# Which statistical distribution could be assumed for the mobility of Silicon substrate material?

Mobility is a key parameter for Hall elements or in transistor models. If we measure the mobility on Si wafers, then we can expect some variation from die to die or sample to sample. Which statistical distribution should be assumed for this? A normal distribution is not a good model because it ranges from -infinite to infinite, but mobility has to be a positive number of course.
It would be good to find references about this topic, or even measurement results from semiconductor fabs. We are not only interested for a good model fit in the body of the data, but also in the distribution tails, because just these extreme samples can ruin a product fabricated on such wafers (not necessarily Silicon). Or are there built-in models with suited mobility distributions in TCAD programs?

Unfortunately I have no raw data, only a Gaussian fit for a MOS transistor model, and often this fit arise from quite noisy data, and also often the mobility model parameter is not fit for physical meaning but just for fitting things not modeled by other parameters.

• Do you have some examples of what the distribution looks like (i.e. p.d.f. plots)? We often use normal distribution even if negative values are non-physical, so long as $\mu\gg\sigma$ so that the portion of the tail falling in the non-physical region is negligible (whatever that means for your situation). – The Photon May 8 at 17:55
• Hi Photon, I do not have any data, because I am no material expert. Indeed the limit for becoming negative is usually around -20 times sigma. But we have a high-yield analysis method which scales up that sigma, and then we really reach these -20 sigma (unfortunately, so that our method becomes inaccurate). – user32038 Jul 2 at 14:14