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 electrons experience weak periodic potential, the interaction of an electron with all other electrons and nuclei is represented by ‘effective potential’ in some average way, in the tight-binding approximation, electrons move in strong periodic potentials which cannot be approximated by an average background. In this situation, we can assume that the atoms are very widely separated and atomic orbitals remain undistorted.
As an example, the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^+_2} problem will be revisited in the second quantization language. States Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1>} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |2>} denote the states of electrons locate on atom 1 and 2, respectively. The state of the two-electron system is thus given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |n_1, n_2>} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |n_i>} is the occupation number for each atom. Define creation and annihilation operators Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c^+_i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_i} , which obey anti-commutation relations: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {c_i, c^+_j}=\delta_i_,_j}
Phonons: crystal vibrations
Lattice Vibration and Phonons in 1D
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^0_{ph} = \sum_{k \lambda}\hbar\omega_{k \lambda}( a^\dagger_{k \lambda} + \frac{1}{2} )}
Acoustical and Optical Phonon in 3D
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^0_{ph} = \sum_{\mathbf{k} \lambda} \hbar \omega_{\mathbf{k} \lambda}( a^\dagger_{\mathbf{k} \lambda} + \frac{1}{2} ), :[a_{\mathbf{k} \lambda} , a^\dagger_{\mathbf{k} \lambda}] = \delta_{\mathbf{k,k^\prime}} \delta_{ \lambda , \lambda^\prime} }
Derivation of Hamiltonian Electron-Phonon Coupling
The Hamiltonian for the electron-phonon interaction can be described as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = H^0_{el} + H^0_{ph} + H_{coul} + H_{int}}
Where
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^0_{el} = \sum_{k \sigma}E_k c^\dagger_{k \sigma} c_{k \sigma}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^0_{ph} = \sum_{k \lambda}\omega_{k \lambda}( a^\dagger_{k \lambda} + \frac{1}{2} )}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{coul} = \frac{1}{2} \sum_{k k^\prime q \atop \sigma \sigma^{\prime} } V(q)c^\dagger_{k ^\prime + q \sigma^\prime } c^\dagger_{k \sigma} c_{k+q \sigma} c_{k^{\prime} \sigma^{\prime}} }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{int} = \sum_{k k^{\prime} \sigma \lambda} g_{k k^{\prime} c^{\dagger}_{k \sigma}} c_{k^{\prime} \sigma} c_{k^\prime \sigma} ( a^\dagger_{-q \lambda} + a_{q \lambda} ) }