Doped Semiconductors — Question 6

Question 6

Doping moves the Fermi level logarithmically

At equilibrium use $n=n_i e^{(E_F-E_i)/(k_BT)}$ and $p=n_i e^{-(E_F-E_i)/(k_BT)}$. At one temperature, $k_BT=.0250\,\mathrm{eV}$, $n_i=10^{16}\,\mathrm{m^{-3}}$, and a donor-dominated sample has $n=10^{21}\,\mathrm{m^{-3}}$.

1. Find the Fermi-level shift and minority density.

2. If n increases by a factor of ten while the sample remains nondegenerate, find the additional Fermi-level shift and the new p.

Question 6 - Solution

Doping moves the Fermi level logarithmically

(a) Method and calculation

$$\boxed{E_F-E_i=k_BT\ln(n/n_i)=.0250\ln10^5=.2878\,\mathrm{eV}}.$$$$\boxed{p=n_i^2/n=10^{11}\,\mathrm{m^{-3}}}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta E_F=k_BT\ln10=.05756\,\mathrm{eV}}.$$$$\boxed{p_{\mathrm{new}}=10^{10}\,\mathrm{m^{-3}}}.$$

Physical check

Each decade of electron density produces the same energy shift at fixed temperature. Degeneracy eventually invalidates the exponential carrier statistics.