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 $H_{2}^{+}$ problem will be revisited in the second quantization language. States $|1>$ and $|2>$ denote the states of electrons locate on atom 1 and 2, respectively. The state of the two-electron system is thus given by $|n_{1},n_{2}>$, where $|n_{i}>$ is the occupation number for each atom. Define creation and annihilation operators $c_{i}^{+}$ and $c_{i}$, which obey anti-commutation relations:

$\left\{c_{i},c_{j}^{+}\right\}=\delta _{i},j$,

$\left\{c_{i},c_{j}\right\}=0$, $\left\{c_{i}^{+},c_{j}^{+}\right\}=0$.

The Hamiltonian has the following form:

$H=E_{0}(c_{1}^{+}c_{1}+c_{2}^{+}c_{2})-t(c_{1}^{+}c_{2}+c_{2}^{+}c_{1})$

Note that this Hamiltonian is non-diagonal. In order to diagonalize it, we could define the following bonding and anti-bonding operators:

$c_{B}=1/{\sqrt {2}}(c_{1}+c_{2})$

$c_{A}=1/{\sqrt {2}}(c_{1}-c_{2})$

Both bonding and anti-bonding operators obey anti-commutation relations. The Hamiltonian can be expressed by these operators as

$H=(E_{0}+t)c_{A}^{+}c_{A}+(E_{0}-t)c_{B}^{+}c_{B}$

Lattice Vibration and Phonons in 1D

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

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Acoustical and Optical Phonon in 3D

In moatomic Bravais lattice.

$U\approx U_{0}+{\frac {1}{2}}\sum _{\mathbf {R} _{1}^{0}\mathbf {R} _{2}^{0}}\sum _{\alpha \beta }u_{\alpha }(\mathbf {R} _{1}^{0}){\frac {\partial ^{2}U}{\partial u_{\alpha }(\mathbf {R} _{1}^{0})\partial u_{\beta }(\mathbf {R} _{2}^{0})}}|_{\mathbf {u} =0}u_{\beta }(\mathbf {R} _{2}^{0})$

$H_{ph}^{0}=\sum _{\mathbf {k} \lambda }\hbar \omega _{\mathbf {k} \lambda }(a_{\mathbf {k} \lambda }^{\dagger }+{\frac {1}{2}}),[a_{\mathbf {k} \lambda },a_{\mathbf {k} \lambda }^{\dagger }]=\delta _{\mathbf {k,k^{\prime }} }\delta _{\lambda ,\lambda ^{\prime }}$

Derivation of Hamiltonian Electron-Phonon Coupling

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

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

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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Kohn anomalies

the coupling function refers to the scattering of a quasi-particle from a point $'''k_{1}'''$ to a point k_2 in momentum space with $'''q'''='''k'''_{1}-'''k'''_{2}$ . Energy and momentum conservation require that both and lie on the Fermi surface. This immediately introduces a restriction on the phonon wave vector  : phonon wave vectors connecting nested parts of the Fermi surface will strongly interact with the electrons leading to a large phonon damping, whereas those which do not span the Fermi surface will not interact with electrons. Kohn pointed out that the interaction of phonons with the conduction electrons in a metal should cause anomalies in the phonon spectra.

$k^{2},$ $f(x)=x^{2}\,$