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

• mzh
In summary, the conversation discusses how to arrive at the expression for the n-type charge carrier concentration at thermal equilibrium by using the charge neutrality condition and the mass action law. The solution involves assuming high temperatures and manipulating the equations to solve for n.

#### 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.

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.