Experimental design
Process Window: From DOE Results to Parameter Limits
How to turn experimental evidence into parameter limits a control plan can use, with a copper-pillar plating example.
Quick answer
A product specification says where the output has to land. A process window says how far the inputs can move before the output stops landing there. The two are often written in the same document and treated as the same kind of thing. They are not.
A window is not derived from the specification by arithmetic. It is drawn from evidence about how the process responds to its inputs, and that evidence usually comes from a designed experiment.
This guide continues the fictional copper-pillar plating case from the experimental design guide. All values are teaching values.
Specification is the output, window is the input
| Product specification | Process window | |
|---|---|---|
| What it limits | A characteristic of the product: edge pillar height, 48–52 µm. | A setting of the process: shield displacement, current density, flow. |
| Who defines it | Product design, customer requirements, downstream assembly needs. | Process engineering, from experimental and verification evidence. |
| Where it is checked | On the product, after the step, by measurement. | On the tool or recipe, before and during the step, by setting and monitoring. |
| What a violation means | The product may not perform. | The product may leave specification. |
A realistic way the two get confused: the plating tool's rectifier has alarm limits at ±2% of the current setpoint. Those limits exist because the rectifier can hold setpoint to that precision, and an excursion beyond it indicates a hardware problem. A control plan then lists "current density: setpoint ±2%" as the process window. But nothing in that number says what happens to pillar height at +2%, or at +8%. The equipment tolerance describes what the tool can do; the process window describes what the product can tolerate. They may coincide. They are not the same thing, and the first does not stand in for the second.
Where the window comes from
Three sources of evidence are commonly cited for a window. They are not equally strong.
| Source | What it tells you | What it does not tell you |
|---|---|---|
| Equipment or chemistry supplier's recommended range | Where the supplier expects the tool or bath to operate correctly. | How your product responds across that range. Supplier ranges are written for the equipment, not for your pillar height. |
| Historical production data | Where the process has been running, and that the output was acceptable there. | Where the edge is. If production has always run at setpoint ±0.3 mm, the data say nothing about ±1 mm. |
| Designed experiment with confirmation | Where the output crosses the specification limit, and whether that point moves with other factors. | Behavior outside the tested range. |
Only the third source locates the boundary. The first two locate the current position. A window built on the first two is an assumption that production has not yet contradicted.
For the plating case, the experimental design guide produced the map: edge height versus shield displacement at two current densities, then a confirmation run at the proposed limits. Fictional confirmed results: at nominal current, edge height stays inside the window up to 2 mm displacement; at +10% current, up to 1 mm. Flow had no measurable effect.
The window is not a rectangle
Write those results as limits and the problem appears:
- Shield displacement ≤ 2 mm if current density is nominal.
- Shield displacement ≤ 1 mm if current density is +10%.
The shield limit depends on the current setting. In two dimensions the acceptable region is a wedge, not a box. That is the interaction from the DOE, now showing up as a control problem.
| Approach | Window written as | What it gives up | When it fits |
|---|---|---|---|
| Conservative rectangle | Shield ≤ 1 mm at any current density. | Half the usable shield range at nominal current. | Current density varies in production and the team wants one limit everyone can remember. |
| Conditional window | Shield ≤ 1.5 mm up to +5% current; ≤ 1 mm above. | Simplicity. Two limits, and the control plan has to say which applies. | Current density is a deliberate setting with a few fixed values. |
| Fix one factor | Current locked at nominal; shield ≤ 2 mm. | Flexibility to raise current. | Nothing requires current above nominal. Most common in practice. |
None of these is wrong. The mistake is writing a single rectangle without noticing the wedge, which usually means writing "shield ≤ 2 mm" and discovering the +10% case in production.
Leave margin, and know who eats it
The confirmed boundary is 1 mm at +10% current. That is not the control limit. Between the boundary and the limit written in the control plan sits a margin, and the margin is consumed by things the experiment did not include.
| Margin consumer | Fictional estimate | Where the estimate comes from |
|---|---|---|
| Measurement of shield position | ±0.2 mm | Gauge R&R on the position check, see Gage R&R. |
| Setting repeatability after maintenance | ±0.2 mm | Ten re-seats of the shield by two technicians, position measured each time. |
| Drift between maintenance events | 0.1 mm | Position measured at the start and end of three maintenance intervals. |
| Boundary uncertainty | 0.1 mm | Assumed for this example, not derived. The confirmation run used two wafers per point, so the confirmed 1 mm is itself uncertain, and this allowance covers that. The 1.7 mm crossing is a straight-line interpolation, not a result, and is not counted as margin. |
Sum: about 0.6 mm. Control limit: the confirmed 1 mm minus 0.6 mm, call it ±0.4 mm from setpoint, rounded to ±0.5 mm only if the team accepts the small loss of margin and documents why.
Two things to notice. First, every line in the table has a source. A margin with no source is a guess dressed as a number. Second, the entries were added, which is the conservative choice; they could be combined as root-sum-of-squares if the team can argue they are independent, and that gives a wider limit. Which rule to use is a decision to record, not a formula to look up.
If the margin consumes the whole boundary (the sum exceeds 1 mm), the window cannot be controlled as written. The options are to reduce one of the consumers (a better position gauge, a fixture that improves re-seat repeatability) or to fix current at nominal and use the 2 mm boundary instead.
In the window is not the same as capable
A window answers one question: will the output leave specification? It does not answer how well the output is distributed inside the specification.
Running every parameter inside its window guarantees, to the extent the evidence holds, that edge height stays between 48 and 52 µm. It does not guarantee a Cpk of 1.33. If shield position sits at +1.5 mm every day at nominal current, edge height sits near 49 µm every day, in specification and off-center. That position is inside the window but far outside the ±0.4 mm control limit worked out above. The distance between the two is the gap between being in the window and being capable. Capability is a separate assessment with its own data requirements; see Cpk vs Ppk.
The reverse also holds. Six months of production data can show three separate things where the process has been running: stability, on a control chart over that period; centering, from the mean compared with the target; and capability, from a high Cpk. It does not show where the window edge is, because production never went there. A process can be capable and still have an unknown window, which is exactly the state the plating line was in before the root cause case.
When the window must be re-verified
A window is evidence about a particular combination of tool, chemistry, product, and recipe. When one of those changes, the evidence may no longer apply. The following situations call for formal re-verification before production resumes under the old limits.
| Change | Why the old window may not hold | What to re-verify |
|---|---|---|
| Transfer to another plating tool | Shield geometry, anode spacing, and flow pattern differ by tool. The DOE was run on tool B only. | Confirmation run at the window limits on the new tool, at each current density in use. |
| Change of bath chemistry supplier or additive package | Throwing power and edge uniformity depend on the additive system. The interaction with current density may change. | Repeat the DOE at the factor levels in use; a confirmation run alone does not show whether the interaction moved. |
| Recipe change to current density or plating time | The window was mapped for nominal and +10%. A new setpoint outside that range is untested. | Extend the DOE to cover the new setpoint, then confirm. |
| Product change to photoresist opening size or plated area | Local current density depends on pattern area. The same shield position may produce a different edge effect. | Confirmation run on the new product at the window limits. |
| Shield replacement or re-machining | A new part may not match the geometry the DOE characterized. | Position check against the original reference, then confirmation run. |
| Major tool overhaul affecting anode, contact ring, or flow | Any change to current distribution invalidates the shield-position evidence. | Treat as a new tool: confirmation run at limits. |
Each row names what to re-verify, not "repeat everything." A transfer to another tool needs a confirmation run; a change of additive system needs the DOE, because the interaction is what might have moved. The PFMEA for the step should list these as change triggers, and the control plan should reference the window's evidence document so that a reviewer can see what the limits rest on.
Common mistakes
| Mistake | What it looks like | Better practice |
|---|---|---|
| Equipment tolerance recorded as process window | "Current: setpoint ±2%" taken from the rectifier alarm. | State where each limit comes from. An equipment limit is a monitoring threshold, not evidence about the product. |
| Supplier range recorded as verified window | Bath temperature 23–27 °C copied from the chemistry datasheet. | Use the supplier range as a starting point for the DOE, not as its result. |
| Rectangular window over an interaction | "Shield ≤ 2 mm" with no condition on current density. | Check the DOE for non-parallel lines before writing one limit. |
| Zero margin | Control limit set at the confirmed boundary. | Allocate margin with a source for each consumer. |
| Window defined but not in the control plan | The DOE report exists; the control plan still says "per recipe." | The control plan carries the limits and references the evidence. |
| Old window carried to a new product | Shield limit kept after a change in pattern area. | Re-verify per the change table. |
| "It has never failed" as window evidence | Six months in specification cited as proof of the limits. | That shows only that the output stayed in specification where the process runs; stability needs a control chart. It does not show where the edge is. |
Before you write it into the control plan
- Each parameter limit names its evidence source.
- Interactions found in the DOE are handled by one of the three approaches, and the choice is written down.
- Margin is allocated with a source for each line, and the combination rule is stated.
- The change triggers that require re-verification are listed, with what each one requires.
- The monitoring method matches the limit: a ±0.5 mm shield limit needs a position check that can resolve 0.2 mm, at a frequency that catches drift between maintenance events.
A window is a claim that the process will stay in specification as long as its inputs stay inside these limits. Write the limits so that someone can check the claim, and write down what would make it expire.
Sources and further reading
- NIST/SEMATECH Engineering Statistics Handbook — Process Improvement. Experimental design, response characterization, and the reasoning behind confirmation runs. (itl.nist.gov)
- AIAG — Control Plan reference material. The control plan is the document that carries process parameter limits and their monitoring methods into production. This guide does not reproduce its format. (aiag.org)