Van Hove singularity

A Van Hove singularity is a discontinuity ("kink") in the density of states of solids . The most common use of the Van Hove Singularity concept occurs in the analysis of optical absorption spectra . The singularities are named after the Belgian physicist Léon Van Hove , who first described the phenomenon in 1953 for the density of states of phonons .

theory

If one considers a one-dimensional lattice, i.e. a chain of length of particles, with neighboring particles at a distance , the result for the magnitude of the wave vector of a standing wave is an expression of the form: ${\displaystyle L}$ ${\displaystyle N}$ ${\displaystyle a}$ ${\displaystyle k}$

${\displaystyle k={\frac {2\pi }{\lambda }}={\frac {2\pi n}{L}}}$

where the wavelength and is an integer. The smallest possible wavelength is . This corresponds to the largest possible wave number and corresponds to the maximum . The density of states is now defined in such a way that there is the number of standing waves whose wave vector lies in the interval from to : ${\displaystyle \lambda }$ ${\displaystyle n}$ ${\displaystyle 2a}$ ${\displaystyle k_{\mathrm {max} }=\pi /a}$ ${\displaystyle |n|:n_{\mathrm {max} }=L/2a}$ ${\displaystyle g(k)}$ ${\displaystyle g(k)dk}$ ${\displaystyle k}$ ${\displaystyle k+dk}$

${\displaystyle g(k)dk={\frac {dn}{2n_{\mathrm {max} }}}={\frac {a}{2\pi }}\,dk}$

If one extends the consideration to three dimensions, the result is:

${\displaystyle g({\vec {k}})d^{3}k={\frac {a^{3}}{(2\pi )^{3}}}\,d^{3}k}$

where a volume element is in space. ${\displaystyle d^{3}k}$ ${\displaystyle k}$

Transition to the density of states per energy

The chain rule applies

${\displaystyle dE={\frac {\partial E}{\partial k_{x}}}dk_{x}+{\frac {\partial E}{\partial k_{y}}}dk_{y}+{\frac {\partial E}{\partial k_{z}}}dk_{z}={\vec {\nabla }}E\cdot d{\vec {k}}}$,

where the gradient is in -space. The set of points in space that correspond to a certain energy form a surface in space; the gradient of is perpendicular to this plane at every point. For the density of states as a function of we get: ${\displaystyle {\vec {\nabla }}}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle E}$ ${\displaystyle k}$ ${\displaystyle E}$ ${\displaystyle E}$

${\displaystyle g(E)dE=\int _{\partial E}g({\vec {k}})\,d^{3}k={\frac {a^{3}}{(2\pi )^{3}}}\int _{\partial E}dk_{x}\,dk_{y}\,dk_{z}}$

where the integral over the surface is to be formed with a constant . Now you introduce coordinates with which is perpendicular to the surface. After this change of coordinates: ${\displaystyle \partial E}$ ${\displaystyle E}$ ${\displaystyle k'_{x},k'_{y},k'_{z}\,}$ ${\displaystyle k'_{z}\,}$

${\displaystyle dE={\vec {\nabla }}E\cdot d{\vec {k}}={\vec {\nabla }}E\cdot d{\vec {k'}}=|{\vec {\nabla }}E|\,dk'_{z}}$.

Inserted into the expression for , we get: ${\displaystyle g(E)}$

${\displaystyle g(E)={\frac {a^{3}}{(2\pi )^{3}}}\ \int \!\int {\frac {dk'_{x}\,dk'_{y}}{|{\vec {\nabla }}E|}}}$

where the term corresponds to a surface element on the equi-energy surface ( const.). ${\displaystyle dk'_{x}\,dk'_{y}}$ ${\displaystyle E=}$

The singularities

g (E) versus E for a simulated three-dimensional solid.

The density of states diverges at points in space at which vanishes and the dispersion relation has an extreme . These points are called Van Hove singularities . ${\displaystyle k}$ ${\displaystyle {|{\vec {\nabla }}E|}}$ ${\displaystyle g(E)}$

A detailed analysis (Bassani 1975) shows that there are four types of Van Hove singularities in three dimensions. These differ in terms of whether the band has a local maximum , a local minimum or a saddle point of the first or second type. The function tends to square-root-like singularities in three dimensions due to the spherical shape of the Fermi surfaces for free electrons . Therefore, although its derivative diverges, the density of states does not diverge, as can be seen in the figure. ${\displaystyle g(E)}$

${\displaystyle E=\hbar ^{2}k^{2}/2m\ }$so that . ${\displaystyle \ |{\vec {\nabla }}E|=\hbar ^{2}k/m=\hbar {\sqrt {\frac {2E}{m}}}}$

In two dimensions the density of states diverges logarithmically, in one dimension it becomes infinite when it is zero. ${\displaystyle {\vec {\nabla }}E}$

literature

• Léon Van Hove: The Occurrence of Singularities in the Elastic Frequency Distribution of a Crystal . In: Physical Review . tape 89 , no. 6 , 1953, pp. 1189–1193 , doi : 10.1103 / PhysRev.89.1189 .
• F. Bassani, G. Pastori Parravicini: Electronic States and Optical Transitions in Solids . Pergamon Press, 1975, ISBN 0-08-016846-9 (with detailed discussion of the different types of van Hove singularities in different dimensions and comparison with experiments on germanium and graphite).
• John Ziman: Principles of Solid State Theory . German, Zurich / Frankfurt am Main 1975, ISBN 3-87144-148-1 (English: Principles of the theory of solids . 1972.).

Individual evidence

1. ^ Léon Van Hove: The Occurrence of Singularities in the Elastic Frequency Distribution of a Crystal . In: Physical Review . tape 89 , no. 6 , 1953, pp. 1189–1193 , doi : 10.1103 / PhysRev.89.1189 .