How do we deduce the temperature dependence of the chemical potential for a doped semiconductor (question 4.6(c)(i) and (ii))? Considering the complication and vagueness of the temperature ranges below, I'm unsure...
For the law of mass action to be valid, we require:
(E_c - mu) >> k_B*T
(mu - E_v) >> k_B*T
For a n-doped semiconductor to show intrinsic behaviour, we require that I>>D, where I=(np)^1/2 and D=n-p. This puts a lower bound on temperature. How do we know that such a temperature (and mu(T)) satisfy the two conditions above? ONLY then can we use the law of mass action to obtain I...
Also, what is the temperature for no freeze-out? It would seem to me for no freeze-out in an n-doped semiconductor:
k_B*T ~ (E_c - E_donor band)
We know that chemical potential at T=0 is midway between E_c and E_donor band, so it better be much below E_donor band at temperatures (including the above) where the donors are ionised (ie. no freeze-out). Otherwise, the law of mass action won't be valid!
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