Weibull distribution

Notes

A beta below one signals infant mortality and above one signals wear-out, making the slope on a Weibull plot a diagnostic in itself.

Overview

The Weibull distribution is a flexible statistical model of time to failure, widely used in reliability engineering to describe how a population of components wears out or fails over time. Its two parameters, shape and scale, let it fit failure rates that fall, stay constant or rise with age. In semiconductors it is the usual model for dielectric breakdown and many other wear-out mechanisms, and for fitting accelerated stress test data.

How it works

The scale parameter, often written eta or alpha, is the characteristic life: whatever the shape, 63.2 per cent of the population has failed by that time. The shape parameter, often written beta (NIST uses gamma), sets how the failure rate changes. A shape below 1 gives a falling failure rate, typical of early failures; a shape of exactly 1 reduces the Weibull to the exponential distribution with a constant failure rate; a shape above 1 gives a rising rate, the signature of wear-out. A three-parameter form adds a shift for a failure-free period at the start. Notation varies between textbooks and software, so parameter names need checking.

Data are commonly analysed on a Weibull probability plot, where failures that follow the distribution fall on a straight line; the line's gradient gives the shape and the scale is read off where it crosses 63.2 per cent. Maximum likelihood methods do the same numerically and handle units that have not yet failed.

The model has a physical basis as a weakest-link distribution: a part fails when the first of many competing flaws gives way. Gate oxide breakdown fits this picture, because breakdown happens when a random percolation path of defects first links gate and channel. Fitted parameters are then combined with voltage and temperature acceleration models to extrapolate from stress conditions to use conditions.

Examples

• Time-dependent dielectric breakdown data from capacitor test structures plotted on Weibull paper to extract the slope.
• A shape well above 1 fitted to high-temperature stress failures, confirming a wear-out mechanism rather than random defects.
• Fitting early field failures with a shape below 1 to confirm an infant-mortality problem that burn-in should screen.
• Comparing the characteristic life of two interconnect processes after the same accelerated stress.

Why it matters

Reliability and qualification engineers use Weibull fits to turn a few hundred hours of accelerated testing into lifetime predictions, to compare process changes, and to decide whether a failure mode is infant mortality or wear-out. Because projections are made at very low failure percentiles, the choice of model and the fitted shape strongly affect the answer.

Related terms

• Bathtub curve
• TDDB
• MTTF
• Useful life

Where it appears on the Hub

Used on 6 pages of the Hub. In each group, the pages that mention it most come first.