Direct Determination of the Electron-Phonon Coupling Matrix Element in a Correlated System

High-resolution electron energy loss spectroscopy measurements have been carried out on an optimally doped cuprate ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$. The momentum-dependent energy and linewidth of an $A_1$ optical phonon were obtained. Based on these data as well as detailed knowledge of the electronic structure, we developed a scheme to determine the electron-phonon coupling (EPC) matrix element related to a specific phonon mode. Such an approach is general and applicable to elucidating the full structure of EPC in a system with anisotropic electronic structure.

Figure 1

(a) STM image of the cleaved BSCCO surface. The left inset shows the $1\times 5$ superstructure with the “$\times 5$” direction aligning to [110] orientation. The right inset shows the low-energy electron diffraction patterns taken with beam energy of 61 eV at room temperature. The line profile along $AB$ is displayed in the lower panel, which indicates a step of 1.5 nm corresponding to the interspacing between two cleavable BiO planes. (b) Schematic drawing of the scattering geometry for the angle-resolved HREELS measurements. (c),(d) The HREELS spectra measured with $\phi =45°$ and 0°, respectively, with an excitation energy of 50 eV at 60 K. The bars (red) are guides to the eye for the phonon energy shift.

Figure 2

(a) ${\Omega }_{\mathit{q}}$ and (b) $\Gamma (\mathit{q})$ of the apical oxygen phonon at 60 K. The upper insets illustrate the azimuth angles of $q_{\parallel }$ with corresponding symbols in $k$ space relative to the Fermi surface, while the lower insets show the data taken at different temperatures across superconducting $T_c$. The dotted lines are guides to the eye. Because of the existence of the ($1\times 5$) superstructure, there are two irreducible nodal directions ($\phi =\pm 45°$). Our measurements show identical momentum dependence along both directions.

Figure 3

(a),(b) The imaginary and real parts of $\chi (\mathit{q},{\Omega }_{\mathit{q}})$ calculated with Eqs. (2, 6), respectively, at 60 K for different $\phi$. (c) The energy contour plot (dotted lines) of the quasiparticle band near $E_F$ of BSCCO. The initial states contributing to the EPC-induced broadening for $q_{\parallel }\sim 0.47{\AA }^{-1}$ at $\phi =45°$ can be determined by superimposing the density plot of the integrating function in Eq. (2) [dark (brown) areas] onto (c). Connecting by $\mathit{q}$ (indicated by the arrows), the final states [light (pink) areas] correspond to an energy gain that equals $\hslash {\Omega }_{\mathit{q}}$.

Figure 4

(a) $|g_{\mathit{q}}{|}^2$ determined by Eqs. (1, 2, 3) and the fitting (solid lines) to Eq. (4) with $a=0.47\mathrm{meV}\AA$ and $b=0.29\mathrm{meV}\AA$. The inset shows the measured $\Gamma (\mathit{q})$ as well as the data calculated with $|g_{\mathit{q}}{|}^2$ for different $\phi$. (b) Measured $\Omega (\mathit{q})$ with data calculated with Eq. (5). Here we assume a bare phonon dispersion ${\Omega }^{\mathrm{bare}}=83.5-1.16[\cos (q_x{a}_0)+\cos (q_y{a}_0)]-1.11\cos (q_x{a}_0)\cos (q_y{a}_0)+0.32[\cos (2q_x{a}_0)+\cos (2q_y{a}_0)](\mathrm{meV})$, as plotted in broken lines.