Van Hove Singularity - Theory

Theory

Consider a one-dimensional lattice of N particles, with each particle separated by distance a, for a total length of L = Na. A standing wave in this lattice will have a wave number k of the form

where 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 and which also corresponds to the maximum possible |n|: . 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 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

dE = \frac{\partial E}{\partial k_x}dk_x + \frac{\partial E}{\partial k_y}dk_y + \frac{\partial E}{\partial k_z}dk_z = \vec{\nabla}E \cdot d\vec{k}

where 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 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 k-prime 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 constant-E surface. The clear implication of the equation for is that at the -points 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 -points.

A detailed analysis shows that 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 (see the Figure) since for a spherical free electron Fermi surface

so that .

In two dimensions the DOS is logarithmically divergent at a saddle point and in one dimension the DOS itself is infinite where is zero.

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