Semiconductors — Question 9

Question 9

An indirect optical transition must account for phonon energy

An ideal indirect semiconductor has gap $E_g=1.10\,\mathrm{eV}$. A phonon of energy $E_{\mathrm{ph}}=.0400\,\mathrm{eV}$ supplies the required momentum. Ignore other losses and take $hc=1240\,\mathrm{eV\,nm}$.

1. Find photon thresholds for transitions accompanied by phonon absorption and phonon emission.

2. Find their threshold wavelengths and explain which process is suppressed as the phonon population tends to zero.

Question 9 - Solution

An indirect optical transition must account for phonon energy

(a) Method and calculation

$$\text{Phonon absorption: }E_\gamma+E_{\mathrm{ph}}=E_g\Rightarrow\boxed{E_{\gamma,a}=1.06\,\mathrm{eV}}.$$$$\text{Phonon emission: }E_\gamma=E_g+E_{\mathrm{ph}}\Rightarrow\boxed{E_{\gamma,e}=1.14\,\mathrm{eV}}.$$

(b) Interpretation and further calculation

$$\boxed{\lambda_a=1240/1.06=1170\,\mathrm{nm},\quad\lambda_e=1240/1.14=1088\,\mathrm{nm}}.$$$$W_a\propto N_{\mathrm{ph}},\quad W_e\propto N_{\mathrm{ph}}+1\Rightarrow\boxed{W_a\to0\text{ as }N_{\mathrm{ph}}\to0}.$$

Physical check

Emission can create a phonon even when none is initially present. The energy thresholds alone do not determine absorption strength or the complete spectrum.