Electron-phonon interactions and Kohn anomalies

Electron-Phonon Interactions

A crystal is a complicated many-body system of electrons and ions that interacts with each other by Coulomb interaction. Although we can write dowm the exact Hamiltonian of the system, it is impossible to find the exact solution to it. Proper approximations have to be used and Hamiltonian has to be divided into an electronic part, a lattice part and interaction terms. The interaction term represents the interactions of the electrons and the quanta of the lattice vibrations, the phonons.

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.

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 \left\{c_i, c^+_j\right\}=\delta_i,j} ,

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 \left\{c_i, c_j\right\}=0} , 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 \left\{c^+_i, c^+_j\right\}=0} .

The Hamiltonian has the following form:

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=E_0(c^+_1c_1+c^+_2c_2)-t(c^+_1c_2+c^+_2c_1)}

Note that this Hamiltonian is non-diagonal. In order to diagonalize it, we could define the following bonding and anti-bonding 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_B=1/\sqrt{2}(c_1+c_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 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

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=(E_0+t)c^+_Ac_A+(E_0-t)c^+_Bc_B}

The state of the 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_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

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

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_{\vec{p},\sigma}=\frac{1}{\sqrt{N}}\sum_{j} e^{i\vec{p}\cdot\vec{R_j}}c_{j,\sigma}}

The momentum 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 \vec{p}} is defined in the first Brillouin zone. Hence, the Hamiltonian can be written in terms of the operators in momentum space 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=\sum_{\vec{p},\sigma}E_{\vec{k}}c^+_{\vec{k},\sigma}c_{\vec{k},\sigma}}

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Phonons: lattice vibrations

In real crystals, atoms are not fixed at rigid sites on a lattice, but vibrate. In a periodic structure the vibrations have a waveform (just like electronic wavefunctions) with a spatial and temporal part:

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 u \left(r, t\right) = u_0 exp \left(ikr \right) exp \left( -i \omega t\right)}

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 k=\frac {2 \pi}{\lambda}} ; 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 \hbar \omega} represents a "quantum" of vibration energy.

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 \omega} vs. 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 k} is the dispersion relation.

Lattice vibrations and phonons in 1D

As a simple example, let us consider a 1D chain of atoms coupled to their nearest neighbors within harmonic approximation. The potential energy is 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 U=\frac{1}2C\sum_n(x_{n+1}-x_n-a)^2}

Source: S. M. Girvin, A. H. MacDonald and Kun Yang

where a is the equilibrium distance between to atoms. Since 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 u_n=x_n-R_n} , we have

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 U=\frac{1}2C\sum_n(u_{n+1}-u_n)^2}

After taking derivatives, we can get the elastic tensor expressed 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 D_{jk}=\frac{\partial^2U}{\partial{u}_k\partial{u}_j}=C(2\delta_{jk}-\delta_{k,j-1}-\delta_{k,j+1})}

Do the Fourier transformation we have <br\>

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 D \left(K \right)= 2C \left( 1-cosKa \right) }

Thus, 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 M\omega^2=D\left(K\right)} , with the solution of

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 \omega(K)=\sqrt{\frac{C}{M}}\sqrt{2(1-cosKa)}=2\sqrt{\frac{C}{M}}|sin\frac{Ka}2|}

Source: Charles Kittel

In the long-wavelength limit, the vibration frequency vanishes linearly as a function of K. The speed of sound wave, which is the slope of the dispersion relation around 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 K\rightarrow{0}} , is 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 \frac{\partial{\omega}}{\partial{K}}=\sqrt{\frac{C a^2}{M}}}

The Hamiltonian is

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_{ph} = \sum^N_{j=1}[\frac{1}{2M}p^2_j + \frac{1}{2}C(u_j-u_{j-1})^2]}

Use the Fourier transforms variables :

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 p_j \equiv \frac{1}{\sqrt{N}}\sum_{k\in FBA} p_ke^{ikR^0_j}}

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 u_j \equiv \frac{1}{\sqrt{N}}\sum_{k\in FBA} u_ke^{ikR^0_j}}

The Hamiltonian becomes

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_{ph} = \sum_k[\frac{1}{2M}p_kp_{-k} + \frac{1}{2}M\omega^2_ku_ku_{-k}]}

combine 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 u_k} 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 p_k} into annihilation and creation 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 b_k } 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 b^\dagger_{-k} }

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 b_k \equiv \frac{1}{\sqrt{2}} ( \frac{u_k}{l_k} + i \frac{p_k}{\hbar / l_k} )} ,

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 b^\dagger_{-k} \equiv \frac{1}{\sqrt{2}} ( \frac{u_k}{l_k} - i \frac{p_k}{\hbar / l_k} )} ,

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 l_k = \sqrt{\frac{\hbar}{M\omega_k}} }

In general case, we introduce index 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 \lambda} as the different branches. The Hamiltonian becomes

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_{ph} = \sum_{k \lambda}\hbar\omega_{k \lambda}( b^\dagger_{k \lambda} b_{k \lambda} + \frac{1}{2} ) }

$[b_{k_{1}\lambda _{1}},b_{k_{2}\lambda _{2}}^{\dagger }]=\delta _{k_{1},k_{2}},\delta _{k_{1},k_{2}}$

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Acoustical and optical phonons in 3D

Lattice vibrations in a 3D crystal is a realistic model for the study of sound waves. Dynamics of lattice vibration generates phonon propagation in crystal. Let us first ignore the the discret properties of crystal and treat it as a continuous media. Let ${\vec {u}}({\vec {r}})$ be the displacement field which describes the elastic deformations of media. For small fluctuations, the kinetic energy is given by

$T={\frac {1}{2}}\rho _{0}\int {d^{3}{\vec {r}}|{\dot {\vec {u}}}({\vec {r}})|^{2}}$

where $\rho _{0}$ is the mass density. The potential energy is given by

$V={\frac {1}{2}}\rho _{0}\int {d^{3}{\vec {r}}}\{\Gamma _{L}({\vec {\nabla }})^{2}+\Gamma _{T}{\vec {\nabla }}\times {\vec {u}}|^{2}\}$

The first term is from longitudinal contribution and the second term is from transverse contribution. However, these expressions are too crude to describe lattice vibrations in general.

If we abandon the ‘continuum’ frame and treat the crystal as a discrete lattice system, the Hamiltonian has the following form

Failed to parse (syntax error): {\displaystyle H=\sum_{j}\frac{P^2_j}{2M}+U(\vec{r}_1,…,\vec{r}_N)}

where j denotes the lattice sites and ${\vec {r}}_{j}={\vec {R}}_{j}+{\vec {u}}_{j}$. Assume that the adiabatic approximation is valid, due to the more massive nuclei than electrons. In this sense, the electrons could follow the movement of nuclei quite fast as if they are fixed on nuclei and move together. Within harmonic approximation, we could expand the potential energy U in Taylor series up to the second order:

Failed to parse (syntax error): {\displaystyle U(\vec{r}_1,…,\vec{r}_N)=U(\vec{R}_1,…,\vec{R}_N)+\sum^{N}_{j=1}\vec{u}_j\cdot\vec{\nabla}_jU|_{\vec{r}=\vec{R}}+\frac{1}2\sum_{j,k}(\vec{u}_j\cdot\vec{\nabla}_j)(\vec{u}_k\cdot\vec{\nabla}_k)U|_{\vec{r}=\vec{R}}+\cdot\cdot\cdot }

Since the atoms do not experience net force at the equilibrium positions, the first order linear term vanishes. Hence,

$U-U_{eq}={\frac {1}{2}}{\vec {u}}_{j}\cdot {\vec {D}}_{jk}\cdot {\vec {u}}_{k}={\frac {1}{2}}u_{j}^{\mu }D_{jk}^{\mu \nu }u_{k}^{\nu }$

where the elastic tensors are defined as

${\vec {D}}_{jk}={\vec {\nabla }}_{j}{\vec {\nabla }}_{k}U$

${\vec {D}}_{jk}^{\mu \nu }=\partial _{r_{j}^{\mu }}\partial _{r_{k}^{\nu }}U$

For a Bravais lattice, the Hamiltonian can be expressed as a set of coupled harmonic oscillators:

$H_{L}=\sum _{j}{\frac {P_{j}^{2}}{2M}}+{\frac {1}{2}}\sum _{j,k}{\vec {u}}_{j}\cdot {\vec {D}}_{jk}\cdot {\vec {u}}_{k}$

We can use classical approach to analyze the normal modes of lattice vibrations. The classical equation of motion is

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 M\ddot{\vec{u}}_j=-\vec{\nabla}_jU=-\vec{D}_{jk}\cdot\vec{u}_k}

It is reasonable to assume that the solution of the above equation has the plane wave form

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 \vec{u}_j=e^{-i\omega t}e^{i\vec{K}\cdot\vec{R}_j}\hat{\epsilon}(\vec{K})\tilde{u}(\vec{K})}

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 \hat{\epsilon}} is the polarization vector 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 \tilde{u}} is the amplitude. Substitute it to the equation of motion, we can get

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 -M\omega^2\tilde{\epsilon}(\vec{K})=-\sum_k\vec{D}_{jk}\cdot\hat{\epsilon}(\vec{K})e^{i\vec{K}\cdot(\vec{R}_k-\vec{R}_j)}}

The Fourier transform of 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 \vec{D}} is

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 \vec{D}(\vec{K})\equiv\sum_k\vec{D}_{jk}e^{i\vec{K}\cdot(\vec{R}_k-\vec{R}_j)}}

Let's introduce the branch index 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 \lambda } to label different branches. The polarization vectors are eigenvectors for the diagonalizied 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 \vec{D}(\vec{K})} ;

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 \mathbf{D}(\mathbf{k})\mathbf{\epsilon}_{\mathbf{k}_\lambda} = K_{\mathbf{k}_\lambda} \mathbf{\epsilon}_{\mathbf{k}_\lambda}}

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 \mathbf{\epsilon}_{\mathbf{k}\lambda}\cdot \mathbf{\epsilon}_{\mathbf{k}\lambda^\prime} = \delta_{\lambda,\lambda^\prime} }

Source: Henrik Bruus, Karsten Flensberg

Thus, the equation of motion can be written 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 M\omega^2\mathbf{\epsilon}_{\mathbf{k}\lambda} = \mathbf{D}(\mathbf{k})\mathbf{\epsilon}_{\mathbf{k}_\lambda} = K_{\mathbf{k}_\lambda} \mathbf{\epsilon}_{\mathbf{k}_\lambda}}

The solution for this problem is

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 \mathbf{u}_{\mathbf{k} \lambda }( \mathbf{R}^0_1, t) = \mathbf{\epsilon}_{\mathbf{k}\lambda}e^{i(\mathbf{k\cdot R}^0-\omega_{\mathbf{k}\lambda }t)} , \omega_{\mathbf{k}\lambda} \equiv \sqrt{\frac{K_{\mathbf{k}\lambda}}{M}} }

Use second quantization,

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 \mathbf{u}_{\mathbf{k}\lambda} \equiv l_{\mathbf{k} \lambda}\frac{1}{\sqrt{2}} ( b^\dagger_{-\mathbf{k},\lambda} + b_{-\mathbf{k},\lambda} )\mathbf{\epsilon}_{\mathbf{k}\lambda} }

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 l_{\mathbf{k}\lambda} \equiv \sqrt{\frac{\hbar}{M \omega_{\mathbf{k}\lambda}}} }

Finally,the Hamiltonian can be written in terms of the 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 H^0_{ph} = \sum_{\mathbf{k} \lambda} \hbar \omega_{\mathbf{k} \lambda}( b^\dagger_{\mathbf{k} \lambda} b_{\mathbf{k} \lambda} + \frac{1}{2} ), [b_{\mathbf{k} \lambda} , b^\dagger_{\mathbf{k} \lambda}] = \delta_{\mathbf{k,k^\prime}} \delta_{ \lambda , \lambda^\prime} }

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Feynman diagrams of electron-phonon coupling

For a system composed of N electrons, the Hamiltonian for electron-phonon interaction has the general form of

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_{ep}=\sum_k\epsilon_kc^+_kc_k+\sum_q\omega_qb^+_qb_q+\sum_{kq}M_qc^+_kc_{k-q}(b^+_{-q}+b_q)}

Introduce an external phonon source term 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 J_{-q}\phi_q} to the Hamiltonian:

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_{ep}=\sum_k\epsilon_kc^+_kc_k+\sum_q\omega_qb^+_qb_q+\sum_{kq}M_qc^+_kc_{k-q}\phi_q+\sum_qJ_{-q}\phi_q}

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 \phi_q=b^+_{-q}+b_q}

The Green function is 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 G(k,t;k\prime,t\prime)=-i<T\{c_k(t)c^+_{k\prime}(t\prime)\}>}

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 D(q,t;k\prime,t\prime)=\frac{\delta<\phi_q(t)>}{\delta J_{q\prime}(t\prime)}=-i(<T\{\phi_q(t)\phi^+_{q\prime}(t\prime)\}>-<\phi_q(t)><\phi^+_{q\prime}(t\prime)>)}

Based on the equation of motion for electron annihilation operator, we can get a differential equation for 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 G(k,t;k\prime,t\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 [i\frac{\partial}{\partial t}-\epsilon_k]G(k,t;k\prime,t\prime)+i\sum_qM(q)<T\phi_q(t)c_{k-q}(t)c^+_{k\prime}(t\prime)>=\delta_{kk\prime}\delta(t-t\prime)}

The expectation value of the time-ordered product is 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 -i<T\phi_q(t)c_{k-q}(t)c^+_{k\prime}(t\prime)>=\frac{\delta<Tc_{k-q}(t)c^+_{k\prime}(t\prime)>}{\delta J_{-q}(t\prime\prime)}-i<\phi_q{t\prime\prime}><Tc_{k-q}(t)c^+_{k\prime}(t\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 =\frac{\delta G(k,t;k\prime,t\prime)}{\delta J_{-q}(t\prime\prime)}-i<\phi_q(t\prime\prime)>G(k,t;k\prime,t\prime)}

Along with the expression for 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 D(q,t;k\prime,t\prime)} , we have

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 [i\frac{\partial}{\partial t}-\epsilon_k]G(k,t;k\prime,t\prime)-\sum_qM(q)<\phi_q(t)>G(k-q,t;k\prime,t\prime)+i\sum_qM(q)G(k-q,t;k_1,t_1)\frac{\delta G^{-1}(k_1,t_1;k_2,t_2)}{\delta<\phi(q_3,t_3)>}G(k_2,t_2;k\prime,t\prime)D(q_3,t_3,-q,t) }

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 =\delta_{k,k\prime}\delta(t-t\prime)}

The free electron Green’s function is 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 G^{-1}_0(k,t;k_2,t_2)=[[i\frac{\partial}{\partial t}-\epsilon_k]\delta_{k_2,k}-M(k-k_2)<\phi_{k-k_2(t)}>]\delta(t_2-t)}

The vertex function is

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 \Gamma(k_1,t_1,k_2,t_2,q_3,t_3)=-\frac{1}{M(q_3)}\frac{\delta G^{-1}(k_1,t_1;k_2,t_2)}{\delta<\phi(q_3,t_3)>}}

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 \Sigma(k,t,k_2,t_2)=i\int dt_1dt_3\sum_{q,k_1,q_3}M(q)M(q_3)G(k-q,t,k_1,t_1)\Gamma(k_1,t_1,k_2,t_2,q_3,t_3)D(q_3,t_3;-q,t)}

If the fact of momentum conservation is taken into consideration, we have

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 G \left(k \right)=G_0\left(k\right)+G_0\left(k\right)\Sigma\left(k\right)G\left(k\right)}

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 D\left(q\right)=D_0\left(q\right)+D_0\left(q\right)\Pi\left(q\right)D\left(q\right)}

Via the vertex function 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 \Gamma} , self energies can be expressed 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 \Sigma(k)=i\int\frac{dq^4}{(2\pi)^3}|M(q)|^2G(k+q)\Gamma(k,q)D(q)}

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 \Pi(q)=-i2|M(q)|^2\int\frac{dk^4}{(2\pi)^3}G(k+q)G(k)\Gamma(k,q)}

The resulting Feynman diagrams are shown in the figure.

Source: Bayo Lau

Since the Green’s function is an infinite series of diagrams itself, the vertex function is also an infinite series of diagrams. In the condition that the variation of phonon field is much smaller compared with external sources, the vertex function can be expressed 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 \Gamma(k,q)=1+i\int\frac{d^4p}{(2\pi)^4}|M(k-p)|^2G(p+q)\Gamma(p,q)G(p)D(k-p)}

To the lowest order in 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 G_0} 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 D_0} , the vertex function can be expressed as follows:

Source: Bayo Lau

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Electron-phonon interaction in the lattice model

The Hamiltonian for the electron-phonon interaction can be split info

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} a_{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}} }

Let's the ion is placed at the position 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 \mathbf{R}_i} , at displacement 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 \mathbf{u}_i } from the ionic equilibrium position, $\mathbf {R} _{j}^{0}$.

$H_{int}=\sum _{i\sigma }\int d^{3}r\psi _{\sigma }^{\dagger }(\mathbf {r} )\psi _{\sigma }(\mathbf {r} )V(\mathbf {r} -\mathbf {R} _{i})$

expand in powers of $\mathbf {u} _{i}$

$V(\mathbf {r} -\mathbf {R} _{i})\approx V(\mathbf {r} -\mathbf {R} _{i}^{0})-\nabla _{\mathbf {R} _{i}}(\mathbf {r} -\mathbf {R} _{i}))$

$H_{int}=\sum _{i\sigma }\int d^{3}r\psi _{\sigma }^{\dagger }(\mathbf {r} )\psi _{\sigma }(\mathbf {r} )V(\mathbf {r} -\mathbf {R} _{i}^{0})+\sum _{i\sigma }\int d^{3}r\psi _{\sigma }^{\dagger }(\mathbf {r} )\psi _{\sigma }(\mathbf {r} )\mathbf {u} _{i}\cdot \nabla _{\mathbf {R} _{i}}(\mathbf {r} -\mathbf {R} _{i})|_{\mathbf {R} _{i}^{0}}+...$

$\psi (\mathbf {r} )=\sum _{\mathbf {k} }c_{\mathbf {k} \sigma }\phi _{\mathbf {k} }(\mathbf {r} )$

with Bloch waves condition

$\phi _{\mathbf {k} \sigma }(\mathbf {r} +\mathbf {R} _{i}^{0})=e^{i\mathbf {k} \cdot \mathbf {R} _{i}^{0}}\phi _{\mathbf {k} }(\mathbf {r} )$

$\int d^{3}r\phi _{\mathbf {k^{\prime }} \sigma }^{*}(\mathbf {r} )\phi _{\mathbf {k} \sigma }(\mathbf {r} )\nabla _{\mathbf {R} _{i}^{0}}V(\mathbf {r} -\mathbf {R} _{i}^{0})=\int d^{3}r\phi _{\mathbf {k^{\prime }} \sigma }^{*}(\mathbf {r} +\mathbf {R} _{j}^{0})\phi _{\mathbf {k} \sigma }(\mathbf {r} +\mathbf {R} _{j}^{0})\nabla _{\mathbf {R} _{i}^{0}}V(\mathbf {r} -\mathbf {R} _{i}^{0})$

$=e^{i(\mathbf {k} -\mathbf {k} ^{\prime })\cdot \mathbf {R} _{j}^{0}}\int d^{3}r\phi _{\mathbf {k^{\prime }} \sigma (\mathbf {r} )}^{*}\phi _{\mathbf {k} \sigma }(\mathbf {r} )\nabla _{\mathbf {R} _{i}^{0}}V(\mathbf {r} -\mathbf {R} _{i}^{0})$

Let's

$\mathbf {W} _{\mathbf {kk} ^{\prime }}=\int d^{3}r\phi _{\mathbf {k^{\prime }} \sigma (\mathbf {r} )}^{*}\phi _{\mathbf {k} \sigma }(\mathbf {r} )\nabla _{\mathbf {R} _{i}^{0}}V(\mathbf {r} -\mathbf {R} _{i}^{0})$

By second quantization from phonon system

$\mathbf {u} _{i}(t)={\frac {1}{\sqrt {NM}}}\sum _{\mathbf {k} \lambda }Q(\mathbf {k} ,t)\mathbf {e} ^{\lambda }(\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {R} _{i}^{0}}$

$Q_{\lambda }(\mathbf {q} )={\frac {1}{2\omega _{\lambda }(\mathbf {q} )}}(b_{\lambda }(\mathbf {q} )+b_{\lambda }^{\dagger }(-\mathbf {q} ))$

The vectors $\mathbf {e} ^{\lambda }(\mathbf {k} )$ are the polarization vectors of the modes with $\sum _{\alpha }\mathbf {e} ^{\lambda }(\mathbf {k} )\cdot \mathbf {e} ^{\lambda ^{\prime }}(\mathbf {k} )=\delta _{\lambda \lambda ^{\prime }}$

Therefore,

$H_{int}=\sum _{\mathbf {kk} ^{\prime }\sigma }c_{\mathbf {k} ^{\prime }\sigma }^{\dagger }c_{\mathbf {k} ^{\prime }\sigma }\sum _{j}\mathbf {W} _{\mathbf {kk} ^{\prime }}e^{i(\mathbf {k} -\mathbf {k} ^{\prime })\cdot \mathbf {R} _{j}^{0}}{\frac {1}{\sqrt {NM}}}\sum _{\mathbf {q} \lambda }Q_{\lambda }(\mathbf {q} )\mathbf {e} ^{\lambda }(\mathbf {q} )e^{i\mathbf {q} \cdot \mathbf {R} _{j}^{0}}$

$=\sum _{\mathbf {kk} ^{\prime }\sigma }\sum _{\lambda }c_{\mathbf {k} ^{\prime }\sigma }^{\dagger }c_{\mathbf {k} ^{\prime }\sigma }(\mathbf {W} _{\mathbf {kk} ^{\prime }}\cdot \mathbf {e} ^{\lambda }(\mathbf {q} ))Q_{\lambda }(\mathbf {q} ){\sqrt {\frac {N}{M}}}$

$\equiv \sum _{\mathbf {kk} ^{\prime }\sigma \lambda }g_{\mathbf {kk} ^{\prime }\lambda }c_{\mathbf {k} ^{\prime }\sigma }^{\dagger }c_{\mathbf {k} ^{\prime }\sigma }(b_{\lambda }(\mathbf {q} )+b_{\lambda }^{\dagger }(-\mathbf {q} ))$

where the electron-phonon coupling constant is

$g_{\mathbf {kk} ^{\prime }\lambda }=(\mathbf {W} _{\mathbf {kk} ^{\prime }}\cdot \mathbf {e} ^{\lambda }(\mathbf {q} )){\sqrt {\frac {N}{2M\omega _{pl}^{ion}(\mathbf {q} )}}}$

Now the phonon wavevector k is restricted to to Brillouin. $\mathbf {p} =\mathbf {q} +\mathbf {G}$ where $q\in FBZ$ and $G\in RL$

The electron-phonon interaction can be described as

$V_{el-ph}=\int d\mathbf {r} \rho _{el}(\mathbf {r} ){\sum _{j}e\mathbf {u} _{j}\cdot \nabla _{\mathbf {r} }V_{ion}(\mathbf {r} -\mathbf {R} _{j}^{0})}$

$\nabla _{\mathbf {r} }V_{ion}(\mathbf {r} -\mathbf {R} _{j}^{0})={\frac {1}{\nu }}\sum _{\mathbf {q} \in FBZ}\sum _{\mathbf {G} \in RL}i(\mathbf {q} +\mathbf {g} )V_{\mathbf {q} +\mathbf {G} }e^{i(\mathbf {q} +\mathbf {G} )\cdot (\mathbf {r} -\mathbf {R} _{j}^{0})}$

$\mathbf {u} _{j}={\frac {1}{N}}\sum _{\mathbf {k} \in FBZ}\sum _{\lambda }{\frac {l_{\mathbf {k} \lambda }}{\sqrt {2}}}(b_{\mathbf {k} ,\lambda }+b_{-\mathbf {k} ,\lambda }^{\dagger })\mathbf {\epsilon } _{\mathbf {k} \lambda }e^{i\mathbf {k} \cdot \mathbf {R} _{j}^{0}}$

By using Fourier transforming and second quantization, The result is

$V_{el-ph}={\frac {1}{\nu }}\sum _{\mathbf {k} \sigma }\sum _{\mathbf {q} \lambda _{l}}\sum _{\mathbf {G} }g_{\mathbf {q} ,\mathbf {G} ,\lambda }c_{\mathbf {k+q+G} ,\sigma }^{\dagger }c_{\mathbf {k} \sigma }(b_{\mathbf {q} ,\lambda }+b_{\mathbf {-q} ,\lambda }^{\dagger })$

where

$g_{\mathbf {q,G} ,\lambda }=ie{\sqrt {\frac {N}{2M\omega _{\mathbf {q} \lambda }}}}(\mathbf {q+G} )\cdot \epsilon _{\mathbf {q} \lambda }V_{\mathbf {q+G} }$

Source: Henrik Bruus, Karsten Flensberg

In the isotropic media, with the normal process (IN) where $\mathbf {G} =0$

$V_{el-ph}^{IN}={\frac {1}{\nu }}\sum _{\mathbf {k} \sigma }\sum _{\mathbf {q} \lambda _{l}}g_{\mathbf {q} ,\lambda _{l}}c_{\mathbf {k+q} ,\sigma }^{\dagger }c_{\mathbf {k} \sigma }(b_{\mathbf {q} ,\lambda _{l}}+b_{\mathbf {-q} ,\lambda _{l}}^{\dagger })$

<br\> There is only the longitudinal acoustical branch for normal acoustical phonon process, the interaction becomes

$V_{el-ph}^{INA}={\frac {1}{\nu }}\sum _{\mathbf {k} \sigma }\sum _{\mathbf {q} }g_{\mathbf {q} ,\lambda _{l}}c_{\mathbf {k+q} ,\sigma }^{\dagger }c_{\mathbf {k} \sigma }(b_{\mathbf {q} }+b_{\mathbf {-q} }^{\dagger })$

If Yukawa potential is used to approxiamte $V_{q}$ The coupling constant is

$g_{\mathbf {q} }={\frac {iZe^{2}}{\epsilon _{0}}}{\frac {q}{q^{2}+k_{s}^{2}}}{\sqrt {\frac {N}{2M\omega _{\mathbf {q} }}}}$

---

Jellium model

In order to describe electron-phonon interactions in solids as well as demonstrate their electronic and phononic properties, Jellium model turns out to be set as a right scheme and could give accurate characterization on the system. The main approximation made in this model is to ignore the discrete nature of the ionic system and treat the nuclei as positively charged, continuous and homogenous fluid, with negatively charged electrons moving among the ‘Jellium sea’.

The Hamiltonian for a charge neutral system with electrons embedded in nuclei background and only the Coulomb interaction is taken into consideration:

$H=-{\frac {\hbar ^{2}}{2m}}\sum _{\sigma =\uparrow ,\downarrow }\int d^{3}r\Psi _{\sigma }^{+}({\vec {r}})\nabla ^{2}\Psi _{\sigma }({\vec {r}})-\sum _{\sigma }\int d^{3}r\int d^{3}r\prime \Psi _{\sigma }^{+}({\vec {r}})\Psi _{\sigma }({\vec {r}}){\frac {e^{2}}{|{\vec {r}}-{\vec {r\prime }}|}}e^{-{\frac {|{\vec {r}}-{\vec {r\prime }}|}{\xi }}}n_{b}(r\prime )$

Failed to parse (syntax error): {\displaystyle +\frac{1}2\sum_{\sigma,\sigma\prime}\int d^3r\int d^3r\prime\frac{e^2}{|\vec{r}-\vec{r\prime}|}e^{-\frac{|\vec{r}-\vec{r}\prime|}{\xi}}\Psi^+_{\sigma}(\vec{r})\Psi^+_{\sigma\prime}(\vec{r}\prime)\Psi_{\sigma\prime}(\vec{r}\prime)\Psi_{\sigma}(\vec{r})+\frac{1}2\int d^3r\int d^3r\prime n_b®\frac{e^2}{|\vec{r}-\vec{r}\prime|}e^{-\frac{|\vec{r}-\vec{r}\prime|}{\xi}}n_b(r\prime)}

For the case of uniform background density,

$n_{b}={\frac {N}{V}}$

where N is the total number of atoms and V is the volume of the system. Monoatomic molecules are assumed in this case.

The last term in the Hamiltonian is

${\frac {1}{2}}({\frac {N}{V}})^{2}e^{2}\int d^{3}r\int d^{3}r\prime {\frac {1}{|{\vec {r}}-{\vec {r}}\prime |}}e^{-{\frac {|{\vec {r}}-{\vec {r}}\prime |}{\xi }}}={\frac {1}{2}}({\frac {N}{V}})^{2}e^{2}V\int d^{3}r{\frac {1}{|{\vec {r}}|}}e^{-{\frac {r}{\xi }}}={\frac {e^{2}}{2}}{\frac {N^{2}}{V}}4\pi \int _{0}^{\infty }dr{\frac {r^{2}}{r}}e^{-{\frac {r}{\xi }}}={\frac {e^{2}}{2}}{\frac {N^{2}}{V}}4\pi \xi ^{2}\int _{0}^{\infty }dxxe^{-x}={\frac {e^{2}}{2}}{\frac {N^{2}}{V}}4\pi \xi ^{2}$

As $\xi$ goes to infinity, this term diverges. So, $\xi$ should be kept finite since now on.

The second term in the Hamiltonian is

$-e^{2}{\frac {N}{V}}\sum _{\sigma }\int d^{3}r\Psi _{\sigma }^{+}({\vec {r}})\Psi _{\sigma }({\vec {r}})\int d^{3}r\prime {\frac {1}{|{\vec {r}}-{\vec {r}}\prime |}}e^{-{\frac {|{\vec {r}}-{\vec {r}}\prime |}{\xi }}}=-e^{2}{\frac {N}{V}}(\sum _{\sigma }\int d^{3}r\Psi _{\sigma }^{+}({\vec {r}})\Psi _{\sigma }({\vec {r}}))4\pi \xi ^{2}=-e^{2}{\frac {N}{V}}{\hat {N}}_{e}4\pi \xi ^{2}$

For the third term in Hamiltonian, perform Fourier transform on $\Psi _{\sigma }({\vec {r}})$:

$\Psi _{\sigma }({\vec {r}})={\frac {1}{\sqrt {V}}}\sum _{\vec {k}}e^{i{\vec {k}}\cdot {\vec {r}}}\Psi _{{\vec {k}}\sigma }$

They obey anti-commutation relation in momentum space:

$\{\Psi _{\vec {\sigma }},\Psi _{{\vec {k}}\prime \sigma \prime }^{+}\}=\delta _{{\vec {k}},{\vec {k}}\prime }\delta _{\sigma ,\sigma \prime }$

Transfer to the center of mass coordinates:

${\vec {r}}={\vec {r}}-{\vec {r}}\prime$, ${\vec {R}}={\frac {{\vec {r}}+{\vec {r}}\prime }{2}}$

Thus the second term in Hamiltonian can be written as

Failed to parse (syntax error): {\displaystyle \frac{1}{2}\sum_{\sigma\sigma\prime}\int d^3r\int d^3r\prime\frac{e^2}{|\vec{r}-\vec{r}\prime|}e^{-\frac{|\vec{r}-\vec{r}\prime|}{\xi}}\frac{1}{V^2}\sum_{\vec{k}_1,…\vec{k}_4}e^{-i\vec{k}_1\cdot\vec{r}}e^{-i\vec{k}_2\cdot\vec{r}\prime}e^{i\vec{k}_3\cdot\vec{r}\prime}e^{i\vec{k}_4\cdot\vec{r}}\Psi^+_{\vec{k}_1,\sigma}\Psi^+_{\vec{k}_2\sigma\prime}\Psi_{\vec{k}_3,\sigma\prime}\Psi_{\vec{k}_4,\sigma} }

Failed to parse (syntax error): {\displaystyle =\frac{1}{2}\sum_{\sigma\sigma\prime}\frac{1}{V^2}\sum_{\vec{k}_1,…\vec{k}_4}\Psi^+_{\vec{k}_1,\sigma}\Psi^+_{\vec{k}_2\sigma\prime}\Psi_{\vec{k}_3,\sigma\prime}\Psi_{\vec{k}_4,\sigma}\int d^3R\int d^3r\frac{e^2}{|\vec{r}|}e^{-\frac{|\vec{r}|}{\xi}}e^{-i(\vec{k}_1-\vec{k}_4)(\vec{R}+\frac{1}{2}\vec{r})}e^{-i(\vec{r}_2-\vec{k}_3)(\vec{R}-\frac{1}{2}\vec{r})} }

Failed to parse (syntax error): {\displaystyle =\frac{1}{2}\sum_{\sigma\sigma\prime}\frac{1}{V}\sum_{\vec{k}_1,…\vec{k}_4}\delta_{\vec{k}_1-\vec{k}_4,\vec{k}_2-\vec{k}_3}\Psi^+_{\vec{k}_1,\sigma}\Psi^+_{\vec{k}_2\sigma\prime}\Psi_{\vec{k}_3,\sigma\prime}\Psi_{\vec{k}_4,\sigma}\int d^3r\frac{e^2}{|\vec{r}|}e^{-\frac{|\vec{r}|}{\xi}}e^{-i(\vec{k}_1-\vec{k}_4-\vec{k}_2+\vec{k}_3)\frac{1}{2}\vec{r}}}

Let ${\vec {k}}_{1}={\vec {k}}$ and ${\vec {k}}_{4}={\vec {k}}-{\vec {q}}$, ${\vec {k}}_{2}={\vec {k}}\prime$ and ${\vec {k}}_{3}={\vec {k}}\prime +{\vec {q}}$

Then

$\sigma _{{\vec {k}}_{1}-{\vec {k}}_{4},{\vec {k}}_{3}-{\vec {k}}_{2}}=\sigma _{{\vec {q}},{\vec {q}}}=1$

The third term in Hamiltonian has the following form:

${\frac {1}{2}}\sum _{\sigma \sigma \prime }{\frac {1}{V}}\sum _{{\vec {k}},{\vec {k}}\prime }\sum {\vec {q}}\Psi _{{\vec {k}}\sigma }^{+}\Psi _{{\vec {k}}\prime \sigma \prime }^{+}\Psi _{{\vec {k\prime }}+{\vec {q}},\sigma \prime }\Psi _{{\vec {k}}-{\vec {q}},\sigma }\int d^{3}r{\frac {e^{2}}{|{\vec {r}}|}}e^{-{\frac {|{\vec {r}}|}{\xi }}}e^{-i{\vec {q}}\cdot {\vec {r}}}$

For the integral over r,

$e^{2}\int _{0}^{2\pi }d\phi \int _{-1}^{1}d(cos\theta )\int _{0}^{\infty }drr^{2}{\frac {1}{r}}e^{-{\frac {r}{\xi }}}e^{-iqrcos\theta }=e^{2}2\pi \int _{0}^{\infty }drre^{-{\frac {r}{\xi }}}{\frac {1}{-iqr}}(e^{-iqr}-e^{iqr})$

$=e^{2}2\pi {\frac {i}{q}}(\int _{0}^{\infty }dre^{-{\frac {r}{\xi }}}e^{-iqr}-\int _{0}^{\infty }dre^{-{\frac {r}{\xi }}}e^{iqr})=e^{2}2\pi {\frac {i}{q}}({\frac {1}{{\frac {1}{\xi }}+iq}}-{\frac {1}{{\frac {1}{\xi }}-iq}})=4\pi {\frac {e^{2}}{q^{2}+\xi ^{-2}}}$

So, this term can be written as

${\frac {1}{2}}\sum _{\sigma \sigma \prime }{\frac {1}{N}}\sum _{{\vec {k}},{\vec {k}}\prime }\Psi _{{\vec {k}}\sigma }^{+}\Psi _{{\vec {k}}\prime \sigma \prime }^{+}\Psi _{{\vec {k}}\prime \sigma \prime }\Psi _{{\vec {k}}\sigma }4\pi e^{2}\xi ^{2}+{\frac {1}{2}}\sum _{\sigma \sigma \prime }{\frac {1}{N}}\sum _{{\vec {k}},{\vec {k}}\prime }\sum _{{\vec {q}}\neq 0}\Psi _{{\vec {k}}\sigma }^{+}\Psi _{{\vec {k}}\prime \sigma \prime }^{+}\Psi _{{\vec {k}}\prime \sigma \prime }\Psi _{{\vec {k}}\sigma }4\pi {\frac {e^{2}}{q^{2}+\xi ^{-2}}}$

$={\frac {1}{2}}e^{2}4\pi \xi ^{2}{\frac {1}{V}}({\hat {N}}_{e}^{2}-{\hat {N}}_{e})+{\frac {1}{2}}\sum _{\sigma \sigma \prime }{\frac {1}{N}}\sum _{{\vec {k}},{\vec {k}}\prime }\sum _{{\vec {q}}\neq 0}\Psi _{{\vec {k}}\sigma }^{+}\Psi _{{\vec {k}}\prime \sigma \prime }^{+}\Psi _{{\vec {k}}\prime \sigma \prime }\Psi _{{\vec {k}}\sigma }4\pi {\frac {e^{2}}{q^{2}+\xi ^{-2}}}$

Assume that the system is charge neutral, ${\hat {N}}_{e}=N$, then the Hamiltonian is finally given by

$H=\sum _{{\vec {k}}\sigma }{\frac {\hbar ^{2}k^{2}}{2m}}\Psi _{{\vec {k}}\sigma }^{+}\Psi _{{\vec {k}}\sigma }+{\frac {1}{2V}}\sum _{{\vec {k}}{\vec {k}}\prime ,\sigma \sigma \prime ,{\vec {q}}\neq 0}{\frac {4\pi e^{2}}{q^{2}}}\Psi _{{\vec {k}}\sigma }^{+}\Psi _{{\vec {k}}\sigma \prime }^{+}\Psi _{{\vec {k}}\prime +{\vec {q}}\sigma \prime }\Psi _{{\vec {k}}-{\vec {q}}\sigma }$

The partition function is given by

$Z=Tr[e^{-\beta ({\hat {H}}-\mu {\hat {N}})}]=Z_{0}<e^{-S_{int}}>=e^{-\beta \Omega }$

where the action function for interaction part is

$S_{int}=\int _{0}^{\beta }d\tau {\frac {1}{2V}}\sum _{{\vec {k}}{\vec {k}}\prime ,\sigma \sigma \prime }\sum _{{\vec {q}}\neq 0}{\frac {4\pi e^{2}}{q^{2}}}\Psi _{{\vec {k}}\sigma }^{\star }(\tau )\Psi _{{\vec {k}}\prime \sigma \prime }^{\star }(\tau )\Psi _{{\vec {k}}\prime +{\vec {q}}\sigma \prime }(\tau )\Psi _{{\vec {k}}-{\vec {q}}\sigma }(\tau )$

Cumulant expansion is applied to the analysis of interactions in Jellium model. Expand the interaction part in partition function and keep to the second order:

$<e^{-S_{int}}>\approx e^{-<S_{int}>}e^{{\frac {1}{2}}(<S_{int}^{2}-<S_{int}>^{2}>)}$

The partition function for non-interacting system is given by

$Z_{0}=\prod _{{\vec {k}}\sigma }(1+e^{-\beta (E_{{\vec {k}}-\mu })})=e^{\beta V{\frac {2}{\beta }}\int {\frac {d^{3}k}{(2\pi )^{3}}}In(1+e^{-\beta (E_{\vec {k}}-\mu )})}=e^{-\beta \Omega ^{(0)}}$

The first order correction to action function is

$<S_{int}>=\int _{0}^{\beta }d\tau {\frac {1}{2V}}\sum _{{\vec {k}}{\vec {k}}\prime ,\sigma \sigma \prime }\sum _{{\vec {q}}\neq 0}{\frac {4\pi e^{2}}{q^{2}}}<\Psi _{{\vec {k}}\sigma }^{\star }(\tau )\Psi _{{\vec {k}}\prime \sigma \prime }^{\star }(\tau )\Psi _{{\vec {k}}\prime +{\vec {q}}\sigma \prime }(\tau )\Psi _{{\vec {k}}-{\vec {q}}\sigma }(\tau )>$

$=\int _{0}^{\beta }d\tau {\frac {1}{2V}}\sum _{{\vec {k}}{\vec {k}}\prime ,\sigma \sigma \prime }\sum _{{\vec {q}}\neq 0}{\frac {4\pi e^{2}}{q^{2}}}(G_{0}(({\vec {k}}-{\vec {q}})\tau ,{\vec {k}}\tau )G_{0}(({\vec {k}}\prime +{\vec {q}})\tau ,{\vec {k}}\prime \tau )-\delta _{\sigma \sigma \prime }G_{0}(({\vec {k}}\prime +{\vec {q}})\tau ,{\vec {k}}\tau )G_{0}(({\vec {k}}-{\vec {q}})\tau ,{\vec {k}}\prime \tau ))$

$=-\sum _{\sigma }\int _{0}^{\beta }d\tau {\frac {1}{2V}}\sum _{\vec {k}}\sum _{{\vec {q}}\neq 0}{\frac {4\pi e^{2}}{q^{2}}}n_{F}(E_{\vec {k}})n_{F}(E_{{\vec {k}}-{\vec {q}}})=\beta \Omega ^{(1)}$

The second order correction to the action function is given by

$\beta \Omega ^{(2)}=-{\frac {1}{2}}[<S_{int}^{2}>_{0}-<S_{int}>_{0}^{2}]$

So,

$\beta \Omega ^{(2)}=-{\frac {1}{2}}\int _{0}^{\beta }d\tau \int _{0}^{\beta }d\tau \prime ({\frac {e^{2}}{2V}})^{2}\sum _{{\vec {k}}{\vec {k}}\prime }\sum _{\sigma \sigma \prime }\sum _{{\vec {q}}\neq 0}\sum _{{\vec {p}}{\vec {p}}\prime }\sum _{{\vec {s}}{\vec {s}}\prime }\sum _{{\vec {q}}\prime \neq 0}{\frac {4\pi }{q^{2}}}{\frac {4\pi }{q\prime ^{2}}}(<\Psi _{{\vec {k}}\sigma }^{\star }(\tau )\Psi _{{\vec {k}}\prime \sigma \prime }^{\star }(\tau )\Psi _{{\vec {k}}\prime +{\vec {q}}\sigma \prime }(\tau )\Psi _{{\vec {k}}-{\vec {q}}\sigma }(\tau )\Psi _{{\vec {p}}s}^{\star }(\tau \prime )\Psi _{{\vec {p}}\prime s\prime }^{\star }(\tau \prime )\Psi _{{\vec {p}}\prime +{\vec {q}}\prime s\prime }(\tau \prime )\Psi _{{\vec {p}}-{\vec {q}}\prime s}(\tau \prime )>_{0}$

$-<\Psi _{{\vec {k}}\sigma }^{\star }(\tau )\Psi _{{\vec {k}}\prime \sigma \prime }^{\star }(\tau )\Psi _{{\vec {k}}\prime +{\vec {q}}\sigma \prime }(\tau )\Psi _{{\vec {k}}-{\vec {q}}\sigma }(\tau )>_{0}<\Psi _{{\vec {p}}s}^{\star }(\tau \prime )\Psi _{{\vec {p}}\prime s\prime }^{\star }(\tau \prime )\Psi _{{\vec {p}}\prime +{\vec {q}}\prime s\prime }(\tau \prime )\Psi _{{\vec {p}}-{\vec {q}}\prime s}(\tau \prime )>_{0})$

After expanding the above expression, there are in total 24 terms. However, 4 of them are disconnected terms which force ${\vec {q}}=0$, thus only 20 terms are left for further calculation.

Since the term $<S_{int}>^{2}$ can be trivially derived from the first order correction $<S_{int}>$, let us focus on the term <S_{int}^2>. Expand $<\Psi _{{\vec {k}}\sigma }^{\star }(\tau )\Psi _{{\vec {k}}\prime \sigma \prime }^{\star }(\tau )\Psi _{{\vec {k}}\prime +{\vec {q}}\sigma \prime }(\tau )\Psi _{{\vec {k}}-{\vec {q}}\sigma }(\tau )\Psi _{{\vec {p}}s}^{\star }(\tau \prime )\Psi _{{\vec {p}}\prime s\prime }^{\star }(\tau \prime )\Psi _{{\vec {p}}\prime +{\vec {q}}\prime s\prime }(\tau \prime )\Psi _{{\vec {p}}-{\vec {q}}\prime s}(\tau \prime )>$ through contraction, we could get three block terms, which are denoted by $\beta \Omega _{1}^{(2)}$, $\beta \Omega _{2}^{(2)}$ and $\beta \Omega _{3}^{(2)}$, respectively. Calculate these three parts one by one.

$\beta \Omega _{1}^{(2)}=-{\frac {1}{2}}\int _{0}^{\beta }d\tau \int _{0}^{\beta }d\tau \prime ({\frac {e^{2}}{2V}})^{2}\sum _{{\vec {k}}{\vec {k}}\prime }\sum _{\sigma \sigma \prime }\sum _{{\vec {q}}\neq 0}\sum _{{\vec {p}}{\vec {p}}\prime }\sum _{{\vec {s}}{\vec {s}}\prime }\sum _{{\vec {q}}\prime \neq 0}{\frac {4\pi }{q^{2}}}{\frac {4\pi }{q\prime ^{2}}}(4\delta _{\sigma s}G_{0}({\vec {k}}-{\vec {q}}\tau ,{\vec {p}}\tau \prime )$

$\delta _{\sigma \sigma \prime }G_{0}({\vec {k}}\prime +{\vec {q}}\tau ,{\vec {k}}\tau )\delta _{\sigma \prime s\prime }G_{0}({\vec {p}}\prime +{\vec {q}}\prime \tau \prime ,{\vec {k}}\prime \tau )(-\delta _{ss\prime })G_{0}({\vec {p}}-{\vec {q}}\prime \tau \prime ,{\vec {p}}\prime \tau \prime ))$

$=(4\pi e^{2})^{2}\int _{0}^{\beta }d\tau \int _{0}^{\beta }d\tau \prime \sum _{\vec {k}}\int {\frac {d^{3}q}{(2\pi )^{3}}}\int {\frac {d^{3}q\prime }{(2\pi )^{3}}}{\frac {1}{q^{2}}}{\frac {1}{q\prime ^{2}}}e^{-(\epsilon _{{\vec {k}}-{\vec {q}}-\mu })(\tau -\tau \prime )}$

$(\Theta (\tau -\tau \prime )-n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}})(-n_{F}(\epsilon _{\vec {k}}))e^{-(\epsilon _{{\vec {k}}-{\vec {q}}-\mu })(\tau \prime -\tau )}(\Theta (\tau \prime -\tau )-n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}}))(-n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}-{\vec {q}}\prime }))$

$=(4\pi e^{2})^{2}\beta ^{2}\sum _{\vec {k}}\int {\frac {d^{3}q}{(2\pi )^{3}}}\int {\frac {d^{3}q\prime }{(2\pi )^{3}}}{\frac {1}{q^{2}}}{\frac {1}{q\prime ^{2}}}n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}})(n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}})-1)n_{F}(\epsilon _{\vec {k}})n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}-{\vec {q}}\prime })$

Upon shifting variables ${\vec {k}}\to {\vec {k}}-{\vec {q}}$, ${\vec {q}}\to {\vec {q}}-{\vec {k}}$, we have

${\frac {\Omega _{1}^{(2)}}{V}}=-(4\pi e^{2})^{2}\beta \int {\frac {d^{3}k}{(2\pi )^{3}}}\int {\frac {d^{3}q}{(2\pi )^{3}}}{\frac {d^{3}q\prime }{(2\pi )^{3}}}{\frac {1}{|{\vec {q}}-{\vec {k}}|^{2}}}{\frac {1}{|{\vec {q}}\prime -{\vec {k}}|^{2}}}n_{F}(\epsilon _{\vec {q}})n_{F}(\epsilon _{{\vec {q}}\prime })n_{F}(\epsilon _{\vec {k}})(1-n_{F}(\epsilon _{\vec {k}}))$

For

${\frac {\Omega _{2}^{(2)}}{V}}=\int _{0}^{\beta }d\tau \int _{0}^{\beta }d\tau \prime ({\frac {e^{2}}{2V}})^{2}\sum _{\sigma }\sum _{\vec {k}}\sum _{{\vec {q}}\neq 0}\sum _{{\vec {q}}\prime \neq 0}{\frac {4\pi }{q^{2}}}{\frac {4\pi }{q\prime ^{2}}}e^{-(\epsilon _{\vec {k}}-\mu )(\tau \prime -\tau )}e^{-(\epsilon _{{\vec {k}}-{\vec {q}}}-\mu )(\tau -\tau \prime )}e^{-(\epsilon _{{\vec {k}}-{\vec {q}}\prime }-\mu )(\tau -\tau \prime )}e^{-(\epsilon _{{\vec {k}}-{\vec {q}}-{\vec {q}}\prime }-\mu )(\tau \prime -\tau )}$

$(\Theta (\tau \prime -\tau )(1-n_{F}(\epsilon _{\vec {k}}))-\Theta (\tau -\tau \prime )n_{F}(\epsilon _{\vec {k}}))(\Theta (\tau -\tau \prime )(1-n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}}))-\Theta (\tau -\tau \prime )n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}}))(\Theta (\tau -\tau \prime )(1-n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}\prime }))$

$-\Theta (\tau \prime -\tau )n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}\prime })(\Theta (\tau \prime -\tau )(1-n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}-{\vec {q}}\prime }))-\Theta (\tau -\tau \prime )n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}-{\vec {q}}\prime }))$

Upon shifting variables ${\vec {k}}\to {\vec {k}}+{\vec {q}}$ and then ${\vec {q}}\to -{\vec {q}}$, we have

${\frac {\Omega _{2}^{(2)}}{V}}=(4\pi e^{2})^{2}\int {\frac {d^{3}k}{(2\pi )^{3}}}\int {\frac {d^{3}q}{(2\pi )^{3}}}\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{q^{2}}}{\frac {1}{({\vec {p}}-({\vec {k}}+{\vec {q}}))^{2}}}{\frac {1}{\epsilon _{{\vec {k}}+{\vec {q}}}+\epsilon _{{\vec {p}}-{\vec {q}}}-\epsilon _{\vec {k}}-\epsilon _{\vec {p}}}}n_{F}(\epsilon _{\vec {k}})n_{F}(\epsilon _{\vec {p}})(1-n_{F}(\epsilon _{{\vec {k}}+{\vec {q}}}))(1-n_{F}(\epsilon _{{\vec {p}}-{\vec {q}}}))$

This term gives finite contribution to the second order correction.

For

${\frac {\Omega _{3}^{(2)}}{V}}=-({\frac {e^{2}}{2V}})^{2}4\int _{0}^{\beta }d\tau \int _{0}^{\beta }d\tau \prime \sum _{\vec {k}}\sum _{{\vec {k}}\prime }\sum _{{\vec {q}}\neq 0}({\frac {4\pi }{q^{2}}})^{2}e^{-(\epsilon _{\vec {k}}-\mu )(\tau \prime -\tau )}(\Theta (\tau \prime -\tau )(1-n_{F}(\epsilon _{\vec {k}}))-\Theta (\tau -\tau \prime )n_{F}(\epsilon _{\vec {k}}))$

$e^{-(\epsilon _{{\vec {k}}-{\vec {q}}}-\mu )(\tau -\tau \prime )}(\Theta (\tau -\tau \prime )(1-n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}}))-\Theta (\tau \prime -\tau )n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}}))$

$e^{-(\epsilon _{{\vec {k}}\prime }-\mu )(\tau -\tau \prime )}(\Theta (\tau -\tau \prime )(1-n_{F}(\epsilon _{{\vec {k}}\prime }))-\Theta (\tau \prime -\tau )n_{F}(\epsilon _{{\vec {k}}\prime }))e^{-(\epsilon _{{\vec {k}}\prime -{\vec {q}}}-\mu )(\tau \prime -\tau )}(\Theta (\tau \prime -\tau )(1-n_{F}(\epsilon _{{\vec {k}}\prime -{\vec {q}}}))-\Theta (\tau -\tau \prime )n_{F}(\epsilon _{{\vec {k}}\prime -{\vec {q}}}))$

Upon shifting variables ${\vec {k}}\to {\vec {k}}+{\vec {q}}$, ${\vec {k}}\prime \to {\vec {k}}\prime +{\vec {q}}$, and then ${\vec {q}}\to -{\vec {q}}$, we have

${\frac {\Omega _{3}^{(2)}}{V}}=-2(4\pi e^{2})^{2}\int {\frac {d^{3}k}{(2\pi )^{3}}}\int {\frac {d^{3}k\prime }{(2\pi )^{3}}}\int {\frac {d^{3}q}{(2\pi )^{3}}}{\frac {1}{q^{4}}}{\frac {1}{\epsilon _{\vec {k}}-\epsilon _{{\vec {k}}-{\vec {q}}}-\epsilon _{{\vec {k}}\prime }+\epsilon _{{\vec {k}}\prime -{\vec {q}}}}}n_{F}(\epsilon _{{\vec {k}}-{\vec {q}}})n_{F}(\epsilon _{{\vec {k}}\prime })(1-n_{F}(\epsilon _{\vec {k}}))(1-n_{F}(\epsilon _{{\vec {k}}\prime -{\vec {q}}}))$

This term is divergent and thus gives infinite contribution to the second order correction.

The electron-phonon interaction in this case can be described as phonon is isotrpic case

$V_{el-ph}^{jel}={\frac {1}{\nu }}\sum _{\mathbf {k} \sigma }\sum _{\mathbf {q} }g_{\mathbf {q} }^{jel}c_{\mathbf {k+q} \sigma }^{\dagger }c_{\mathbf {k} \sigma }(b_{\mathbf {q} }+b_{\mathbf {-q} }^{\dagger })$

$g_{\mathbf {q} }^{jel}={\frac {iZe^{2}}{\epsilon _{0}}}{\frac {1}{q}}{\sqrt {\frac {N}{2M\Omega }}}$

the ion plasme frequence $\Omega$

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Polarons

The electrical resistivity is rooted in the electron-phonon interaction via a temperature-dependent manner. Electrons are scattered by phonons. With the increase of temperature, more phonons are generated and thus more scattering events occur. When electrons moving in crystal, they drag heavy ion core with them, as a result of which, the apparent electron mass increases. This effect is larger in ionic crystals than in covalent crystals, since the Coulomb interaction is much stronger in ionic crystals compared with covalent crystals. The electron, together with its strain field, is called polaron. Below are figures demonstrating the formation of polarons in a rigid lattice of an ionic crystal.

Source: Kittel, Charles

Source: Kittel, Charles

In the limit of low electron density and when Fermi momentum $k_{F}\to 0$, the problem can be reduced to a 3D system with one electron inside. The Hamiltonian of this system is composed of two parts:

$H=H_{0}+H_{el-ph}$

where $H_{0}$ is for free electron-phonon system and $H_{el-ph}$ is for electron-phonon interaction. In perturbation theory, the first order correction to the ground state of $H_{0}$ vanishes due to the absence of phonon number conservation and finite expectation value. The second order correction to the ground state energy is given by

$\delta E=\sum _{n\neq 0}{\frac {|<n|H_{el-ph}|0>|^{2}}{E_{0}^{0}-E_{n}^{0}}}$

where $H_{0}|n>=E_{n}^{0}|n>$

The state with only one electron can be created from vacuum via

$a_{-{\vec {p}}}^{+}|0>=|-{\vec {p}}>$

The energy of the excited state is

$E_{n}=E_{0}^{0}-E_{\vec {k}}+E_{{\vec {k}}+{\vec {p}}}+\hbar \omega _{0}$

For each value of ${\vec {k}}$, the second order correction to ground state energy is

$\delta E_{\vec {k}}=-2{\frac {\alpha ^{2}}{L^{d}}}\sum _{\vec {p}}{\frac {1}{{\vec {p}}^{2}}}{\frac {\Theta (k_{F}-|{\vec {k}}|)\Theta (|{\vec {p}}+{\vec {k}}|-k_{F})}{E_{{\vec {k}}+{\vec {p}}}-E_{\vec {k}}+\hbar \omega _{0}}}$

where the electron-phonon coupling constant $\alpha$ is given by

$\alpha =4\pi e^{2}\kappa {\sqrt {\frac {\hbar }{2M\rho _{s}\omega _{0}}}}$

Evaluate the above summation via integrals, we have

$\delta E_{\vec {k}}=-{\frac {4m\alpha ^{2}}{\hbar ^{2}}}\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{{\vec {p}}^{2}}}{\frac {1}{({\vec {p}}+{\vec {k}})^{2}-k^{2}+P_{0}^{2}}}$

$=-{\frac {4m\alpha ^{2}}{(2\pi )^{3}\hbar ^{2}}}\int _{0}^{\pi }d\theta sin\theta \int _{0}^{+\infty }dp{\frac {1}{p^{2}+2pkcos\theta +P_{0}^{2}}}$

$=-{\frac {4m\alpha ^{2}}{(2\pi )^{3}\hbar ^{2}}}\int _{-1}^{1}du\int _{0}^{+\infty }dp{\frac {1}{p^{2}+2pku+P_{0}^{2}}}$

$=-{\frac {4m\alpha ^{2}}{(2\pi )^{3}\hbar ^{2}P_{0}}}\int _{-1}^{1}du\int _{0}^{+\infty }dx{\frac {1}{x^{2}+1+2(k/P_{0})xu}}$

where

$P_{0}={\sqrt {2m\omega _{0}/\hbar }}$

In the limit of low energies, the electron momentum is small. Expansion of the denominator in terms of $k/P_{0}$ in the integrant leads to

$\delta E_{\vec {k}}=-{\frac {4m\alpha ^{2}}{(2\pi )^{3}\hbar ^{2}P_{0}}}\int _{0}^{+\infty }dx\int _{-1}^{1}du({\frac {1}{x^{2}+1}}+{\frac {8}{3}}(k/P_{0})^{2}u^{2}){\frac {x^{2}}{(1+x^{2})^{3}}}=-{\frac {4m\alpha ^{2}}{(2\pi )^{3}\hbar ^{2}P_{0}}}(\pi +{\frac {\pi }{6}}(k/P_{0})^{2})$

Therefore, the ground state energy to the second order in perturbation theory is given by

$E_{\vec {k}}\approx -E_{0}+{\frac {\hbar ^{2}k^{2}}{2m^{\star }}}$

with

$E_{0}={\frac {m\alpha ^{2}}{2\pi \hbar ^{2}P_{0}}}$

and

$m^{\star }=m(1+{\frac {2mE_{0}}{3\hbar ^{2}P_{0}^{2}}})$

which is the effective mass due to dragging of nuclei by electron motion. The effective mass $m^{\star }$ of the polaron can be measured from cyclotron resonance experiments. The following relation demonstrates how the effective mass of polaron relates to the effective band mass of electron:

$m_{pol}^{\star }\approx m_{band}^{\star }({\frac {1-0.0008\alpha ^{2}}{1-{\frac {1}{6}}\alpha +0.0034\alpha ^{2}}})$

Since the electron-phonon coupling constant $\alpha$ is a positive number, it is obvious that the effective mass of polaron is greater than the bare mass.

Polarons can be described as large polarons and small polarons, based on their effective mass. For electrons associated with large polarons moving in the band, their mass is increased, as we discussed above. For electrons associated with small polarons, they are trapped in single ions for most of the time. At high temperatures, electrons hop from site to site due to thermal excitation. However, at low temperatures, electrons slowly tunnel inside the crystal, as if they moving in a band with large effective mass.

When the band edge is degenerate and the crystal is polar, holes or electrons can be self-trapped due to asymmetric local deformation of the lattice. The valence band edge is more likely degenerate than conduction band edge, therefore, holes are more likely to become self-trapped than electrons.

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Linear response calculations of electron-phonon interactions

In the strong-coupling theory of superconductivity, the electron-phonon spectral distribution function $\alpha ^{2}F(\omega )$ and its first reciprocal moment $\lambda$, are two most essential quantities which are fully demonstrated by electron-phonon interactions. There exist several approaches to calculate the first reciprocal moment. One is only to calculate electronic contribution [2] and the information about phonon frequencies and eigenvectors is got from inelastic neutron-scattering data [3]. Alternatively, the electron-phonon interaction may be calculated via frozen-phonon total-energy approach. However, these listed approaches are too crude to estimate some critical quantities, like critical temperature ($T_{c}$) in phase transition.

Source: S. Y. Savrasov, D. Y. Savrasov and O. K. Andersen

Linear response method is based on solid-state Sternheimer method [4] and can produce accurate phonon dispersion relations as well as eigenvectors for arbitrary ${\vec {q}}$. In solid-state Sternheimer method, the electron-phonon spectral distribution function is given by

$\alpha ^{2}F(\omega )={\frac {1}{2\pi N(0)}}\sum _{{\vec {q}}\nu }{\frac {\gamma _{{\vec {q}}\nu }}{\omega _{{\vec {q}}\nu }}}\delta (\omega -\omega _{{\vec {q}}\nu })$

where $\sum _{\vec {q}}$ is the summation over the first Brillouin zone, and $N(0)$ is the density of states for electrons at Fermi level. $\gamma _{{\vec {q}}\nu }$ is the phonon line-width which is derived from Fermi golden rule:

$\gamma _{{\vec {q}}\nu }=2\pi \omega _{{\vec {q}}\nu }\sum _{{\vec {k}}jj\prime }\delta (E_{{\vec {k}}j})\delta (E_{{\vec {k}}+{\vec {q}}j\prime })|g_{{\vec {k}}+{\vec {q}}j\prime ,{\vec {k}}j}^{{\vec {q}}\nu }|^{2}$

where $E_{{\vec {k}}j}$ denotes energies with respect to Fermi level, and $g_{{\vec {k}}+{\vec {q}}j\prime ,{\vec {k}}j}^{{\vec {q}}\nu }$ is the electron-phonon interaction matrix element whose physical meaning is the probability of scattering from state $|{\vec {k}}j>$ to state $|{\vec {k}}+{\vec {q}}j\prime >$ via phonon ${\vec {q}}\nu$.

Let $|\chi _{\alpha }^{\vec {k}}>$ to be the finite orbital basis, the electron-phonon interaction matrix element is thus given by

$g_{{\vec {k}}+{\vec {q}}j\prime ,{\vec {k}}j}^{{\vec {q}}\nu }=<{\vec {k}}+{\vec {q}}j\prime |\delta ^{{\vec {q}}\nu }V|{\vec {k}}j>+<\sum _{\alpha }\delta ^{{\vec {q}}\nu }\chi _{\alpha }^{{\vec {k}}-{\vec {q}}}A_{\alpha }^{{\vec {k}}+{\vec {q}}j\prime }|H-E_{{\vec {k}}j}|{\vec {k}}j>+<{\vec {k}}+{\vec {q}}j\prime |H-E_{{\vec {k}}j}|\sum _{\alpha }\delta ^{{\vec {q}}\nu }\chi _{\alpha }^{\vec {k}}A_{\alpha }^{{\vec {k}}j\prime }>$

where $A_{\alpha }^{{\vec {k}}j\prime }$ is the coefficient of basis$|\chi _{\alpha }^{\vec {k}}>$ in the basis $|{\vec {k}}j>$, and $\delta ^{{\vec {q}}\nu }V$ is the changes in one electron potential, $\delta ^{{\vec {q}}\nu }\chi _{\alpha }^{\vec {k}}$ is the changes of basis due to electron-phonon interaction. This expression gives a correct behavior in the long-wavelength limit.

Source: S. Y. Savrasov, D. Y. Savrasov and O. K. Andersen

Due to some phase-space reasons, such as Fermi surface nesting, the phonon line-width $\gamma _{{\vec {q}}\nu }$ could show violent variations in the first Brillouin zone, as a result of which, the calculation for electron-phonon spectral distribution function will be performed on a rather dense mesh. However, $\omega _{{\vec {k}}\nu }$, the phonon frequency, is a relatively smooth function of ${\vec {q}}$ and it is not necessary to calculate it on such a dense mesh. Via transforming the dynamical matrix from momentum space to real space, together with diagonalization, the phonon frequencies and eigenvectors for arbitrary ${\vec {q}}$ can be reached.

The linear response calculations of electron-phonon interactions have a general agreement with experimental results, which accounts for the validity of this method in analysis electron-phonon coupling problem. For instance, for the most abundant metal on earth, aluminum (Al) displays a nice consistence between theoretical calculation and tunneling measurements on electron-phonon spectral distribution function. Fig 5 shows the comparison.

Niobium (Nb) is one the well-studied elemental superconductors, with a relatively high critical temperature $T_{c}=9.25K$. Compared with tunneling measurements, the theoretical calculations yield stronger electron-phonon coupling, and the shape of electron-phonon spectral distribution function is broadened, as shown in Fig 6.

Kohn Anomalies

Screening

Source: Yun-Chieh Peng

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, which is very hard to calculate. Therefore, we often adopt the free electron model which treats the ions as a uniform background of positive charge even for real metals. This is called a free electron gas.

Screening is an important phenomena in a free electron gas. When an external positive charge density $\rho _{\text{ext}}\left({\vec {r}}\right)$ is applied, the electrons will be attracted to surround the positive charge. The re-arrangement of the electrons generates an induced charge distribution $\rho _{\text{el}}\left({\vec {r}}\right)$. Therefore, the total charge density $\rho \left({\vec {r}}\right)$ = $\rho _{\text{ext}}\left({\vec {r}}\right)$ + $\rho _{\text{el}}\left({\vec {r}}\right)$ is less positive than $\rho \left({\vec {r}}\right)$. Thus, The total potential $\phi \left({\vec {r}}\right)$ is weaker than the external potential $\phi _{\text{ext}}\left({\vec {r}}\right)$ caused by the positive charge only. The phenomenon is called screening. By assuming that the applied charge is weak enough so that the total potential and external potential are linearly related, we get the following function in momentum space.

$\varepsilon \left({\vec {k}}\right)=1-{\frac {4\pi }{{\vec {k}}^{2}}}{\frac {\rho _{\text{el}}\left({\vec {k}}\right)}{\phi _{\text{ext}}\left({\vec {k}}\right)}}$

$\varepsilon$ is the Fourier transform of the dielectric constant, ${\vec {k}}$ is the wave vector of the field, $\rho _{\text{el}}\left({\vec {r}}\right)$ is the Fourier transform of induced charge density, and $\phi _{\text{ext}}\left({\vec {r}}\right)$ is the Fourier transform of the total potential.

---

Kohn anomalies

The Hamiltonian of the electron-phonon interaction is:

$H_{el-ph}=\sum _{{\vec {k}},{\vec {q}}j}g\left({\vec {k}}_{1},{\vec {k}}_{2};{\vec {q}}j\right)c_{{\vec {k}}_{1}j}^{\dagger }c_{{\vec {k}}_{2}j}\left(a_{{\vec {-q}}j}^{\dagger }+a_{{\vec {q}}j}\right)$

where $c_{{\vec {k}}_{1}j}^{\dagger }$ and $c_{{\vec {k}}_{2}j}$ are the creation and annihilation operators for the quasiparticles with wave vectors ${\vec {k}}_{1}={\vec {k}}+{\vec {q}}$ and ${\vec {k}}_{2}={\vec {k}}$, respectively; $a_{{\vec {q}}j}^{\dagger }$ and a_{\vec{q} j} are the creation and the annihilation operators of the phonon of energy $a_{{\vec {\omega }}j}$ and wave vector ${\vec {q}}$. The latter two operators in the Hamiltonian mean that two interactions are possible, one in which a phonon with wave vector ${\vec {q}}$ in branch $j$ is created, and a second in which a phonon ${\vec {q}}$, $j$ is annihilated. Both processes are accompanied by an electron transition from an initial state ${\vec {k}}_{1}$ into a final state ${\vec {k}}_{2}$. The matrix element $g\left({\vec {k}}_{1},{\vec {k}}_{2};{\vec {q}}j\right)$ describes the electron-phonon coupling and is defined as:

$g\left({\vec {k}}_{1},{\vec {k}}_{2};{\vec {q}}j\right)=-i\varepsilon \left({\vec {q}},j\right)\cdot {\vec {q}}V\left({\vec {q}}\right)$

where $V\left({\vec {q}}\right)$ is the matrix element of the electron-phonon pseudopotential and $\varepsilon \left({\vec {q}},j\right)$ is the phonon polarization.

The lattice vibration of the ions in a metal are partly screened by the conduction electrons. We shall see that this screening changes rather rapidly on certain surfaces in the space of phonon ${\vec {q}}$ vectors and that therefore on surfaces the frequencies $\omega$ vary abruptly with ${\vec {q}}$. And $\omega \left({\vec {q}}\right)$ is a continuous function of ${\vec {q}}$ but that on the surfaces,

$\left|\bigtriangledown _{\vec {q}}\omega \left({\vec {q}}\right)\right|=\infty$

The location of these surfaces is entirely determined by the shape of the electronic Fermi surface.

Here we use the above free electron model for calculation. For an embedded charge distribution,

$\rho _{\text{ext}}\left({\vec {r}}\right)=\rho _{0}e^{i{\vec {q}}\cdot {\vec {r}}}$,

the induced electronic charge density is:

$\rho _{\text{el}}\left({\vec {r}}\right)=-F\left(q\right)\rho _{0}e^{i{\vec {q}}\cdot {\vec {r}}}$,

where

$F\left(q\right)={\frac {1}{\pi a_{0}q^{2}}}\left[1+{\frac {k_{F}}{q}}\left(1-{\frac {q^{2}}{4k_{F}^{2}}}\right)\ln {\left|{\frac {q+2k_{F}}{q-2k_{F}}}\right|}\right]$,

here $a_{0}$ is the Bohr radius. Note that near $q=2k_{F}$,

$F\left(q\right)={\frac {1}{2\pi a_{0}k_{F}}}\left[1+{\frac {1}{2k_{F}}}\left(q-2k_{F}\right)\ln {\left|{q-2k_{F}}\right|}\right]$,

and

${\frac {dF\left(q\right)}{dq}}={\frac {1}{4\pi a_{0}k_{F}^{2}}}\ln {\left|{q-2k_{F}}\right|}\approx -\infty$,

Source: Kohn

this equation shows an abrupt decrease of the ability of the electrons to screen the embedded charge distribution as soon as $q$ exceeds $2k_{F}$.

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

Source: Pegor Aynajian

As long as $q<2k_{F}$, $\rho _{\text{ext}}\left({\vec {r}}\right)$ causes virtual excitations of some electrons with conservation of energy. When $q<2k_{F}$ such excitations are no longer possible.

Now a lattice vibration of wave vector ${\vec {q}}$ produces a change of ionic charge density of the form

$\rho _{\text{ion}}\left({\vec {r}}\right)=\sum A_{\nu }exp\left[i\left({\vec {q}}+{\vec {K_{\nu }}}\right)\cdot {\vec {r}}\right]$,

where ${\vec {K_{\nu }}}$ are the reciprocal lattice vectors.

Therefore we expect an abrupt change of the restoring force whenever ${\vec {q}}$ is such that, for some reciprocal lattice vector ${\vec {K_{\nu }}}$,

$\left|{\vec {q}}+{\vec {K_{\nu }}}\right|=2k_{F}$,

on the surfaces in ${\vec {q}}$ space defined by this equation.

The strength of the Kohn anomaly depends on the joint density of occupied and unoccupied electronic states. If their quantity is large, conduction electrons may, become unstable with respect to a spatially inhomogeneous perturbation. To first order, the response of the electrons to such a perturbation is measured by the generalized susceptibility $\chi _{\vec {q}}$.