linearity
**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**
| Metric | Acceptable | Concern |
|--------|-----------|---------|
| Linearity (slope) | Close to 0 | Significantly non-zero |
| Bias at all points | Within specification | Exceeds tolerance at extremes |
| R² of regression | >0.7 (strong relationship) | Indicates systematic non-linearity |
**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.