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For a fixed allowed band/subband pair, parabolic dispersions give $E_c(\mathbf{k})-E_v(\mathbf{k})=E_{j,n}^{\mathrm{edge}}+{\hslash }^2{k}_{\parallel }^2/(2{\mu }_j)$. If the photon energy exceeds the edge, the contour is a circle of radius $k_{\parallel }$ and the gradient magnitude is ${\hslash }^2{k}_{\parallel }/{\mu }_j$. ThusBelow threshold the contour is absent. With the conventional $d^{2}k/(2\pi {)}^2$ normalization, the two-dimensional joint density of states per area and independent channel isWith a locally energy-independent optical matrix element, each allowed pair therefore adds a constant step to the continuum dielectric function in the approximation of the supplied equation. Different hole families need not have equal step heights: reduced masses, spin multiplicities and Bloch-state matrix elements differ. Frequency-dependent overall optical prefactors, nonparabolicity, broadening and excitonic Coulomb enhancement can modify the literal step shape outside this approximation.

The absorptive dielectric function counts optically accessible electron-hole pairs. In the electric dipole approximation the perturbation is proportional to $\mathbf{a}\cdot \hat{\mathbf{p}}$, with $\mathbf{a}$ the light polarization and $\hat{\mathbf{p}}$ the electronic momentum operator. The squared matrix element between occupied valence and empty conduction states is the transition strength; it includes polarization and band-symmetry selection rules. The sums enumerate initial and final subbands. photon momentum is negligible on the crystal-momentum scale, so the transition is approximately vertical in the in-plane wavevector $\mathbf{k}$.

For each subband pair, $C$ is the constant-transition-energy contour in the two-dimensional Brillouin zone. Its line element is $dl$, and the reciprocal gradient weights the states satisfying energy conservation. Specifically, the level-set representation of the joint density of states isA small energy gradient means many nearly resonant states; a vanishing gradient can produce a Van Hove singularity or an edge. The displayed expression suppresses overall optical prefactors and assumes the momentum matrix element is approximately constant along each contour. More generally it stays inside that integral. Occupation factors $f_v-f_c$ are also implicit; at low temperature in an undoped well the relevant valence states are full and the conduction states empty. This is the independent-particle continuum contribution from Fermi's golden rule. Bound excitons add optical resonances below the corresponding free-pair thresholds.

The GaAs four-band model is rooted in bonding and spin-orbit coupling. Near the zone centre, the conduction edge is predominantly s-like and antibonding, while the upper valence manifold is predominantly p-like. Combining orbital angular momentum $l=1$ with spin $s=1/2$ splits the six p-derived states into a $j=3/2$ quartet and a lower $j=1/2$ doublet. The latter is the split-off valence band. Within the upper quartet, the $m_j=\pm 3/2$ and $m_j=\pm 1/2$ branches, defined relative to a propagation or quantization direction, have different curvatures and are called heavy holes and light holes. They coincide at the unstrained bulk zone centre but need not remain degenerate in a well. The conduction edge is a spin doublet. Thus “four bands” counts conduction, heavy-hole, light-hole and split-off branches, including their spin multiplicities; it does not count four individual states. Positive hole masses describe the downward valence curvature, whereas the conduction curvature is upward.

For the final exciton comparison, define the reduced mass ${\mu }_j$ and screened exciton Rydberg $R_j^{\ast }$ byThe resulting $({\mu }_j/m_e,R_j^{\ast })$ are approximately $(0.05948,4.937\mathrm{meV})$ for heavy holes, $(0.03646,3.027\mathrm{meV})$ for light holes and $(0.04631,3.844\mathrm{meV})$ for split-off holes. In the ideal two-dimensional exciton model, the first two optically active internal states lie $4R_j^{\ast }$ and $4R_j^{\ast }/9$ below their continuum edge. Their separation is $32R_j^{\ast }/9$, independent of a common threshold offset. The corresponding three-dimensional separation would be only $3R_j^{\ast }/4$. The listed peak spacings specified for this theoretical spectrum giveHere each pair comprises the stronger lower-energy peak and its weaker nearby companion, in the order associated with the continuum edges calculated in part (c). The approximately 18-meV and 11-meV pair splittings support the two-dimensional series and are incompatible with the three-dimensional series at the supplied masses and permittivity. Equivalently, for a two-dimensional pair with energies $E_1,E_2$, the inferred edge is $E_2+(E_2-E_1)/8$; the observed pair splittings predict ground-state bindings around 20 and 12 meV, close to $4R_j^{\ast }$.

Absolute agreement should not be overstated. For example, the stated square-well parameters give the first heavy-hole edge $1.56390$ eV and the first light-hole edge $1.59161$ eV. Their ideal two-dimensional peak predictions are approximately $(1.54415,1.56170)$ eV and $(1.57950,1.59027)$ eV, respectively. The listed peaks are lower by about 1–2 meV across the families; rounded parameters and this simple effective-mass model do not reproduce every absolute energy exactly. No gap, width or mass has been silently adjusted to fit them. The peak-spacing comparison remains useful because it avoids a common edge offset. The split-off transitions predicted by the four-band data are discussed separately in part (c), rather than assigned fictitious extra peaks.

At 300 K, $k_{B}T\simeq 25.9$ meV, whereas the ideal ground-state binding energies are only about $19.7$, $12.1$ and $15.4$ meV; excited states are still more weakly bound. None of these series is expected to remain a clearly resolved, robust bound-exciton peak at room temperature in this simple estimate. Thermal ionization and phonon broadening work against their visibility. This is a scale argument, not a theorem that the exciton population or all Coulomb-enhanced absorption becomes exactly zero: actual observability also requires temperature-dependent gaps, linewidths, screening and carrier density, which are not supplied.

The static independent-electron density response of a uniform three-dimensional, two-spin Fermi gas, combined with self-consistent Poisson equation screening, gives the displayed dielectric function. For bare electron mass and vacuum background, ${\kappa }^2=m_e{e}^2{k}_F/({ϵ}_0{\pi }^2{\hslash }^2)$ andHere $F(0)=1$, $F(1)=1/2$ by continuous extension, and $F(x)\sim 1/(3x^2)$ at large $x$. A host dielectric background and band effective mass require replacing the vacuum parameters appropriately. This nonlocal response contains Friedel oscillations, which its long-wavelength Thomas-Fermi screening limit misses.