Van-Hove-Singularität

A 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 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 Dutch physicist Léon van Hove in 1953 for the case of phonon densities of states.

Theory

A particle in an interval of length $L$ must have wavevector ${\frac {2\pi n}{L}}$ where $n$ is an integer. In a box of volume $V$ there are by extension ${\frac {V}{(2\pi )^{3}}}$ possible $k$ vectors or ${\frac {2}{(2\pi )^{3}}}$ electron states, where the factor of 2 in the numerator allows for two possible spins. Therefore the density of states in k-space $g(k)={\frac {2}{(2\pi )^{3}}}$. The DOS in energy space can be derived by considering the Taylor expansion $E=E_{0}+{\vec {\nabla }}E\cdot d{\vec {k}}$. Then $d{\vec {k}}=dE/{\vec {\nabla }}E$. Since for a given energy band $i$, $g_{i}(E)=\int d{\vec {k}}\,g(k)\,\delta \left(E-E_{i}(k)\right)$, it follows that

$g_{i}(E)={\frac {1}{4\pi ^{3}}}\int _{E=E_{i}}{\frac {dE}{{\vec {\nabla }}E}}$. Equation 1

Note that the restriction $E=E_{i}$ constrains the integral to a constant energy surface. Equation 1 may also be simply derived without the Taylor expansion by noting $\int _{-\infty }^{\infty }dx\,f(x)\,\delta (g(x))=\sum _{i}{\frac {f(x_{i})}{|g'(x_{i})|}}$ where x_i are the roots of g(x) (see thedelta function article).

The clear implication of Equation 1 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 (see Bassani book cited below) shows that there are three types of van Hove singularities in three-dimensional space, as illustrated in the Figure. In three dimensions, the DOS itself is not divergent although its derivative is. In two dimensions the DOS is logarithmically divergent and in one dimension the DOS itself is infinite where ${\vec {\nabla }}E$ 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 to be evaluated is the dipole operator ${\vec {A}}\cdot {\vec {p}}$ where ${\vec {A}}$ is the vector potential and ${\vec {p}}$ 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 JDOS exhibits singular behavior wherever the individual DOS of either band has a van Hove singularity.

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) and Bechgaard salts (quasi-1D) show anomalies in spectroscopic measurements that are attributable to the van Hove singularities.

External links

• van Hove's original paper
• Vorlage:Book reference This book contains an exhaustive but accessible discussion of optical absorption with extensive comparison between calculations and experiment.