Semiconductors: Doping and Devices Question 2.3

Physics Q&A
1

I cannot seem to get the correct equation in which E_D - E_C are involved according to the answers. Could you please explain to me how this is obtained?

I can prove the intrinsic version stated in the lecture quite well, but have no idea how to involve the doping parameters or where even the E_D comes from. Instead of
E_D - E_C I am able to get E_G, but this seems wrong.
Where do I even involve n_D?

2

the question asks you to consider when the number of non-ionized dopants is significant (say e.g. 50%). Try to write down an expression for this number. It should depend on the Fermi Energy.

4

Yes, it is dependent on E_F as follows: assuming room temperature
n_D = N_D * exp(-(E_D - E_F)/kT) ~ N_D

Using our expression for E_F = E_C - kT*log((N_C)/(N_D - N_A)) we get our desired expression. However I don’t see why I am allowed this formula.
The formula has two implicit assumptions: extrinsic regime and n_D ~= 0; n_A ~= 0.

  1. Why should I suppose n_A ~= 0?
  2. Why is n_D ~= 0 valid when we also suppose it to be approximately N_D?
    It really seems like I am invalidating my own argument I used to obtain the solution?