Van Hove Singularity

 

Van Hove singularity singularity - Wikipedia, the free encyclopedia

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Van Hove singularity From Wikipedia, the free encyclopedia

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. (The critical point found in phase diagrams is a completely separate  phenomenon.) For three-dimensional crystals, they take the form of kinks (where the density of states is 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 [1] Belgian physicist Léon Van Hove in 1953 for the case of phonon densities of states.

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

where is wavelength, and n is an integer. (Positive integers will denote forward waves, negative ne gative integers will denote reverse waves.) The smallest wavelength possible is 2a which 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  states  g(k)dk  as  as the number of standing waves with [2]

wave vector k   to to k+dk :

Extending the analysis to wavevectors wavevec tors in three dimensions the density of states in a box will be

where is a volume element in k -space, -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

is the gradient in k-space.

The set of points in k -space -space which correspond to a particular energy E  energy E  form  form a surface in k -space, -space, and the gradient of E  of E  will  will be a vector perpendicular to this surface at every point. [3] The density of states as a function of this energy E  energy E  is:  is:

 

http://en.wikipedia.org/wiki/Van_Hove_singularity Van Hove singularity singularity - Wikipedia, the free encyclopedia

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where the integral is over the surface of constant E  constant E . We can choose a new coordinate system such that is perpendicular to the surface and therefore parallel to the gradient of of E   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:  as:

and, substituting into the expression for g(E) for  g(E) we  we have:

where the equation for

term is an area element on the constant- E   E  surface.  surface. The clear implication of the is that at the -points where the dispersion relation has an extremum, the

integrand in the DOS expression diverges. The Th e Van Hove singularities are the features that occ occur ur in the DOS function at these -points. A detailed analysis[4] 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.

Experimental observation The 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 elemen t 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 density of states states, which is the

 

http://en.wikipedia.org/wiki/Van_Hove_singularity Van Hove singularity singularity - Wikipedia, the free encyclopedia

2013-09-20 3 3

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 d ipole operator matrix element (also known as the oscillator strength) and the JDOS. The divergences in the two- and one-dimensional 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 (quasi-1D) show anomalies in spectroscopic measurements that are attributable to the Van Hove singularities. Van Hove singularities play a significant role in understanding optical intensities in single-walled nanotubes (SWNTs) which are also quasi-1D systems. The Dirac point in graphene is a Van-Hove singularity that can be seen directly as a  peak in electrical resistance, when the graphene is chargeneutral.

A sketch of the DOS g(E) versus energy E for a simulated threedimensional solid. The Van Hove singularities occur where dg(E)/dE diverges.

Notes 1. ^ L. Van Hove, "The Occurrence of Singularities in the Elastic Frequency Distribution of a Crystal," (http://dx.doi.org/10.1103/PhysRev.89.1189) Phys. Rev. 89, 1189–1193 (1953). 2. ^ *M. A. Parker(1997-2004)"Introduction Parker(1997-2004)"Introduction to Density of States" States" Marcel-Dekke  Marcel-Dekkerr Publishing  (http://www.ece.rutgers.edu/~maparker/cla (http://www.ece.rutger s.edu/~maparker/classes/582-Chapters/Ch07-Sol-S sses/582-Chapters/Ch07-Sol-StatetateCarriers/Ch07S16DensityStates.pdf) Carriers/Ch07S16Den sityStates.pdf) p.7. 3. ^ *Ziman, John (1972). Principles (1972). Principles of the Theory of Solids. Solids. Cambridge University Press. ISBN B0000EG9UB. 4. ^ *Bassani, F.; Pastori Parravicini, G. (1975). Electronic (1975). Electronic States and Optical Optical Transitions in Solids. Solids. Pergamon Press. ISBN 0-08-016846-9. This book contains an extensive discussion of the types of Van Hove singularities in different dimensions and illustrates the concepts with detailed theoretical-versus-experimental theoretical-versus-experimental comparisons for Ge and graphite.

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