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How to deal with time-dependent factors

Nov 14, 2016 11:30 AM
(3271 views)

How to deal with time-dependent factors

I am designing an experiment in a semiconductor context. Briefly: the performance (i.e. etched material) of a tool (CMP - chemical mechanical polishing) used in the manufacturing of silicon wafers strongly depends on the lifetime of certain consumable parts - i.e. carrier and pad. These parts get consumed quite independently from one another and once they reach a target lifetime, they get replaced. Unfortunately we believe that also some other factors linked to the fabrication of each sample of these parts and not only to the lifetime can affect performance of CMP, such that two physically different carriers with same lifetime can have different impact on response.

The purpose of the study would be to link the performance of CMP to the lifetime of these parts (which would be quite easy) but taking into account the random variation coming from part-to-part construction differences. The main challenge is that lifetime of a given part can only increase with time, so full randomization is not possible. To explain: I cannot perform the initial runs with fresh carrier, perform other runs with the same carrier near lifetime end and finish the experiment with the fresh carrier! On the other hand, if I physically change the part , I'm afraid to confuse response with other factors coming from part-to-part variation.

I felt initially that split-plot design would help, but I cannot address the above issue within this frame. Let me also add that changing these parts is quite time consuming and expensive, therefore minimization of changes is part of the challenge.

I also read some interesting advice by **Bradley Jones** dated 2012 in the **JMP Blog** about using time as covariate, but my case above seems to be a bit more convoluted than the one presented in that post.

It would be really nice if JMP could take into account the kind of constraints I've described above.

I hope my description is clear enough, any suggestion would be appreciated!

thanks

Matteo

5 REPLIES 5

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Re: How to deal with time-dependent factors

Matteo,

I responded to your inquiry in the general LinkedIn DOE discussion group. Because you won't be able to control age, but are still interested in accessing the effects of age, you won't be able to include it in the design of the experiment, but certainly will be able to use it in the modeling of the experimental data. Let's assume, simplistically, that the only things that change are the carrier, the carrier pads, and age (of the pad and carrier). For now, let's disreguard the number of pads and carriers you'll need. Over the course of the experiment, for a given observation, you'll record your responses of interest, plus the age of the pad, the pad ID (any value that will allow you do discriminate the different pads), the age of the carrier (assuming you think that's important), and the carrier ID. Once you've finished, a simple model (but not the only possible model) would be to treat the two age variables as fixed effects, and Pad[Carrier] - pad nested in carrier - and Carrier as random effects. Here, I'm assuming that there may be multiple pad changes for a given carrier.

Don

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Re: How to deal with time-dependent factors

Thanks Don, I think I understand your suggestion, but I would need some further inputs:

1) How should I make sure that age factors will be "orthogonal enough" with each other and to pad/carrier to avoid confusion - I'm not sure how this concept applies to random factors by the way..

2) how many pads/carriers should I use as thumb of rule (I guess this depends on many things, but let's say I'd like to see if they can explain at least half of the total variance).

3) in the design you propose should I still use the split plot approach for pad/carrier, like the following obtained with JMP, setting pad to hard and carrierhead to easy?

Whole_plot CARRIER_HEAD PAD

1 L2 L1

1 L3 L1

1 L1 L1

2 L1 L2

2 L3 L2

2 L2 L2

3 L3 L3

3 L1 L3

3 L2 L3

4 L1 L2

4 L3 L2

4 L2 L2

5 L1 L3

5 L3 L3

5 L2 L3

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Re: How to deal with time-dependent factors

Matteo,

What other age factors are there beside the age of the pad and carrier?

Don

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Re: How to deal with time-dependent factors

At the moment, we are interested only in these two age factors.

br

Matteo

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Re: How to deal with time-dependent factors

Matteo,

Sounds more like a passive data collection than a designed experiment. You have little control over the age of the carriers and pads (only when you can collect data, and only in the forward in time direction). I'll have to give the sample size question a bit of thought, it's a hard question given you have a mixed model and involves some guess as to the pad-to-pad and carrier-to-carrier variability. Hopefully, someone with a more extensive background in this area can chime in.

Don