= Solution
For a representative unstrained <zinc blende structure> <III-V semiconductor> such as <gallium arsenide>, take the valence-band maximum at the <Gamma point> as zero. The near-centre ordering is an s-like $\Gamma_{6c}$ <conduction band> doublet above a p-like $\Gamma_{8v}$ valence quartet. <Spin-orbit interaction> puts the p-like $\Gamma_{7v}$ doublet below the quartet by $\Delta_{\rm so}$. These are the <zone-centre bands of a zinc-blende III-V semiconductor>.
The $j=3/2$ quartet produces the <heavy-hole and light-hole bands>. Along a chosen symmetry direction, their components have $|m_j|=3/2$ and $1/2$ respectively; their different curvatures define different positive hole <effective masses>. They are degenerate at zero wavevector in the unstrained cubic crystal, but separate away from it. The $j=1/2$ doublet is the <split-off valence band>. To leading isotropic order,
$$
\boxed{E_c(k)\simeq E_g+\frac{\hbar^2k^2}{2m_c},\quad
E_{v,\rm hh/lh}(k)\simeq-\frac{\hbar^2k^2}{2m_{\rm hh/lh}},\quad
E_{v,\rm so}(k)\simeq-\Delta_{\rm so}-\frac{\hbar^2k^2}{2m_{\rm so}}.}
$$
Valence-band curvature is negative for electrons; the corresponding hole masses are positive. Real valence bands warp with direction, so these are illustrative effective-mass parabolas rather than an exact spherical band structure.
The <k·p perturbation theory> couples the s-like and p-like states through momentum matrix elements. It controls the conduction-band mass, nonparabolicity and mixing of hole characters. Strain or quantum confinement can split the heavy/light degeneracy even at zero in-plane wavevector. The lack of inversion symmetry in zinc blende also permits spin splitting away from the zone centre; time-reversal symmetry still protects the zone-centre doublets.
The s–p electric-dipole transitions near $\Gamma$ are optically important and produce structures at $E_g$ and $E_g+\Delta_{\rm so}$ with polarization-dependent weights. For <direct band gap> compounds the conduction minimum is also at $\Gamma$, allowing near-edge absorption and emission without a momentum-supplying phonon. \b[Not every III-V compound is direct-gap], and wurtzite materials have an additional crystal-field splitting; the cubic scheme should not be imposed on every III-V semiconductor. <Optical Properties of Solids by Mildred S. Dresselhaus>[§3.4, spin-orbit interaction in solids;§6.2, band-structure optical features;§7.7, heavy/light GaAs valence states and confinement splitting]{ref}
\Image[/past-exam-of-the-advanced-quantum-condensed-matter-physics-course-of-the-university-of-cambridge/2012/paper-1-aqc2012-iii-v-bands.png]
{title=Zone-centre III-V bands}
{source=paper-1-aqc2012-iii-v-bands.py}
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