Energy Levels in a Crystalline Solid — Question 6

Question 6

A lattice Fourier component opens a gap at a Brillouin-zone boundary

Two degenerate free-electron states at a zone boundary are coupled by a periodic potential. In their basis $H=\begin{pmatrix}E_B&U_G\\U_G^*&E_B\end{pmatrix}$. Let $|U_G|=.350\,\mathrm{eV}$.

1. Derive the two energies and gap.

2. Find the longest photon wavelength able to bridge that gap if a direct optical transition is allowed, using $hc=1240\,\mathrm{eV\,nm}$.

Question 6 - Solution

A lattice Fourier component opens a gap at a Brillouin-zone boundary

(a) Method and calculation

$$\det(H-EI)=(E_B-E)^2-|U_G|^2=0.$$$$\boxed{E_\pm=E_B\pm|U_G|,\quad E_g=2|U_G|=.700\,\mathrm{eV}}.$$

(b) Interpretation and further calculation

$$hc/\lambda\ge E_g\Rightarrow\lambda\le hc/E_g.$$$$\boxed{\lambda_{\max}=1240/.700=1771\,\mathrm{nm}}.$$

Physical check

The gap depends on the magnitude of the coupling, not its complex phase. The wavelength condition assumes momentum and selection rules allow the stated transition.