Doping and Devices question 2.3

Physics Q&A
1

Dear course team,

I would like some help with the question: Determine at what donor concentration one cannot assume anymore that all donors are ionized in germanium at room temperature.

My approach was first taking the approximation of the expression of 2.2 such that
n_D = N_D*e^{\frac{(E_F-E_D)}{K_B T}} since |E_F-E_D| >>K_B T .

However I don’t really know how to continue from there since I then get (using the charge balance):

N_D(-e^{\frac{(E_F-E_D)}{K_B T}}+1)-N_A( -e^{\frac{(E_A-E_F)}{K_B T}}+1) = n_e-n_h = N_c e^{-\frac{(E_C-E_F)}{K_B T}}- N_v e^{-\frac{(E_v-E_F)}{K_B T}}

I understand that I should somehow get n_D * n_e since then the fermi energy will drop out of the exponent but I would not know how to proceed can anyone help me along?

2

Hi,

The answer to this question comes from the sections “Adding an impurity to a semiconductor” and “Density of states with donors and acceptors”.

If you re-read them you will see how they describe how to model the extra electron or hole as a hydrogen atom, and how this model is limited because you need the hydrogen-like structures to be independent. The size of these structures is given by a formula similar to that of the bohr radius and from there you obtain a limit to the concentration.

Hope this helps. Let me know if you need more clarification :smiling_face: