Insulators — Question 2

Question 2

An optical edge tests whether photons can create band carriers

An ideal direct-gap insulator has $E_g=4.80\,\mathrm{eV}$. Ignore excitons, defects and multiphoton processes. Use $hc=1240\,\mathrm{eV\,nm}$.

1. Find the longest wavelength that can excite a valence electron across the gap.

2. Classify 250 nm and 400 nm photons and find the excess energy above the gap for any allowed case.

Question 2 - Solution

An optical edge tests whether photons can create band carriers

(a) Method and calculation

$$hc/\lambda\ge E_g\Rightarrow\boxed{\lambda\le\lambda_g=hc/E_g}.$$$$\boxed{\lambda_g=1240/4.80=258.3\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$250\,\mathrm{nm}:\quad E=4.96\,\mathrm{eV}\Rightarrow\boxed{E-E_g=.160\,\mathrm{eV}}.$$$$400\,\mathrm{nm}:\quad E=3.10\,\mathrm{eV}<E_g\Rightarrow\boxed{\text{no single-photon band excitation}}.$$

Physical check

The gap provides an energy threshold, while matrix elements determine absorption strength. Transparency below this ideal edge can be spoiled by other allowed absorption channels.