Quantum mechanical picture of the coupling between interlayer electronic excitations and infrared active phonons in bilayer cuprate superconductors

The formula frequently used to describe the $c$-axis infrared response of the coupled electron-phonon system of bilayer cuprate superconductors and providing important insights into the physics of these materials has been originally obtained at the level of the phenomenological multilayer model. Here we derive it using diagrammatic perturbation theory.

Figure 1

(a) Schematic representation of the multilayer model, reproduced from Ref. 26. The shaded planes represent the copper-oxygen planes. The distance between the closely spaced planes forming a bilayer (between the bilayers) is denoted by $d_{\mathrm{bl}}$ ($d_{\mathrm{int}}$), the $c$-axis lattice parameter $d$ is given by $d=d_{\mathrm{bl}}+d_{\mathrm{int}}$. If an ac electric field $E_{\mathrm{ext}}$ is applied perpendicular to the planes, currents flow between the planes. The corresponding current densities are denoted by $j_{\mathrm{bl}}$ and $j_{\mathrm{int}}$. They are not equal and the planes thus become charged. The resulting charge density alternates from one plane to the other ($\rho$ and $-\rho$). As a consequence, the average electric field of the bilayer region $E_{\mathrm{bl}}$ differs from that of the interbilayer region (spacing layer) $E_{\mathrm{int}}$. The current densities are related to the fields via $j_{\mathrm{bl}}={\sigma }_{\mathrm{bl}}{E}_{\mathrm{bl}}$ and $j_{\mathrm{int}}={\sigma }_{\mathrm{int}}{E}_{\mathrm{int}}$ with the local conductivities ${\sigma }_{\mathrm{bl}}$ and ${\sigma }_{\mathrm{int}}$. Feynman diagrams representing the correlators defined in the text [(b) and (c)], the interaction vertices corresponding to $H_C$ of Eq. (13) and $H_{\mathrm{e}-\mathrm{p}}$ of Eq. (16) [(d)], and two segments that appear in the series of part (f) [(e)]. (f) Series of the diagrams corresponding to the quantities ${\chi }_A$, ${\chi }_B$, ${\chi }_C$, and ${\chi }_D$ defined in the text. The sum of the contributions of the diagrams is to be multiplied by $d_{\mathrm{bl}}^2/d^2$ in case of ${\chi }_A$ and by $d_{\mathrm{bl}}/d$ in case of ${\chi }_C$ and ${\chi }_D$.