## 3 Electron-Phonon Interactions: Theoretical Background
### The Frohlich Hamiltonian  
This model takes for granted the concept of screening and it assumes that the ions interact with each other and with the electrons only through a short-range screened potential (dashed line in Figure 6) and it treats the electrons themselves as independent fermions. Also it neglects electron-electron interactions. For a Bravais lattice the unperturbed Hamiltonian, where the electrons and phonons are treated separately, is [1]:  
\[H_{0}=\sum_{\mathbf{k}}\varepsilon_{\mathbf{k}}c^{\dagger}_{\mathbf{k}}c_{ \mathbf{k}}+\sum_{\mathbf{q},\mathbf{s}}\hbar\omega_{\mathbf{q},\mathbf{s}}a^{ \dagger}_{\mathbf{q},\mathbf{s}}a_{\mathbf{q},\mathbf{s}} \tag{2}\]  
where \(c^{\dagger}_{k}\) (\(c_{\mathbf{k}}\)) is the creation (annihilation) operator for the electrons with energy \(\varepsilon_{\mathbf{k}}\) and momentum k, and \(a^{\dagger}_{\mathbf{q}}\) (\(a_{\mathbf{q}}\)) is the creation (annihilation) operator for the phonons with energy \(\hbar\omega_{\mathbf{q}}\), momentum q and direction of polarization s. If it happens that s is parallel to q, we say that it is a longitudinally polarized phonon in the crystal. If s\(\cdot\)q = 0, the phonon is transversely polarized. To the unperturbed Hamiltonian, we add the interaction \(H_{1}\) of the electrons with the screened ions:  
\[H_{1}=\sum_{\mathbf{k},\mathbf{k}^{\prime},\mathbf{l}}\left\langle\mathbf{k} |V(\mathbf{r}-\mathbf{l}-\mathbf{y}_{\mathbf{l}})|\mathbf{k}^{\prime}\right\rangle c ^{\dagger}_{\mathbf{k}}c_{\mathbf{k}^{\prime}} \tag{3}\]  
where \(|\mathbf{k}\rangle\) and \(|\mathbf{k}^{\prime}\rangle\) are the initial and final electronic state, respectively. Here, it is assumed that at any point the potential \(V(\mathbf{r}-1-\mathbf{y}_{l})\) due to a particular ion depends only on the distance from the center of the ion, where \(\mathbf{y}_{l}\) is the ion displacement from the equilibrium position l. We can use the Fourier transform of the potencial, and Equation (3) can be written as:  
\[H_{1}=\sum_{\mathbf{k},\mathbf{k}^{\prime},\mathbf{l}}e^{i(\mathbf{k}^{\prime }-\mathbf{k})\cdot(\mathbf{l}+\mathbf{y}_{\mathbf{l}})}\;V_{\mathbf{k}- \mathbf{k}^{\prime}}\;c^{\dagger}_{\mathbf{k}}c_{\mathbf{k}^{\prime}} \tag{4}\]  
If we assume that the displacement \(\mathbf{y}_{l}\) of the ion is sufficiently small that \((\mathbf{k}^{\prime}-\mathbf{k})\cdot\mathbf{y}_{\mathbf{l}}\ll\mathbf{1}\), we can separate \(H_{1}\) into two parts, \(H_{1}=H_{\mathrm{Bloch}}+H_{\mathrm{e-ph}}\). The first term  
\[H_{\mathrm{Bloch}}=\sum_{\mathbf{k},\mathbf{k}^{\prime},\mathbf{l}}e^{i( \mathbf{k}^{\prime}-\mathbf{k})\cdot\mathbf{l}}\;V_{\mathbf{k}-\mathbf{k}^{ \prime}}\;c^{\dagger}_{\mathbf{k}}c_{\mathbf{k}^{\prime}} \tag{5}\]  
is independent of the lattice displacements. The second term can be written as:  
\[H_{\mathrm{e-ph}}=iN^{1/2}\sum_{\mathbf{k},\mathbf{k}^{\prime}}(\mathbf{k}^{ \prime}-\mathbf{k})\cdot\mathbf{y}_{\mathbf{q}}\;V_{\mathbf{k}-\mathbf{k}^{ \prime}}\;c^{\dagger}_{\mathbf{k}}c_{\mathbf{k}^{\prime}} \tag{6}\]  
where q = k \(-\) k\({}^{\prime}\). The displacement \(\mathbf{y}_{q}\) can be written in the harmonic approximation as a function of the phonon creation and annihilation operators [1] and the \(H_{\mathrm{e-ph}}\) then becomes:  
\[H_{\mathrm{e-ph}}=i\sum_{\mathbf{k},\mathbf{k}^{\prime},\mathbf{s}}\sqrt{ \frac{N\hbar}{2M\omega_{\mathbf{k}-\mathbf{k}^{\prime},\mathbf{s}}}}\;( \mathbf{k}^{\prime}-\mathbf{k})\cdot\mathbf{s}\;V_{\mathbf{k}-\mathbf{k}^{ \prime}}\;(a^{\dagger}_{\mathbf{k}^{\prime}-\mathbf{k},\mathbf{s}}+a_{\mathbf{ k}-\mathbf{k}^{\prime},\mathbf{s}})\;c^{\dagger}_{\mathbf{k}}c_{\mathbf{k}^{ \prime}} \tag{7}\]  
For simplicity we shall assume the phonon spectrum to be isotropic, so that the phonons will be either longitudinally or transversely polarized. Then, only the longitudinal modes, for which s is parallel to k\({}^{\prime}-\) k, will enter \(H_{\mathrm{e-ph}}\). Also, since the \(H_{\mathrm{Bloch}}\) is not dependent of the displacement, we shall neglect its effects for the e-ph interaction.