Van Hove singularity¶

A Van Hove singularity is a kink in the Density of states(DOS) of a solid. The wave vectors at which Van Hove singularities occur are often referred to as critical points of the Brillouin zone. The most common application of the Van Hove singularity concept comes in the analysis of optical absorption spectra. The occurrence of such singularities was first analyzed by the Belgian physicist Léon Van Hove in 1953 for the case of phonon densities of states.

We have the following expression for density of states as function of energy (see Van Hove singularity - Wikipedia for derivation),

\(g(E) = \frac{L^3}{(2\pi^3)} \int \int \frac{dk_x^\prime dk_y^\prime}{|\vec{\nabla E}|}\)

Where, \(dk_x^\prime dk_y^\prime\) term is an area element on the constant-E surface. The clear implication of the equation for \(g(E)\) is that at the k-points where the dispersion relation \(E({\vec {k}})\) has an extremum, the integrand in the DOS expression diverges. The Van Hove singularities are the features that occur in the DOS function at these k-points.

There are four types of Van Hove singularities in three-dimensional space, depending on whether the band structure goes through a local maximum, a local minimum or a saddle point. In three dimensions, the DOS itself is not divergent, although its derivative is. The function g(E) tends to have square-root singularities, since for a spherical free electron gas Fermi surface.

\({\displaystyle E={\frac {\hbar ^{2}k^{2}}{2m}}}\)

So that,

\(|{\vec {\nabla }}E|={\frac {\hbar ^{2}k}{m}}=\hbar {\sqrt {\frac {2E}{m}}}\)

In two dimensions the DOS is logarithmically divergent at a saddle point and in one dimension the DOS itself is infinite where \(\vec{\nabla}E\) is zero.

The Van Hove singularities occur where \(dg(E)/dE\) diverges.