Maclaurin's inequality

A Kohn anomaly is an anomaly in the dispersion relation of a phonon branch in a metal. For a specific wavevector, the frequency—and thus the energy—of the associated phonon is considerably lowered, and there is a discontinuity in its derivative. They have been first proposed by Walter Kohn in 1959. In extreme cases (that can happen in low-dimensional materials), the energy of this phonon is zero, meaning that a static distortion of the lattice appears. This is one explanation for charge density waves in solids. The wavevectors at which a Kohn anomaly is possible are the nesting vectors of the Fermi surface, that is vectors that connect a lot of points of the Fermi surface (for a one dimensional chain of atoms this vector would be $2k_F$).

In the phononic spectrum of a metal a Kohn anomaly is a discontinuity in the derivative of the dispersion relation that occurs at certain high symmetry points of the first Brillouin zone, produced by the abrupt change in the screening of lattice vibrations by conduction electrons. Kohn anomalies arise together with Friedel oscillations when one considers the Lindhard approximation instead of the Thomas-Fermi approximation in order to find an expression for the dielectric function of a homogeneous electron gas. The expression for the real part $\mathrm{\Re }(ϵ(\mathbf{q},\omega ))$ of the reciprocal space dielectric function obtained following the Lindhard model includes a logarithmic term that is singular at $\mathbf{q}=2{\mathbf{k}}_F$, where ${\mathbf{k}}_F$ is the Fermi wavevector. Although this singularity is quite small in reciprocal space, if one takes the Fourier transform and passes into real space, the Gibbs phenomenon causes a strong oscillation of $\mathrm{\Re }(ϵ(\mathbf{r},\omega ))$ in the proximity of the singularity mentioned above. In the context of phonon dispersion relations, these oscillations appear as a vertical tangent in the plot of ${\omega }^2(\mathbf{q})$, the so-called Kohn anomalies.

Many different systems exhibit Kohn anomalies, including graphene, bulk metals, and many low-dimensional systems (the reason involves the condition $\mathbf{q}=2{\mathbf{k}}_F$, which depends on the topology of the Fermi surface). However, it is important to emphasize that only materials showing metallic behaviour can exhibit a Kohn anomaly, as we are dealing with approximations that need a homogeneous electron gas.

Notes

For experimental results, one can turn to Observation of Giant Kohn Anomaly in the One-Dimensional Conductor K2Pt(CN)4Br0.3· 3H2O, Renker et al., Phys. Rev. Lett. 30, 1144