Linearity in metrology is the consistency of measurement accuracy across the entire operating range of an instrument — verifying that a semiconductor metrology tool is equally accurate when measuring thin films as thick films, small features as large features, and low temperatures as high temperatures, not just at the calibration point.

What Is Linearity?

• Definition: The difference in bias (systematic error) values throughout the expected operating range of the measurement system — a perfectly linear gauge has the same bias at every measurement point.
• Problem: A gauge might be perfectly accurate at its calibration point but increasingly inaccurate at the extremes of its range — linearity studies detect this.
• Study: Part of the AIAG MSA analysis — measures reference parts spanning the full operating range and compares gauge readings to reference values.

Why Linearity Matters

• Range-Dependent Errors: An ellipsometer calibrated at 100nm film thickness might read accurately at 100nm but show 2% error at 10nm and 3% error at 500nm — linearity quantifies this behavior.
• Process Window Coverage: Semiconductor processes operate across a range of parameter values — measurements must be trustworthy across the entire range, not just at a single point.
• Specification Compliance: If bias changes across the range, parts at one end of the specification may be systematically accepted or rejected differently than parts at the other end.
• Calibration Strategy: Linearity results determine whether single-point or multi-point calibration is needed.

Linearity Study Method

• Step 1: Select 5+ reference parts (or standards) spanning the full operating range — from minimum to maximum expected measurement values.
• Step 2: Measure each reference part 10+ times to establish the gauge's average reading at each level.
• Step 3: Calculate bias at each level: Bias = Average measured value - Reference value.
• Step 4: Plot bias vs. reference value — a perfectly linear gauge shows a flat horizontal line (zero bias everywhere) or a consistent slope.
• Step 5: Perform regression analysis — the slope of the bias-vs.-reference line indicates non-linearity; the R² value indicates consistency.

Acceptance Criteria

Correcting Non-Linearity

• Multi-Point Calibration: Calibrate at multiple reference points across the range — the instrument applies correction factors.
• Lookup Table: Instrument firmware applies point-by-point corrections based on characterized non-linearity.
• Range Restriction: Limit the instrument's operating range to the region where linearity is acceptable.
• Replace/Upgrade: If non-linearity exceeds correction capability, upgrade to a more linear instrument.

Linearity is the assurance that semiconductor metrology tools are trustworthy across their entire operating range — not just at the single calibration point, but everywhere the measurement is needed to support process control and product quality decisions.