Solving n_{n0} Using Charge Neutrality & Mass Action Law

[mzh]

Dear PF users
Would be great if somebody could point me out how to arrive at n_{n0} = \frac{1}{2} \left[ (N_D - N_A) + \sqrt{ (N_D - N_A)^2 + 4n_i^2} \right] (n-type charge carrier concentration at thermal equilibrium) by using the expression for the charge neutrality n+N_A = p+N_D and the mass action law np=n_i^2.

I understand I should assume that N_D > N_A, but I can't work it out.

Any comments are very welcomed.

 

[mzh]

Dear PF users
Would be great if somebody could point me out how to arrive at n_{n0} = \frac{1}{2} \left[ (N_D - N_A) + \sqrt{ (N_D - N_A)^2 + 4n_i^2} \right] (n-type charge carrier concentration at thermal equilibrium) by using the expression for the charge neutrality n+N_A = p+N_D and the mass action law np=n_i^2.

I think I found the solution to this. The important point to note is that we assume relatively high temperatures. Given the relationship for N_D^+ = \frac{N_D}{1+2\exp\left[\frac{E_F - E_D}{kT}\right]}, we can assume that E_F - E_D is much lower than zero. Then, when dividing by kT = 0.025 \mbox{eV} at room temperature, the exponential term becomes approximately zero and so N_D^+ = N_D. Then, N_D can be inserted into the charge neutrality condition and, after expressing p=\frac{n_i^2}{n}, the resulting quadratic equation can be solved for n. Great.