Joint density of states ($J(\Omega )={\sum }_{v,c}\int \frac{d^{d}k}{(2\pi {)}^d}\delta (E_c(\mathbf{k})-E_v(\mathbf{k})-\Omega )$)

For filled valence and empty conduction energy bands, the joint density of states counts pairs of states at a prescribed interband energy difference. In a direct optical transition the photon momentum is usually negligible compared with a Brillouin zone dimension, so initial and final states have nearly the same crystal momentum. The absorption spectrum weights this count by transition matrix elements; its onset and structure can constrain the band gap and electronic band structure. Indirect transitions additionally involve phonons to supply crystal momentum.

For isotropic direct-gap parabolic bands, $m_r^{-1}=m_e^{\ast -1}+m_h^{\ast -1}$ and transition energy is $E_g+{\hslash }^2{k}^2/(2m_r)$. Differentiating the two-spin state count $k^3/(3{\pi }^2)$ per volume gives the displayed density for $E>E_g$, and zero below. Density per angular frequency is $\hslash g_j(\hslash \omega )$, not the same dimensioned quantity.

For a local transition-energy model $E_c-E_v=E_s+A(k_x^2-k_y^2)$ with $A>0$, positive detuning has two hyperbola branches opening along x. Negative detuning opens along y; the zero-detuning limiting contour is a pair of intersecting lines. The coarea integrand on one positive-detuning branch reduces to $d{k}_y/[2A\sqrt{k_y^2+\delta /A}]$. The gradient vanishes at the saddle, so this is a limiting description as nonzero detuning approaches zero.

For positive detuning inside a strict square region $|k_x|,|k_y|\le \Lambda$, substituting $k_x^2=k_y^2+\delta /A$ bounds $k_y$ by the displayed value, provided $0<\delta /A<{\Lambda }^2$. Integrating to $\Lambda$ instead extends the contour slightly outside that square. Either prescription has the same leading logarithmic singularity near zero detuning; it does not give the same exact finite-detuning result.

Parabolic electron-hole continua with $d$ unconfined directions have cumulative state count proportional to $(E-E_{\mathrm{th}}{)}^{d/2}$. Differentiating gives the threshold joint density proportional to $(E-E_{\mathrm{th}}{)}^{d/2-1}$: a square root for bulk, a step for a well, and an inverse square root for a wire. A fully confined dot instead has discrete lines. Optical matrix elements, occupations, excitons and broadening determine the actual absorption beyond this counting argument.

For a smooth two-dimensional transition energy $D(\mathbf{k})$ and a regular level $D=E$, the coarea identity gives $\int d^{2}k\delta (D(\mathbf{k})-E)={\int }_{D=E}dl/|\nabla D|$. Critical points require a limiting or distributional treatment; the apparent small contour at a parabolic edge is compensated by its small energy gradient.

For two isotropic parabolic subbands, the joint density of states per area and per independent channel is $\mu /(2\pi {\hslash }^2)\Theta (\hslash \omega -E_{\mathrm{edge}})$. The constant is a two-dimensional phase-space effect. Degeneracy and optical matrix elements determine the actual step weights.