Electron-phonon interactions and Kohn anomalies

Electron-phonon interactions

The study of interactions between electrons and phonons, is an interesting and classical topic in quantum many body theory as well as condensed matter physics. The electron-phonon interaction leads to many novel properties in metals, for instance, electrical resistance, thermal resistance, superconductivity and the renormalization of linear electronic specific heat. [1]

Free electrons in lattice

In contrary to the independent electron model, where In the tight-binding approximation,

Lattice Vibration and Phonons in 1D

$H_{ph}^{0}=\sum _{k\lambda }\omega _{k\lambda }(a_{k\lambda }^{\dagger }+{\frac {1}{2}})$

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Acoustical and Optical Phonon in 3D

$H_{ph}^{0}=\sum _{k\lambda }{\bar {h}}\omega _{k\lambda }(a_{k\lambda }^{\dagger }+{\frac {1}{2}})$

Derivation of Hamiltonian Electron-Phonon Coupling

The Hamiltonian for the electron-phonon interaction can be described as

$H=H_{el}^{0}+H_{ph}^{0}+H_{coul}+H_{int}$

Where

$H_{el}^{0}=\sum _{k\sigma }E_{k}c_{k\sigma }^{\dagger }c_{k\sigma }$

$H_{ph}^{0}=\sum _{k\lambda }\omega _{k\lambda }(a_{k\lambda }^{\dagger }+{\frac {1}{2}})$

$H_{coul}={\frac {1}{2}}\sum _{kk^{\prime }q \atop \sigma \sigma ^{\prime }}V(q)c_{k^{\prime }+q\sigma ^{\prime }}^{\dagger }c_{k\sigma }^{\dagger }c_{k+q\sigma }c_{k^{\prime }\sigma ^{\prime }}$

$H_{int}=\sum _{kk^{\prime }\sigma \lambda }g_{kk^{\prime }c_{k\sigma }^{\dagger }}c_{k^{\prime }\sigma }c_{k^{\prime }\sigma }(a_{-q\lambda }^{\dagger }+a_{q\lambda })$

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