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If we assume the semi-conductor is doped by a variable amount, is there some way I can look up carrier charge density for the material in a reference somewhere? What about carrier mobility?

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You can look it up for Si or GaAs or other well characterized semiconductors. If you want to look at the relationship between doping level and charge density, you can use the equations $$n_e = 2(2\hbar^{-2} \pi m_e^*kT)^{3/2} e^{\frac{E_F - E_C}{kT}}$$ $$n_v = 2(2 \hbar^{-2}\pi m_v^*kT)^{3/2} e^{\frac{E_v - E_F}{kT}}$$

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No, the carrier charge densities and mobilities are not constant with temperature. While the charge concentration increases with temperature, the mobility, in general, reduces with temperature due to scattering of the charge carriers due to phonons and ionized impurities. This is because scattering increases with temperature.

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