Past exam of the quantum condensed matter course of the University of Cambridge / 2022 / Section B / 6 / a / Solution

Count wavevectors in a large volume with periodic boundary conditions. One spin-degenerate spherical band suppliesFor the conduction band, $k=\sqrt{2m_e^{\ast }(ϵ-{ϵ}_c)}/\hslash$ and $dk/dϵ=m_e^{\ast }/({\hslash }^{2}k)$. Transforming the state count to energy givesFor the valence band use $|dk/dϵ|=m_h^{\ast }/({\hslash }^{2}k)$ and $k=\sqrt{2m_h^{\ast }({ϵ}_v-ϵ)}/\hslash$, obtainingEach density of states is zero beyond its indicated band edge. The factor two counts spin; an extra valley degeneracy was not assumed in these stipulated isotropic bands.