## 3 Electron-Phonon Interactions: Theoretical Background
### Phonon Frequencies and the Kohn Anomaly
For fermions (electrons) we have:  
\[c_{\mathbf{k}}^{\dagger}c_{\mathbf{k}} = n_{\mathbf{k}} \tag{12}\] \[c_{\mathbf{k}}c_{\mathbf{k}}^{\dagger} = 1-n_{\mathbf{k}} \tag{13}\]  
and for bosons (phonons):  
\[a_{\mathbf{q}}^{\dagger}a_{\mathbf{q}} = n_{\mathbf{q}} \tag{14}\] \[a_{\mathbf{q}}a_{\mathbf{q}}^{\dagger} = 1+n_{\mathbf{q}} \tag{15}\]  
Rearranging the creation and annihilation operators in Equation (11) into the form of number operators and assuming that \(\omega_{\mathbf{q}}=\omega_{-\mathbf{q}}\), and hence that in equilibrium \(\left\langle n_{\mathbf{q}}\right\rangle=\left\langle n_{-\mathbf{q}}\right\rangle\), and \(\left\langle n_{\mathbf{q}}n_{\mathbf{k}}n_{\mathbf{k}^{\prime}}\right\rangle=0\) by symmetry, the total energy of the system is then given by:  
\[\varepsilon=\hbar\omega_{\mathbf{q}}^{0}\left\langle n_{\mathbf{q}}\right\rangle +\sum_{\mathbf{k},\mathbf{k}^{\prime}}|M_{\mathbf{k},\mathbf{k}^{\prime}}|^ {2}\ \left\langle n_{\mathbf{k}}\right\rangle\left[\frac{2\left\langle n_{ \mathbf{q}}\right\rangle\left(\varepsilon_{\mathbf{k}}-\varepsilon_{\mathbf{k} ^{\prime}}\right)}{(\varepsilon_{\mathbf{k}}-\varepsilon_{\mathbf{k}^{\prime }})^{2}-(\hbar\omega_{\mathbf{q}}^{0})^{2}}+\frac{1-\left\langle n_{\mathbf{k} ^{\prime}}\right\rangle}{(\varepsilon_{\mathbf{k}}-\varepsilon_{\mathbf{k}^{ \prime}}-\hbar\omega_{\mathbf{q}}^{0})}\right] \tag{16}\]  
The effect of the e-ph interaction on the phonon spectrum is contained in the term proportional to \(\left\langle n_{\mathbf{q}}\right\rangle\) in Equation (16). Now the perturbed phonon energy \(\hbar\omega_{\mathbf{q}}^{p}\) is the energy required to increase \(\left\langle n_{\mathbf{q}}\right\rangle\) by unit  
\[\hbar\omega_{\mathbf{q}}^{p}=\frac{\partial\varepsilon}{\partial\left\langle n _{\mathbf{q}}\right\rangle}=\hbar\omega_{\mathbf{q}}^{0}+\sum_{\mathbf{k}}|M_ {\mathbf{k},\mathbf{k}^{\prime}}|^{2}\ \frac{2\left\langle n_{\mathbf{k}}\right\rangle( \varepsilon_{\mathbf{k}}-\varepsilon_{\mathbf{k}^{\prime}})}{(\varepsilon_{ \mathbf{k}}-\varepsilon_{\mathbf{k}^{\prime}})^{2}-(\hbar\omega_{\mathbf{q}}^ {0})^{2}} \tag{17}\]  
If we neglect the phonon energy in the denominator in comparison with the electron energies we have  
\[\hbar\omega_{\mathbf{q}}^{p}=\hbar\omega_{\mathbf{q}}^{0}-\sum_{\mathbf{k}}2 |M_{\mathbf{k},\mathbf{k}^{\prime}}|^{2}\ \left\langle n_{\mathbf{k}}\right\rangle( \varepsilon_{\mathbf{k}^{\prime}}-\varepsilon_{\mathbf{k}})^{-1} \tag{18}\]  
One may picture the origin of this change in phonon frequency by Figure 8a, in which the first interaction is represented as the creation of an electron-hole (e-h) pair by the absorption of a phonon. One can then say that it is the fact that the phonon spends part of its lifetime (in the order of picoseconds) in the form of an e-h pair that modifies its energy. One interesting consequence of Equation (18) occurs in metals when q has a value close to the diameter of the Fermi surface \(2k_{F}\). In this case, the states k and k\({}^{\prime}\) =k - q are connected by q in  
\[\varepsilon_{2}=\langle\Phi|\sum_{\mathbf{k},\mathbf{k}^{\prime}}|M_{\mathbf{ k},\mathbf{k}^{\prime}}|^{2}\;\left[\frac{(a^{\dagger}_{-\mathbf{q}}a_{- \mathbf{q}})\;c^{\dagger}_{\mathbf{k}}c_{\mathbf{k}^{\prime}}\;c^{\dagger}_{ \mathbf{k}^{\prime}}c_{\mathbf{k}}}{(\varepsilon_{0}-H_{0})}+\frac{(a_{ \mathbf{q}}a^{\dagger}_{\mathbf{q}})\;c^{\dagger}_{\mathbf{k}}c_{\mathbf{k}^ {\prime}}\;c^{\dagger}_{\mathbf{k}^{\prime}}c_{\mathbf{k}}}{(\varepsilon_{0} -H_{0})}\right]\;|\Phi\rangle \tag{11}\]0 The Frohlich Hamiltonian includes an interaction term in which an electron is scattered from \(k^{\prime}\) to \(k\) with either (**a**) emission or (**b**) absorption of a phonon. In each case the total momentum is conserved. the same Fermi surface, which causes a logarithmic divergence in Equation (18), and thus a kink in the phonon dispersion is observed. This divergence when \(q=2k_{F}\) is the so-called _Kohn anomaly_[1]. For graphene, the Fermi surface is formed by the six K points.