Semiconductors — Question 1

Question 1

Band-edge populations determine the intrinsic carrier density

In a nondegenerate semiconductor, $n=N_c e^{-(E_c-E_F)/(k_BT)}$ and $p=N_v e^{-(E_F-E_v)/(k_BT)}$. Let $E_g=E_c-E_v$.

1. Derive the equilibrium mass-action relation and intrinsic density.

2. Evaluate $n_i$ for $N_c=4.00\times10^{25}\,\mathrm{m^{-3}}$, $N_v=1.00\times10^{25}\,\mathrm{m^{-3}}$, $E_g=1.00\,\mathrm{eV}$, and $k_BT=.0250\,\mathrm{eV}$.

Question 1 - Solution

Band-edge populations determine the intrinsic carrier density

(a) Method and calculation

$$np=N_cN_v e^{-(E_c-E_v)/(k_BT)}\equiv n_i^2.$$$$n=p=n_i\Rightarrow\boxed{n_i=\sqrt{N_cN_v}\,e^{-E_g/(2k_BT)}}.$$

(b) Interpretation and further calculation

$$\sqrt{N_cN_v}=2.00\times10^{25}\,\mathrm{m^{-3}},\quad E_g/(2k_BT)=20.$$$$\boxed{n_i=2.00\times10^{25}e^{-20}=4.12\times10^{16}\,\mathrm{m^{-3}}}.$$

Physical check

The Fermi level cancels from np at thermal equilibrium. Illumination or carrier injection can produce distinct quasi-Fermi levels and invalidate this equilibrium product.