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My textbook (Jaeger's Microelectronic Circuit Design) uses what appears to be this equation to describe the hole and electron mobility in doped silicon, essentially showing that as something is lightly doped the mobility is much higher. Doping concentrations cause a rapid drop off in mobility rapidly. Here are some resources from Indiana University and Colorado University that describe the equations my textbook uses in a bit more detail than my textbook does (it only covers the subject abruptly and not much detail).

$$\mu=\mu_{min}+\frac{\mu_{max}-\mu_{min}}{1+(\frac{N_T}{N_R})^{\alpha}}$$

I'm trying to understand how these equations work in greater detail so that I can potentially use them to solve problems and think about semiconductors.

My questions: Is alpha the product of the "Base Transport Factor" and "Emitter Injection Efficiency" or was the alpha that the Colorado University used in this equation just a fitting parameter that has no relationship to that product? (the alpha they had and my book uses was less than 1 so I assumed that it was plausible that they are the same alpha)

$$\alpha = \alpha_T\gamma = \frac{I_{collector}}{I_{eliminator}}$$

I'm asking all of this because I would like to be able to solve problems such as the one below in ways that are conducive to learning. The way my book solved this problem was by using the relationship if conductivity is equal to the product of the electron charge, hole mobility and hole concentration and solving it iteratively using Matlab by trying values within a range where they know it is and using the one with the least error then using the equations I talked about above they calculate the mobility from that. They come to different equations, presumably based on the doping of the material, but neither show how one might calculate or find those equations. I don't really care so much about getting to their answer but more on how this works and how I might calculate it on my own. If this is a fitted equation you can only obtain through experimentation that's fine, I would like to know that too. Currently this process appears to be a black box to me and I would like to get into the guts of this concept.

$$\sigma =q\mu_{p}p$$

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The \$\alpha\$ in your mobility equation is a fitting parameter determined experimentally. It has no relation to the \$\alpha\$ in your second equation which is a BJT gain parameter. The entire mobility equation you provide is not 100% accurate but instead a "good enough" fit to most mobility curves, which have to be found experimentally.

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