Understanding underlying statistical assumptions in Monte Carlo simulations

Uncorrelated devices have a lower variance. They vary independently, also in different directions, so that the variations partly cancel each other. For 100% correlated devices, the variations always add up. This is probably also the reason why the mirror ratio makes no difference for your version (c): the relative variation is always the same for all values of m.

Hi

I would like to bring up this topic again.
First, thanks to Frank for his response.

If you blindly apply the mismatch theory, meaning that you use the same Pelgrom coefficients in all settings, a big transistor has less variance than a set of smaller, uncorrelated devices connected in parallel with the same area.

Edit: This is wrong. The outcome is the same in both scenarios.

Clearly, that is not what is happening in the simulator. So what does Specter exactly do in each setting? How does it sample parameter variations in each scenario? If somebody could elaborate how it's done, that would be great.

I didn't see either of the earlier posts in this thread (something is afoot with the email notifications not always being sent; hopefully that will get fixed when we upgrade to a newer version of the forum software; with a bit of luck that will also fix the sporadic image uploading issues that some are facing too).

Please take a look at Recommended Spectre Monte Carlo modeling methodology (I think the same paper is on the Cadence support site somewhere too, but I knew where it was on the Designer's Guide site!). Note that whilst this talks about modelling of the effects of reducing mismatch by close proximity and use of correlation coefficients, in practice the "mismatch" part of the statistical models from most foundries are just modelling the residual local variation of a device and are not taking into account that you might or might not use good layout practice to minimise the differences. Put another way, the term "mismatch" in the model is slightly misleading - it's more about modelling global (die to die) or local (device to device) variation - trying to figure out appropriate correlation coefficients (which are on often meaningless parameters which have variation) is pretty hard and will not have been characterised by the foundries. Note too that the nullmfactorcorrelation doesn't always make sense (depends on the foundry) as sometimes they will have had subckt or other models which have already taken m-factor into account.

The general principle of what Spectre does is that it just generates random values for each of the statistical parameters according to the specified distributions. There's a value picked randomly once per simulation point (using the "process" distribution), and then that is locally varied once per (subckt) instance (see the paper) (using the "mismatch" distribution). Those parameters are then used within the models to adjust various model parameters (exactly how is down to the foundry who produced the models).

Andrew

Dear Andrew

Thank you very much for your response. I carefully exanimated the linked whitepaper find it indeed helpful. I have two follow-up questions:

1. The guide does not talk about device segmentation, but if I understood the paper correctly, that has no impact on how the simulator behaves. Spectre will always sample from the same distribution, regardless of whether I split my device into smaller instances. Correct or not?

2. The paper then further states that "However, process experts are completely free to decide upon (measure) a ratio for their own process, and communicate the corresponding correlation coefficient for such matched devices to their design community." This is actually new for me. My understanding - so far - has been that local, random fluctuations are always assumed to be fully uncorrelated, no matter the situation. I have two PDKs installed and checked some matching reports of different devices (MOSFETs, CAPs, ...); I couldn't find a word about (recommended) correlation values. So as you stated in your answer, it is not characterized by the foundry. Therefore, the fully uncorrelated assumption is true?

My understanding is that foundries usually characterize the mismatch parameters with layouts that use best practices for good matching (small distance, same orientation, same environment). For layouts that don't take matching into account, the mismatch variation of the simulation models might be optimistic. For details, you will probably have to take a look at the documentation from the foundry or ask them.

Frank has said what I would have said for point 2. For point 1, spectre is indeed sampling from the same distribution (I'm not sure how it could do otherwise - there will be more samples if there are more instances, but the distribution remains the same) - but of course the impact on the device will depend on how the varied parameter is used in the models.

Andrew

Dear Andrew, dear Frank

Thanks for clarifying the simulator's behaviour and cleaning up some misunderstandings.