Impact ionization with a hole at zero momentum ($E_{\mathrm{threshold}}=2E_g$)

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 ${\mathbf{k}}_1={\mathbf{k}}_2+{\mathbf{k}}_3$, while energy conservation gives $E_1=E_g+\alpha (k_2^2+k_3^2)$ with $\alpha ={\hslash }^2/(2m^{\ast })$. The Cauchy-Schwarz inequality gives $k_2^2+k_3^2\ge k_1^2/2$, so $E_1\ge E_g+E_1/2$ 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.