Van Hove singularity

A Van Hove singularity is a singularity (non-smooth point) in the density of states (DOS) of a crystalline solid. The wavevectors at which Van Hove singularities occur are often referred to as critical points of the Brillouin zone. For three-dimensional crystals, they take the form of kinks (where the density of states is not differentiable). 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. A Van Hove singularity is a singularity (non-smooth point) in the density of states (DOS) of a crystalline solid. The wavevectors at which Van Hove singularities occur are often referred to as critical points of the Brillouin zone. For three-dimensional crystals, they take the form of kinks (where the density of states is not differentiable). 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. Consider a one-dimensional lattice of N particle sites, with each particle site separated by distance a, for a total length of L = Na. Instead of assuming that the waves in this one-dimensional box are standing waves, it is more convenient to adopt periodic boundary conditions: where λ {displaystyle lambda } is wavelength, and n is an integer. (Positive integers will denote forward waves, negative integers will denote reverse waves.) The shortest wavelength needed to describe a wavemotion in the lattice is equal to 2a which then corresponds to the largest needed wave number k m a x = π / a {displaystyle k_{max}=pi /a} and which also corresponds to the maximum possible |n|: n m a x = L / 2 a {displaystyle n_{max}=L/2a} . We may define the density of states g(k)dk as the number of standing waves with wave vector k to k+dk: Extending the analysis to wavevectors in three dimensions the density of states in a box will be where d 3 k {displaystyle d^{3}k} is a volume element in k-space, and which, for electrons, will need to be multiplied by a factor of 2 to account for the two possible spin orientations. By the chain rule, the DOS in energy space can be expressed as where ∇ → {displaystyle {vec { abla }}} is the gradient in k-space. The set of points in k-space which correspond to a particular energy E form a surface in k-space, and the gradient of E will be a vector perpendicular to this surface at every point. The density of states as a function of this energy E is: where the integral is over the surface ∂ E {displaystyle partial E} of constant E. We can choose a new coordinate system k x ′ , k y ′ , k z ′ {displaystyle k'_{x},k'_{y},k'_{z},} such that k z ′ {displaystyle k'_{z},} is perpendicular to the surface and therefore parallel to the gradient of E. If the coordinate system is just a rotation of the original coordinate system, then the volume element in k-prime space will be We can then write dE as: