Density of states ($g(E)$)

The density of states $g(E)$ is defined so that $g(E)dE$ counts one-particle states with energies in an interval of width $dE$. A sum over closely spaced states then becomes an integral weighted by $g(E)$.

Two spin states per wavevector and parabolic dispersion give $n=2v_d{k}_F^d/(2\pi {)}^d$, where $v_d$ is the unit-ball volume. Thus $E_F={\hslash }^2[n{a}_d{]}^{2/d}/(2m)$ with $a_1=\pi /2$, $a_2=2\pi$, $a_3=3{\pi }^2$. Differentiating the cumulative count yields $g(E_F)=dn/(2E_F)$, normalized per length, area or volume as appropriate.

For the continuum dispersion ${\epsilon }_{\mathbf{k}}={\hslash }^2(k_x^2+k_y^2+k_z^2/\gamma )/(2m)$ with $\gamma >0$, substituting $q_z=k_z/\sqrt{\gamma }$ makes the energy isotropic and gives Jacobian $\sqrt{\gamma }$. Every momentum-state counting integral therefore acquires that factor. Per internal state and unit volume, $N=\sqrt{\gamma }(2m/{\hslash }^2{)}^{3/2}\sqrt{\epsilon }/(4{\pi }^2)$. The result assumes an unrestricted three-dimensional continuum; increasing this mass anisotropy is not the same operation as imposing a two-dimensional Brillouin zone and a finite interlayer band width.

For an area $A$ with periodic in-plane boundary conditions, one polarization contributes $A2\pi kdk/(2\pi {)}^2$ normal modes of the electromagnetic field. For linear dispersion relation $\omega =ck$ and effective physical volume $Ad$, this gives the density of states in the title. No extra factor two for polarization is allowed. Opposite in-plane directions are already counted in the full circle of wavevectors.

For a spinless nonrelativistic particle in a central potential, $g(ϵ)=(2m{)}^{3/2}(4{\pi }^2{\hslash }^3{)}^{-1}{\int }_{U<ϵ}\sqrt{ϵ-U}{d}^{3}r$. Spherical symmetry reduces this to a radial integral over every allowed shell. It is a leading phase-space approximation.

Periodic boundaries on a cube give one spinless state per $(2\pi /L{)}^3$ in wave-vector space. Counting the full spherical shell gives $d\mathcal{N}/V=k^{2}dk/(2{\pi }^2)$. Hard-wall counting in the positive octant gives the same leading volume term.

The number of lattice-vibration modes per unit angular frequency. For a three-dimensional isotropic monatomic crystal with three linear acoustic branches of a common speed $v_s$, wavevector-shell counting gives $g(\omega )=3V{\omega }^2/(2{\pi }^2{v}_s^3)$ below a suitable cutoff. Different branch speeds replace $3v_s^{-3}$ by $v_L^{-3}+2v_T^{-3}$.

For the nearest-neighbour monatomic-chain dispersion relation, ${\omega }_m=2\sqrt{\alpha /M}$ and $d\omega /dq=(a/2)\sqrt{{\omega }_m^2-{\omega }^2}$ on the positive half-zone. The folded wave-number count gives the displayed density of states, including both propagation directions. Its upper-edge Van Hove singularity is integrable: ${\int }_0^{{\omega }_m}g(\omega )d\omega =N$. A finite chain has discrete modes; the continuous density describes its large-size limit.

A single acoustic branch $\omega =v|k|$ has thermal energy per length $U_T/\ell =\int dk\hslash v|k|/[2\pi (e^{\beta \hslash v|k|}-1)]=\pi (k_{B}T{)}^2/(6\hslash v)$. Differentiating gives the displayed linear heat capacity. The regime requires temperatures below the microscopic phonon cutoff but above the finite-size level spacing, with the translational zero mode treated separately.

A nonanalyticity of the density of states associated with a stationary point of a band dispersion relation. Its form depends on dimension and on whether the stationary point is an extremum or a saddle.

A single two-dimensional quadratic saddle with $A={\hslash }^2/(2m^{\ast })$ contributes this leading joint density of states when both contour branches are included and $g_s$ counts independent spin channels. The contour integral is proportional to ${\int }_{-\Lambda }^{\Lambda }dy/\sqrt{y^2+\delta /A}=2\mathrm{arsinh}(\Lambda \sqrt{A/\delta })$ above the saddle. Its asymptote gives the logarithm with $E_{\Lambda }$ of order $A{\Lambda }^2$. The cutoff-dependent constant is not universal. This is a density per area per energy; changing to a density per frequency requires a Jacobian. ^[ref]

A spin factor and disconnected contour branches count different independent contributions. For the transition energy $E_s+{\hslash }^2(k_x^2-k_y^2)/(2m^{\ast })$, one branch contributes $g_s{m}^{\ast }I/(4{\pi }^2{\hslash }^2)$, with $I=\int d{k}_y/\sqrt{k_y^2+\delta /A}$. The full two-branch saddle doubles this. A formula for one branch with spin degeneracy two can therefore equal a full-contour formula with spin degeneracy one, but those are distinct counting conventions. An extra valley or inequivalent saddle requires its own multiplicity.

For a one-dimensional subband $E=E_0+\alpha k^2$ with $\alpha >0$, the density of states varies as $(E-E_0{)}^{-1/2}$ just above its edge. Optical peaks also depend on the joint density of states, selection rules and excitons.

The density of available one-particle states multiplied by their mean occupation. For equilibrium electrons, $Z(E)=D(E)f(E)$ with the Fermi-Dirac distribution $f$. The integral of $Z$ counts electrons; the integral of $D$ counts available states.

A many-to-one change of energy variable requires summing every inverse branch. For $E(\xi )=\sqrt{{\xi }^2+{\Delta }^2}$ and constant normal density of states $g_F$ on both sides of zero, there are two roots for $E>\Delta$, giving $D_{+}(E)=2g_{F}E/\sqrt{E^2-{\Delta }^2}$. A single branch instead gives $g_{F}E/\sqrt{E^2-{\Delta }^2}$. The distinction persists when $\Delta \to 0$: folding $\xi$ into $|\xi |$ doubles a two-sided constant density on the positive axis. Spin degeneracy is a separate normalization choice; it must be applied consistently to both normal and transformed densities. The BCS electronic density of states has the single-factor expression because it is spectrally weighted rather than an unweighted branch count.

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.

For a length $L$ and parabolic free-electron dispersion relation $E={\hslash }^2{k}^2/(2m)$, the displayed total density of states includes both signs of $k$ and both spin states. Per spin it is half as large. Its $E^{-1/2}$ dependence follows by transforming the constant signed-$k$ density $L/(2\pi )$ to energy and including both momenta. The continuum singularity at the band bottom is integrable; a finite specimen still has discrete levels.

The local density of states resolves the density of states by position as well as energy. With a consistent counting of spin states, its integral over the specimen's volume gives the total density of states, $g(E)=\int \rho (\mathbf{r},E)d^3\mathbf{r}$. It describes the spatial distribution of available states, rather than their occupation.

For a spin-degenerate band, the total density of states includes both spin species and is twice the single-spin density of states. Per lattice site, $\nu (ϵ)=2N^{-1}{\sum }_k\delta (ϵ-{ϵ}_k)$. Using a separate explicit spin sum then requires a compensating factor one half when converting momentum sums. The normalization matters in the Stoner criterion.

The single-spin density of states counts momentum states for one spin species. For degenerate up and down species, the spin-summed density of states is twice this value. Specify separately whether the density is per unit volume or extensive. In the BCS gap equation, the momentum sum labels opposite-spin pairs once and therefore uses the single-spin density per volume when the interaction has normalization $V/\mathcal{V}$.