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

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

The state of the system is thus given by $|n_{A},n_{B}>$.

It is easy to generalize the above single molecule problem to electrons moving in crystal. Assume that the tight binding approximation is valid and only one atomic orbital (say, orbital A) is taken into consideration. Moreover, assume that only nearest neighbor hopping term is significant. The Hamiltonian of the system is given by

$H=E_{A}\sum _{i,\sigma }c_{i,\sigma }^{+}-t_{A}\sum _{<i,j>,\sigma }(c_{i,\sigma }^{+}c_{j,\sigma }+c_{j,\sigma }^{+}c_{i,\sigma })$

where $<i,j>$ means that the summation goes over only nearest neighbor sites. $R_{i}$ is the site position and $c_{i,\sigma }$ is the annihilation operator which destroys one electron in orbital A at site i with spin $\sigma$. Apply Fourier transformation on electron operator

$c_{{\vec {p}},\sigma }={\frac {1}{\sqrt {N}}}\sum _{j}e^{i{\vec {p}}\cdot {\vec {R_{j}}}}c_{j,\sigma }$

The momentum ${\vec {p}}$ is defined in the first Brillouin zone. Hence, the Hamiltonian can be written in terms of the operators in momentum space as

$H=\sum _{{\vec {p}},\sigma }E_{\vec {k}}c_{{\vec {k}},\sigma }^{+}c_{{\vec {k}},\sigma }$

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

Screening

What is a free electron gas? A metal can be considered as an ionic lattice embedded inside an electron sea. The periodic ion lattice causes a periodic potential. In the free electron model we treat the ions as a uniform background of positive charge.

Screening is an important phenomena in a free electron gas. When an external positive charge density $\rho _{e}xt\left(\mathbf {r} \right)$

is applied, the electrons will be attracted to surround the positive charge (Fig. 1). The re-arrangement of the electrons generates an induced charge distribution ρind(r). Therefore, the total charge density ρ(r) = ρext(r)+ρind(r) is less positive than ρext(r). Thus, The total potential φ(r) is weaker than the external potential φext(r) caused by the positive charge only. The phenomenon is called screening.

Kohn anomalies

the coupling function refers to the scattering of a quasi-particle from a point $\mathbf {k} _{1}$ to a point $\mathbf {k} _{2}$ in momentum space with $\mathbf {p} =\mathbf {k} _{1}-\mathbf {k} _{2}$. Energy and momentum conservation require that both $\mathbf {k} _{1}$ and $\mathbf {k} _{2}$ lie on the Fermi surface. This immediately introduces a restriction on the phonon wave vector $\mathbf {q}$: 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.

$\mathbf {p} =\mathbf {k} _{1}-\mathbf {k} _{2}$