## 3 Electron-Phonon Interactions: Theoretical Background
### Phonon Frequencies and the Kohn Anomaly  
To calculate the effect of the e-ph interaction on the phonon spectrum, we may use perturbation theory to calculate the total energy \(\varepsilon\) of the system described by the Frohlich Hamiltonian (Equation (8)) to second order in \(H_{\mathrm{e-ph}}\):  
\[\varepsilon=\varepsilon_{0}+\langle\Phi|H_{el-ph}|\Phi\rangle+\big{\langle} \Phi|H_{el-ph}(\varepsilon_{0}-H_{0})^{-1}H_{el-ph}|\Phi\big{\rangle} \tag{10}\]  
with \(\varepsilon_{0}=\varepsilon_{\mathbf{k}}\;c^{\dagger}_{\mathbf{k}}c_{ \mathbf{k}}+\hbar\omega^{0}_{\mathbf{q}}\;a^{\dagger}_{\mathbf{q}}a_{\mathbf{q}}\) being the unperturbed energy of the state \(\Phi\) having \(n_{\mathbf{q}}\) phonons in the longitudinally polarized mode \(\mathbf{q}\) and \(n_{\mathbf{k}}\) electrons in state \(\mathrm{k}\). Since the components of \(H_{\mathrm{e-ph}}\) act on \(\Phi\) either to destroy or create one phonon, the first-order term vanishes from this expression because the resulting wavefunction must be orthogonal to \(\Phi\). In second order there is a set of nonvanishing terms, as the phonon destroyed by the first factor of \(H_{\mathrm{e-ph}}\) to act on \(\Phi\) can be replaced by the second factor of \(H_{\mathrm{e-ph}}\), and vice versa. We then find the contribution of the second-order terms \(\varepsilon_{2}\) to be  
\[\varepsilon_{2}=\langle\Phi|\sum_{\mathbf{k},\mathbf{k}^{\prime}}|M_{\mathbf{ k},\mathbf{k}^{\prime}}|^{2}\;\left[\frac{(a^{\dagger}_{-\mathbf{q}}a_{- \mathbf{q}})\;c^{\dagger}_{\mathbf{k}}c_{\mathbf{k}^{\prime}}\;c^{\dagger}_{ \mathbf{k}^{\prime}}c_{\mathbf{k}}}{(\varepsilon_{0}-H_{0})}+\frac{(a_{ \mathbf{q}}a^{\dagger}_{\mathbf{q}})\;c^{\dagger}_{\mathbf{k}}c_{\mathbf{k}^ {\prime}}\;c^{\dagger}_{\mathbf{k}^{\prime}}c_{\mathbf{k}}}{(\varepsilon_{0} -H_{0})}\right]\;|\Phi\rangle \tag{11}\]  
all other terms having zero matrix element since the resulting wavefunction will be orthogonal to \(\Phi\). The first term in the brackets in Equation (11) can be represented as in Figure 8a. An electron is first scattered from \(\mathrm{k}\) to \(\mathrm{k}^{\prime}\) with the absorption of a phonon with momentum \(-\mathrm{q}=\mathrm{k}^{\prime}-\mathrm{k}\). The factor \((\varepsilon_{0}-H_{0})^{-1}\) measures the amount of time the electron is allowed by the Uncertainty Principle to stay in the intermediate state \(\mathrm{k}^{\prime}\) and can be written as the energy difference between the initial and intermediate states, \((\varepsilon_{\mathbf{k}}+\hbar\omega^{0}_{-\mathbf{q}}-\varepsilon_{\mathbf{ k}^{\prime}})^{-1}\). The electron is then scattered back into its original state with the re-emission  
Figure 7: The Frohlich Hamiltonian includes an interaction term in which an electron is scattered from \(\mathrm{k}^{\prime}\) to \(\mathrm{k}\) with either (**a**) emission or (**b**) absorption of a phonon. In each case the total momentum is conserved. of the phonon. We can represent the second term in Equation (11) by Figure 8b, and then, we find an energy denominator of \(\varepsilon_{\mathbf{k}}-\hbar\omega_{\mathbf{q}}^{0}-\varepsilon_{\mathbf{k}^{ \prime}}\). We can write the creation and annihilation operators in terms of occupation numbers (\(n_{\mathbf{k}}\) or \(n_{\mathbf{q}}\)).