Van Hove singularity

A Van Hove singularity is a discontinuity ( " kink" ) in the density of solids. The most common application of the concept of the Van Hove singularity occurs in the analysis of the optical absorption spectra. Named are the singularities introduced by the Belgian physicist Léon Van Hove, who first described the phenomenon in 1953 for the density of states of phonons.

Theory

If we consider a one-dimensional lattice, ie a chain of length L of N particles, with adjacent particles have a distance a, a standing wave is obtained for the magnitude of the wave vector k is an expression of the form:

Wherein the wavelength and n is an integer. The smallest possible wavelength 2a. This corresponds to the maximum wave number and corresponds to the maximum | n |: . The state density g (k) is now defined as: g (k) dk indicates the number of standing waves, the wave vector is in the interval of k to k dk:

We extend the consideration to three dimensions, we have:

Wherein a volume element in the k- space.

Transition to the density of states per energy

After applies the chain rule

Wherein the gradient in k- space. The amount of points in k-space corresponding to a given energy E, form a surface in k-space; the gradient of E is available at any point perpendicular to this plane. For the density of states as a function of E, this results in:

Where the integral is to be formed over the surface with a constant e. Now is introduced a coordinate in which is perpendicular to the surface. After this change of coordinates is:

In the expression for g ( E) which gives:

Where the term a surface element on the Äquienergie - surface (E = const. ) corresponds.

The singularities

Of points in k-space, in which the dispersion relation and disappears thus has an extremum, the diverging state density. These points are called Van Hove singularities.

A detailed analysis ( Bassani 1975) shows that there are four types of the Van Hove singularities in three dimensions. These differ to the extent that the tape has a local maximum, a local minimum or a saddle point of the first or second type. The function g (E) tends in three dimensions on the basis of the spherical shape of the Fermi surface for free electrons to square -like singularities. Although its derivative diverges, the density is therefore not diverge, as seen in the figure.

In two dimensions, the density of states diverges logarithmically, in one dimension it is infinite if is zero.