Density of States

This card is the compact lookup. Density of States is the canonical quantum-matter article for crystalline-band normalization, dimensional thresholds, van Hove singularities, projected densities, thermodynamics, spectroscopy, and numerical evaluation.

Formula

For a finite spectrum, the exact density of states is the distribution

$$D(E) = \sum_\alpha \delta(E-E_\alpha),$$

where $\alpha$ labels independent states.

Equivalently, if $N(E)$ counts states below energy $E$,

$$D(E) = \frac{dN}{dE}.$$

For a periodic single-particle band model in $d$ dimensions, a common density per volume is

$$\frac{D(E)}{V} = \sum_n \int_{\mathrm{BZ}} \frac{d^d k}{(2\pi)^d} \, \delta(E-E_n(\mathbf k)).$$

When the energy shell is smooth and $\nabla_{\mathbf k}E_n\ne0$, this can be written as

$$\frac{D(E)}{V} = \sum_n \int_{E_n(\mathbf k)=E} \frac{dS_{\mathbf k}}{(2\pi)^d} \frac{1}{\lVert\nabla_{\mathbf k}E_n\rVert}.$$

Assumptions

• The state-counting convention is specified: finite volume, per volume, per unit cell, per spin, or including internal degeneracies.
• The delta functions are distributions or are replaced by a stated broadening in numerical work.
• The Brillouin-zone normalization uses the convention compatible with the chosen unit-cell volume.
• The energy-shell form assumes a smooth constant-energy surface away from critical points.

Validity

Density of states is a counting object. It becomes a smooth function only after a thermodynamic limit, coarse graining, or explicit broadening. Singularities can appear at band extrema, saddle points, flat bands, or thresholds where the energy-shell formula requires care.

Common Mistakes

• Forgetting spin, valley, orbital, or band degeneracy factors.
• Mixing total density of states with density per volume or per unit cell.
• Treating a broadened numerical histogram as an exact formula.
• Using the energy-shell expression at a critical point where $\nabla_{\mathbf k}E_n=0$.
• Confusing density of states with occupation probability.

Quick Check

Why does a flat band produce a large density-of-states contribution?

Solution

If many independent states share the same energy, the sum $\sum_\alpha\delta(E-E_\alpha)$ has many contributions at that energy. In the band expression, a nearly flat $E_n(\mathbf k)$ also makes $\lVert\nabla_{\mathbf k}E_n\rVert$ small on the energy shell, enhancing the density of states.

Canonical Links

• Density of States in Quantum Matter
• Tight-Binding Model
• Density of States: First Encounter
• Green Functions and Density of States
• Density of States in Transition Rates
• Plane Waves and Delta Normalization
• Fermi Golden Rule
• Landau-Level Degeneracy

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976.
• R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Academic Press, 2011.
• S. H. Simon, The Oxford Solid State Basics, Oxford University Press, 2013.