= Impact ionization with a hole at zero momentum
{title2=$E_{\rm threshold}=2E_g$}
For a direct-gap semiconductor with equal parabolic conduction-electron effective masses, restrict the final valence hole to zero crystal momentum and consider a normal process. Momentum conservation gives $\boldsymbol k_1=\boldsymbol k_2+\boldsymbol k_3$, while energy conservation gives $E_1=E_g+\alpha(k_2^2+k_3^2)$ with $\alpha=\hbar^2/(2m^*)$. The <Cauchy-Schwarz inequality> gives $k_2^2+k_3^2\geq k_1^2/2$, so $E_1\geq E_g+E_1/2$ and the threshold is $2E_g$. Equal final wavevectors attain the bound; opposite spins allow the two electrons to occupy that wavevector. This is the threshold under the stated hole constraint, not the general threshold when the final hole can carry momentum.
Back to article page
