

10.2 Why Weibull Works for Semiconductors

1. Physical meaning of beta — directly indicates failure mechanism type
2. Area/volume scaling — derives from extreme value theory (weakest-link)
3. Censored data handling — essential since most test units don't fail
4. Acceleration compatibility — seamlessly integrates with physics-based models
5. Competing risks framework — models complex multi-mechanism devices

Gamma Function Values

Common values of Gamma(1 + 1/beta) for mean life calculations:

Common Activation Energies