To use all functions of this page, please activate cookies in your browser.
my.chemeurope.com
With an accout for my.chemeurope.com you can always see everything at a glance – and you can configure your own website and individual newsletter.
 My watch list
 My saved searches
 My saved topics
 My newsletter
Van Hove singularityA Van Hove singularity is a kink in the density of states (DOS) of a solid. The wavevectors at which Van Hove singularities occur are often referred to as critical points of the Brillouin zone. (The critical point found in phase diagrams is a completely separate phenomenon.) 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. Additional recommended knowledgeTheoryConsider a onedimensional lattice of particles, with each particle separated by distance a, with a total of N particles, for a total length of L=Na. A standing wave in this lattice will have a wave number k of the form where λ is wavelength, and n is an integer. (Positive integers will denote forward waves, negative integers will denote reverse waves.) The smallest wavelength possible is 2a which corresponds to the largest possible wave number k_{max} = π / a and which also corresponds to the maximum possible n: 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 is a volume element in kspace, 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 is the gradient in kspace. The set of points in kspace which correspond to a particular energy E form a surface in kspace, and the gradient of E will be a vector perpendicular to this surface at every point (Ziman, 1972). The density of states as a function of this energy E is: where the integral is over the surface of constant E. We can choose a new coordinate system such that 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 kprime space will be We can then write dE as: and, substituting into the expression for g(E) we have: where the term is an area element on the constantE surface. The clear implication of the equation for g(E) is that at the kpoints where the dispersion relation 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 kpoints. A detailed analysis (Bassani 1975) shows that there are four types of Van Hove singularities in threedimensional 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 squareroot singularities (see the Figure) since for a spherical free electron Fermi surface
In two dimensions the DOS is logarithmically divergent and in one dimension the DOS itself is infinite where is zero.
Experimental observationThe optical absorption spectrum of a solid is most straightforwardly calculated from the electronic band structure using Fermi's Golden Rule where the relevant matrix element to be evaluated is the dipole operator where is the vector potential and is the momentum operator. The density of states which appears in the Fermi's Golden Rule expression is then the joint density of states, which is the number of electronic states in the conduction and valence bands that are separated by a given photon energy. The optical absorption is then essentially the product of the dipole operator matrix element (also known as the oscillator strength) and the JDOS. The divergences in the two and onedimensional DOS might be expected to be a mathematical formality, but in fact they are readily observable. Highly anisotropic solids like graphite (quasi2D) and Bechgaard salts (quasi1D) show anomalies in spectroscopic measurements that are attributable to the Van Hove singularities. References

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Van_Hove_singularity". A list of authors is available in Wikipedia. 