Maclaurin's inequality: Difference between revisions
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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.<ref>W. Kohn, Image of the [[Fermi surface]] in the vibration spectrum of a metal, ''Phys. Rev. Lett'' '''2''', 393 (1959)</ref> 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 [[Spin density wave|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 <math>2k_F</math>). | |||
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]] <math> \operatorname{Re}(\epsilon (\mathbf{q}, \omega)) </math> of the [[Reciprocal lattice#Reciprocal space|reciprocal space]] [[dielectric function]] obtained following the Lindhard model includes a logarithmic term that is singular at <math> \mathbf{q} = 2\mathbf{k}_F </math>, where <math> \mathbf{k}_F </math> is the [[Fermi energy#Related quantities|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 <math> \operatorname{Re}(\epsilon (\mathbf{r}, \omega)) </math> in the proximity of the singularity mentioned above. In the context of phonon [[dispersion relation]]s, these oscillations appear as a vertical [[tangent]] in the plot of <math> \omega ^2(\mathbf{q}) </math>, the so-called Kohn anomalies. | |||
Many different systems exhibit Kohn anomalies, including [[graphene]],<ref name="kohngraphene_prl2004">S. Piscanec, M. Lazzeri, F. Mauri, A. C. Ferrari, and J. Roberston, [http://dx.doi.org/10.1103/PhysRevLett.93.185503 Kohn Anomalies and Electron-Phonon Interactions in Graphite], ''Phys. Rev. Lett.'', '''93''', 185503 (2004)</ref> bulk metals,<ref name="pdkohn">D. A. Stewart, [http://dx.doi.org/10.1088/1367-2630/10/4/043025 Ab initio investigation of phonon dispersion and anomalies in palladium], ''New J. Phys.'', '''10''', 043025 (2008) ''Open Access article''</ref> and many [[low-dimensional systems]] (the reason involves the condition <math> \mathbf{q} = 2 \mathbf{k}_F </math>, 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.<ref>'''R. M. Martin''', ''Electronic Structure, Basic Theory and Practical Methods'', Cambridge University Press, 2004, ISBN 0-521-78285-6</ref> | |||
==Notes== | |||
<references/> | |||
For experimental results, one can turn to | |||
[http://prl.aps.org/abstract/PRL/v30/i22/p1144_1 Observation of Giant Kohn Anomaly in the One-Dimensional Conductor K2Pt(CN)4Br0.3· 3H2O, Renker ''et al.'', ''Phys. Rev. Lett.'' 30, 1144] | |||
{{DEFAULTSORT:Kohn Anomaly}} | |||
[[Category:Condensed matter physics]] | |||
Revision as of 12:13, 1 March 2013
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.[1] 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 ).
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 of the reciprocal space dielectric function obtained following the Lindhard model includes a logarithmic term that is singular at , where 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 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 , the so-called Kohn anomalies.
Many different systems exhibit Kohn anomalies, including graphene,[2] bulk metals,[3] and many low-dimensional systems (the reason involves the condition , 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.[4]
Notes
- ↑ W. Kohn, Image of the Fermi surface in the vibration spectrum of a metal, Phys. Rev. Lett 2, 393 (1959)
- ↑ S. Piscanec, M. Lazzeri, F. Mauri, A. C. Ferrari, and J. Roberston, Kohn Anomalies and Electron-Phonon Interactions in Graphite, Phys. Rev. Lett., 93, 185503 (2004)
- ↑ D. A. Stewart, Ab initio investigation of phonon dispersion and anomalies in palladium, New J. Phys., 10, 043025 (2008) Open Access article
- ↑ R. M. Martin, Electronic Structure, Basic Theory and Practical Methods, Cambridge University Press, 2004, ISBN 0-521-78285-6
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