- Van Hove singularity
A

**Van Hove singularity**is a kink in thedensity of states (DOS) of asolid . Thewavevector s at which Van Hove singularities occur are often referred to as critical points of theBrillouin zone . (The critical point found inphase diagram s is a completely separate phenomenon.) The most common application of the Van Hove singularity concept comes in the analysis ofoptical absorption spectra. The occurrence of such singularities was first analyzed by the Belgian physicistLéon Van Hove in 1953 for the case ofphonon densities of states.**Theory**Consider a one-dimensional 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:$k=frac\{2pi\}\{lambda\}=frac\{2pi\; n\}\{L\}$

where $lambda$ 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 $k\_\{max\}=pi/a$ 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": [

*Parker, p.7.*]:$g(k)dk\; =\; dn\; =\; frac\{L\}\{2pi\},dk$

Extending the analysis to

wavevector s in three dimensions the density of states in a box will be:$g(vec\{k\})d^3k\; =frac\{L^3\}\{(2pi)^3\},d^3k$

where $d^3k$ 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:$dE\; =\; frac\{partial\; E\}\{partial\; k\_x\}dk\_x\; +frac\{partial\; E\}\{partial\; k\_y\}dk\_y\; +frac\{partial\; E\}\{partial\; k\_z\}dk\_z\; =vec\{\; abla\}E\; cdot\; dvec\{k\}$

where $vec\{\; abla\}$ 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 (Ziman, 1972). The density of states as a function of this energy "E" is:

:$g(E)dE=int\_\{partial\; E\}g(vec\{k\}),d^3k\; =\; frac\{L^3\}\{(2pi)^3\}int\_\{partial\; E\}dk\_x,dk\_y,dk\_z$

where the integral is over the surface $partial\; E$ of constant "E". We can choose a new coordinate system $k\text{'}\_x,k\text{'}\_y,k\text{'}\_z,$ such that $k\text{'}\_z,$ 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

:$dk\text{'}\_x,dk\text{'}\_y,dk\text{'}\_z\; =\; dk\_x,dk\_y,dk\_z$

We can then write "dE" as:

:$dE=|vec\{\; abla\}E|,dk\text{'}\_z$

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

:$g(E)=frac\{L^3\}\{(2pi)^3\}intintfrac\{dk\text{'}\_x,dk\text{'}\_y\}$

where the $dk\text{'}\_x,dk\text{'}\_y$ term is an area element on the constant-"E" surface. The clear implication of the equation for $g(E)$ is that at the $k$-points where the

dispersion relation $E(vec\{k\})$ 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 $k$-points.A detailed analysis (Bassani 1975) 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 , alocal minimum or asaddle 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 sphericalfree electron Fermi surface :$E\; =\; hbar^2\; k^2/2m$ so that $|vec\{\; abla\}E|\; =\; hbar^2\; k/m\; =\; hbar\; sqrt\{\; frac\{2E\}\{m$.

In two dimensions the DOS is logarithmically divergent and in one dimension the DOS itself is infinite where $vec\{\; abla\}E$ is zero.

**Experimental observation**The optical absorption spectrum of a solid is most straightforwardly calculated from the

electronic band structure usingFermi's Golden Rule where the relevant matrix element to be evaluated is thedipole operator $vec\{A\}\; cdot\; vec\{p\}$ where $vec\{A\}$ is thevector potential and $vec\{p\}$ is themomentum 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 one-dimensional DOS might be expected to be a mathematical formality, but in fact they are readily observable. Highly anisotropic solids like

graphite (quasi-2D) andBechgaard salt s (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 NTs (SWNTs) which are also quasi-1D systems.**Notes****References***L. Van Hove, [

*http://dx.doi.org/10.1103/PhysRev.89.1189 "The Occurrence of Singularities in the Elastic Frequency Distribution of a Crystal,"*] Phys. Rev. 89, 1189–1193 (1953).

*cite book | last=Bassani | first=F. | coauthors = Pastori Parravicini, G. | title=Electronic States and Optical Transitions in Solids | publisher=Pergamon Press | year=1975 | id=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 comparisons for Ge andgraphite .

*cite book

first = John | last = Ziman | authorlink = John Ziman | year = 1972

title = Principles of the Theory of Solids | publisher = Cambridge University Press

id = ISBN B0000EG9UB

*M. A. Parker(1997-2004) [*http://www.ece.rutgers.edu/~maparker/classes/582-Chapters/Ch07-Sol-State-Carriers/Ch07S16DensityStates.pdf "Introduction to Density of States" "Marcel-Dekker Publishing"*] ]

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