Anisotropic Electron-Phonon Interaction in the Cuprates

We explore manifestations of electron-phonon coupling on the electron spectral function for two phonon modes in the cuprates exhibiting strong renormalizations with temperature and doping. Applying simple symmetry considerations and kinematic constraints, we find that the out-of-plane, out-of-phase O buckling mode ($B_{1g}$) involves small momentum transfers and couples strongly to electronic states near the antinode while the in-plane Cu-O breathing modes involve large momentum transfers and couples strongly to nodal electronic states. Band renormalization effects are found to be strongest in the superconducting state near the antinode, in full agreement with angle-resolved photoemission spectroscopy data.

Figure 1

Plots of the electron-phonon coupling $|g(\mathbf{k},\mathbf{q}){|}^2$ for initial $\mathbf{k}$ and scattered ${\mathbf{k}}^{\prime }=\mathbf{k}-\mathbf{q}$ states on the Fermi surface for the buckling mode (left panels) and breathing mode (right panels) for initial fermion $\mathbf{k}$ at an antinodal (top panels) and a nodal (bottom panels) point on the Fermi surface, as indicated by the arrows. The red or blue color indicates the maximum or minimum of the el-ph coupling vertex in the BZ for each phonon.

Figure 2

Plots of the electron-phonon coupling ${\lambda }_{\mathbf{k}}$ in the first quadrant of the BZ for the buckling mode (right top panel) and the breathing mode (right bottom panel). The color scale is shown on the right for each phonon. The left panel shows energy contours for the band structure used [3].

Figure 3

Image plots of the calculated spectral functions in the normal (a1,b1,c1) and superconducting (a2,b2,c2) states compared to the spectral functions in the normal (a3,b3,c3) and superconducting (a4,b4,c4) states measured in ${\mathrm{B}\mathrm{i}}_2{\mathrm{S}\mathrm{r}}_2{\mathrm{C}\mathrm{a}}_{0.92}{\mathrm{Y}}_{0.08}{\mathrm{C}\mathrm{u}}_2{\mathrm{O}}_{8+\delta }$ (Bi-2212) [6] for momentum cuts $a,b,c$ shown in the rightmost panel and in Fig. 2. The same color scale is used for the normal or superconducting pairs within each cut, but the scaling for the data and the calculation are separate. The red markers indicate 70 meV in the superconducting state.