Doped Semiconductors — Question 10

Question 10

Quasi-Fermi separation quantifies injected carriers in a doped sample

An n-type sample has dark $n_0=10^{21}\,\mathrm{m^{-3}}$, $p_0=n_i^2/n_0$, and $n_i=10^{16}\,\mathrm{m^{-3}}$. Illumination adds equal excess densities $\Delta n=\Delta p=10^{19}\,\mathrm{m^{-3}}$. Use $np=n_i^2e^{(F_n-F_p)/(k_BT)}$ and $k_BT=.0250\,\mathrm{eV}$.

1. Find n and p and test whether low-level injection applies.

2. Calculate the electron-hole quasi-Fermi-level separation.

Question 10 - Solution

Quasi-Fermi separation quantifies injected carriers in a doped sample

(a) Method and calculation

$$\boxed{n=1.01\times10^{21},\quad p=10^{19}+10^{11}\simeq10^{19}\,\mathrm{m^{-3}}}.$$$$\boxed{\Delta n/n_0=.0100\ll1}\quad\text{low-level injection}.$$

(b) Interpretation and further calculation

$$F_n-F_p=k_BT\ln(np/n_i^2).$$$$\boxed{F_n-F_p=.0250\ln(1.01000001\times10^8)=.4608\,\mathrm{eV}}.$$

Physical check

The minority population can rise enormously while the majority population changes little. A single equilibrium Fermi level cannot describe this illuminated steady state.