inc. dual response surface methods (rsm) to make ......rsm: when to apply it (flowchart)when to...

Dual Response Surface Methods (RSM) to Make Processes More Robust*

*Posted at www.statease.com/webinar.html

Presented by Mark J. Anderson(Email: [email protected] )

July 2008 Webinar 1Timer by Hank Anderson

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AgendaAgenda

k d S f i i iBackground on RSM for optimizing processesDetail on propagation of error (POE) for finding

the flats & How I applied it for driving to workThe dual response approach – mean & std devCase study on dual response with POESummary & conclusionsySuggested reading & how to get help applying

these powerful statistical methods for yourself p y

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Response Surface Methods (RSM)When to Apply ItWhen to Apply It

1. Fractional factorials for screening ac o a ac o a s o sc ee g2. High-resolution fractional or full factorial to

understand interactions (add center points at this stage to test for curvature)

3. Response surface methods (RSM) to optimize.

Contour maps (2D) and 3D surfaces guide you to the peakguide you to the peak.

“Prediction is very hard, especially when it’s about the “Prediction is very hard, especially when it’s about the future.”future.”Yogi Berra

July 2008 Webinar

- Yogi Berra

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RSM:When to Apply It (Flowchart)When to Apply It (Flowchart)

Time is short, so Time is short, so hang on for a hang on for a quick overview. quick overview.


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RSM: Role of Center PointsRole of Center Points

If average response at the center points significantly differs from that of the outer points add a block of axial runs tofrom that of the outer points, add a block of axial runs to model curvature. This is a central composite design (CCD).


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RSM: When to Apply It (Topography)When to Apply It (Topography)

Region of InterestUse factorial design to t l t th kget close to the peak.

Then RSM to climb it.

Region of OperabilityRegion of Operability


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RSM: Model FittingModel Fitting

Subject Matter KnowledgeSubject Matter Knowledge(Plus Factorial Screening)(Plus Factorial Screening)

Vital Few Factors Vital Few Factors ((x’sx’s))

d ( )d ( ) ( (( ( ))))ProcessProcess Measured Response(s) Measured Response(s) (y(s(y(s))))

FittingFittingUncontrolled Factors (Uncontrolled Factors (z’sz’s))

Polynomial ModelPolynomial Model

FittingFittingUncontrolled Factors (Uncontrolled Factors (z sz s))

Polynomial ModelPolynomial Model

Response SurfaceResponse Surface


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RSM: Surfaces from PolynomialsSurfaces from Polynomials

Q d i b l id Simple Cubic Sheet(Contains 3rd order terms)

Quadratic Parabaloidy = β0 + β1 x12 + β2 x22

Quadratic generally suffices, provided you Quadratic generally suffices, provided you get a good focus on the optimal region.get a good focus on the optimal region.


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Propagation of Error (POE)Propagation of Error (POE)

Via the tools of calculus, POE measures the variation transmitted from input factors to the response as a function of the shape of the surface.

It facilitates finding the flats stable spots to locate It facilitates finding the flats – stable spots to locate your process, for example a high plateau of wafer yield.


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Propagation of ErrorMathematical Detail

Greek for the stat geeks (just kidding – good stuff):

Mathematical Detail

22

This calculation of variance (sigma-squared) transmitted to the predicted

∂ ∂σ = σ + σ + σ = σ ∂ ∂ i j

222 2 2 2 2

ˆ ˆx z residy yi ji j

f f POEx z

This calculation of variance (sigma squared) transmitted to the predicted response (y-hat) requires knowing the model (transfer function – f(x)), typically a 2nd order polynomial from RSM. It stems from the lack of control of the input factors (x) and known noise factors (z),* thus one

t k ( ti t b “ ”) th t d d d i ti ( i )must know (or estimate by a “swag”) the standard deviations (sigma). The normal process variation (sigma-squared residual) is included. For convenience sake, POE is expressed as standard deviation in original units of measure – hence the square root. q

*Consider gaining control long enough to model the impact of “z” factors!

Goal: Minimize the propagated error (POE)Goal: Minimize the propagated error (POE)Goal: Minimize the propagated error (POE)Goal: Minimize the propagated error (POE)


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Propagation of ErrorExample Calculation (p 1/2)Example Calculation (p. 1/2)

Find regions where variation in the control factors transmits the least i i h I hi i b di d d lvariation to the response. In this case it can be predicted adequately

by a quadratic function.

= β + β + β 20 1 1 11 1

2

ŷ x x

ˆ 15 25 0 7= + − 21 1y 15 25x 0.7x

Goal: Minimize the slope computed by the 1Goal: Minimize the slope computed by the 1stst derivative of the function.derivative of the function.July 2008 Webinar 11

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Propagation of ErrorExample Calculation (p 2/2)

Assume σx = 1 and σresid = 0

Example Calculation (p. 2/2)

= β + β + β

= + −

20 1 1 11 1

21 1

ŷ x x

ŷ 15 25x 0.7x+

∂ σ = σ + σ ∂

1 1

22 2 2ˆ x residy

y 15 25x 0.7x

yx

( )

∂

σ = − σ + σ2 2 2ˆ 1 x residy

x

25 1.4x

As the slope of the relationship between x and y decreases, the variation transmitted to y also decreasesthe variation transmitted to y also decreases.


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POE in Real Life: Driving to WorkDriving to Work

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POE of drive time by city 45

e tim

e

slicker gets magnified by the accelerating rush of traffic at T2.

35y: D

rive

0 10 20 30 40 50

25 T1 T2

x: Departure

PS Thi f bi d l 32 1 1 7 0 1 2 0 00016 3

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PS. This wavy curve comes from a cubic model: y = 32.1 + 1.7x – 0.1x2 + 0.00016x3

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Driving to Work:Plotting the POEPlotting the POE

POE minimized at flats: one high the other low

15one high, the other low on response plot.

When should commuter10

ive

time)

When should commuter depart?

5PO

EPO

E(Dr

i

0 10 20 30 40 50

0

0 10 20 30 40 50

x: Departure

PS This plot was computer generated using ANOVA standard deviation as the floorDrive time

July 2008 Webinar

PS. This plot was computer-generated using ANOVA standard deviation as the floor value (see screen shot), but the POE equation could be easily calculated as detailed in referenced textbook RSM Simplified, p198.

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Dual Response ApproachAnalyze Mean and Standard DeviationAnalyze Mean and Standard Deviation

The dual response approach suggests collecting repeated samples for each run The average is entered as one response and thefor each run. The average is entered as one response, and the standard deviation is entered as a second response.

GoalGoal: – Use the average response to find settings that make the

target producth f h– Use the standard deviation response to find settings that are

robust to uncontrolled factors.

This is the ONLY method that can find settings robust to This is the ONLY method that can find settings robust to unidentified sources of noise.unidentified sources of noise.


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Dual Response ApproachHow Many Samples?How Many Samples?

The variability within the sample collection should represent theThe variability within the sample collection should represent the long-term variability of the process. The following case study by Montgomery, et al, works with only about a dozen batches, but as f l l b d d dfew as 3 samples per experimental run may be needed to provide adequate precision. However, no matter what the sample size (n), if the study conditions are not representative of true manufacturing conditions, this method may underestimate the overall variation.

Fewer samples can be used if taken over a time period that encompasses long-term variation.


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Case Study to Illustrate Dual RSM with POE:Single Wafer Etching *Single-Wafer Etching *

Experimenters desired a more robust result for resistivity (the response output “y”) as a function of three key factors (the input “x”s) known to affect their single-wafer etching process:

A. Gas flow rateB. TemperatureC. Pressure

Other variables for example radio frequency (RF) power cannot beOther variables, for example radio frequency (RF) power, cannot be controlled very well. To measure the resulting variation over time, batches of wafers were collected over 11 different days from each of 17 runs in a central composite design (CCD).p g ( )

The process engineers hoped to hit a target resistivity of 350 ohm-cm with minimal variation.

July 2008 Webinar

* Data from Robinson, Wulff and Montgomery, “Robust Parameter Design Using Generalized Linear Mixed Models,” Journal of Quality Technology, Vol. 38, No. 1, Jan. 2006, p. 70, Table 1.

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Resistivity Results(F l l d d)(Factor levels coded)

IdA:

Gas FlowB:

TempC:

Pressure Mean Std dev1 1 1 1 263 99 107 421 -1 -1 -1 263.99 107.422 1 -1 -1 389.94 96.123 -1 1 -1 205.84 66.924 1 1 -1 292.53 110.065 1 1 1 290 10 141 335 -1 -1 1 290.10 141.336 1 -1 1 302.32 147.247 -1 1 1 164.29 79.958 1 1 1 160.37 82.639 -1.68 0 0 211.04 57.15

10 1.68 0 0 272.08 53.4211 0 -1.68 0 293.78 68.9312 0 1.68 0 147.13 39.4013 0 0 -1.68 418.55 221.9614 0 0 1.68 273.06 193.8915 0 0 0 268.38 64.2916 0 0 0 236.46 81.86

July 2008 Webinar

17 0 0 0 250.02 73.980 0 0 315.56 99.11

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Predictive Models for Resistivity(aka transfer functions)(aka transfer functions)

Mean = 255 71 + 23 69 A − 49 06 B − 35 14 C − 25 54 AC − 16 57 B2 + 27 75 C2 Mean = 255.71 + 23.69 A 49.06 B 35.14 C 25.54 AC 16.57 B + 27.75 C(p

Contour Plot of Resistivity Mean(Temp vs Pressure with Gas Flow at +1)*( p )

*(This factor A most linear set high to achieve target of y 350)

July 2008 Webinar

*(This factor, A, most linear – set high to achieve target of y = 350)PS. Best to stay within the ‘box’ of factorial settings in CCD – not axials.

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Surfaces of Resistivity Mean (left) and POE* (right) (Temp vs Pressure with Gas Flow at +1)( p )

Assumes these std Assumes these std devsdevs of factors (in coded scale):of factors (in coded scale):Gas flow (A) 0.2, Temp (B) 0.1, Pressure (C) 0.3.Gas flow (A) 0.2, Temp (B) 0.1, Pressure (C) 0.3.

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Wafer

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Response Surface of Resistivity Standard Deviation(Temp vs Pressure with Gas Flow at +1)( p )

Note: Generally standard deviation will be log linear, so then POE is moot. However, even when it does exhibit second-order behavior like this, it adds little

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or nothing to make use of the POE if it is minimized at minimal standard deviation, and nonsensical to trade it off for consistently greater variation.

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Most Desirable* Process Settingsg

* For details, see Derringer’s “A Balancing Act: Optimizing a Product's Properties,” Quality Progress,

Criterion: Target mean at 350 at minimum POE Criterion: Target mean at 350 at minimum POE with std dev (dual response) minimizedwith std dev (dual response) minimized

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, g g p g p , Q y g ,posted at www.statease.com/pubs/derringer.pdf © 2002 American Society for Quality (ASQ).

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3D View of Desirable Combinations of Temperature vs Pressure (gas flow at +1 level )p (g )

PS. Authors of this case study recommend coordinate (1.18, -0.80, -0.57) vs our i l i ( 0 ) b d l id h CC ’ b

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optimal point at (1,-1,-0.5), but we do not extrapolate outside the CCD’s box.

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Summary and ConclusionsSummary and Conclusions

Response surface methods (RSM) provide statistically-validated predictive p ( ) p y pmodels, sometimes referred to as “transfer functions,” that can then be manipulated for finding optimal process configurations.

Variation transmitted to responses from poorly-controlled process factors can b d f b h h l h f fbe accounted for by the mathematical technique of propagation of error (POE), which facilitates ‘finding the flats’ on the surfaces generated by RSM.

The dual response approach to RSM captures the standard deviation of the output(s) as well as the average It accounts for UNKNOWN sources ofoutput(s) as well as the average. It accounts for UNKNOWN sources of variation.

Dual response plus POE provides a more useful model of overall response variation.

The end-result of applying these statistical tools for design and analysis of experiments will be inin--specification products that exhibit minimal variabilityspecification products that exhibit minimal variability –the ultimate objective of robust design.


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Further Reading for More Detail on Methodologyfor More Detail on Methodology

1. Mark J. Anderson and Patrick J. Whitcomb, RSM Simplified – Optimizing Process Using Response Surface Methods for Design of Experiments Productivity PressUsing Response Surface Methods for Design of Experiments, Productivity Press, NY, NY, 2005.

2. Mark J. Anderson and Patrick J. Whitcomb, “Robust Design – Reducing Transmitted Variation,” Proceedings from the 50th Annual Quality Congress, 1996, pages 642-651. Milwaukee: American Society of Quality. (Write-up of talk presented by MJA at the 13th SEMATECH Statistical Methods Symposium in San Antonio, TX, on April 24, 1996.)

3 Wayne A Taylor “Comparing Three Approaches To Robust Design: Taguchi3. Wayne A. Taylor, Comparing Three Approaches To Robust Design: Taguchi Versus Dual Response Versus Tolerance Analysis,” presented at 1996 Fall Technical Conference (FTC) of the American Society of Quality (ASQ) and American Statistical Association (ASA).

4. Geoff G. Vining and Raymond H. Myers, “Combining Taguchi and Response Surface Philosophies: A Dual Response Surface Approach, Journal of Quality Technology, Vol. 22, No. 1, January 1990, pp. 38-45.


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How to get helpg p

Search publications posted at www.statease.com.

Use Screen Tips in Stat-Ease software, view reports in annotated mode check for context sensitive Help (right click)annotated mode, check for context-sensitive Help (right-click) or search the main Help system.

E-mail [email protected] for answers from Stat-Ease’s pstaff of statistical consultants.

Call 612.378.9449 and ask for “statistical help.”

Thanks for attending!Thanks for attending!

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