Doping dependence of the electron-phonon coupling in two families of bilayer superconducting cuprates

While electron-phonon coupling (EPC) is crucial for Cooper pairing in conventional superconductors, its role in high-$T_c$ superconducting cuprates is debated. Using resonant inelastic x-ray scattering at the oxygen $K$ edge, we study the EPC in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ (Bi2212) and ${\mathrm{Nd}}_{1+x}{\mathrm{Ba}}_{2-x}{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ (NBCO) at different doping levels ranging from heavily underdoped ($p=0.07$) to overdoped ($p=0.21$). We analyze the data with a localized Lang-Firsov model that allows for the coherent excitations of two phonon modes. While electronic band dispersion effects are non-negligible, we are able to perform a study of the relative values of EPC matrix elements in these cuprate families. In the case of NBCO, the choice of the excitation energy allows us to disentangle modes related to the CuO chains and the $\mathrm{Cu}{\mathrm{O}}_2$ planes. Combining the results from the two families, we find the EPC strength decreases with doping at ${\mathbf{q}}_{\parallel }=(-0.25,0)$ r.l.u., but has a nonmonotonic trend as a function of doping at smaller momenta. This behavior is attributed to the screening effect of charge carriers. We also find that the phonon intensity is enhanced in the vicinity of the charge-density-wave excitations while the extracted EPC strength appears to be less sensitive to their proximity. By performing a comparative study of two cuprate families, we are able to identify general trends in the EPC for the cuprates and provide experimental input to theories invoking a synergistic role for this interaction in $d$-wave pairing.

Figure 1

Overview of data on NBCO and Bi22212 at the O $K$ edge. (a), (b) XAS spectra for different doping values of Bi2212 and NBCO, respectively. The legends indicate the critical temperature of the samples. The relevant resonances are indicated with blue and red arrows. (c), (d) Overview of RIXS spectra in Bi2212 OP91 and NBCO OP92, respectively. The excitation energies for the different spectra are highlighted as ticks in the XAS, using the same color code. (e) Example of RIXS spectrum at oxygen $K$ edge collected on NBCO OP92 at 528.2 eV. Labels indicate the different excitations. (f) Zoom-in of the low-energy regions of RIXS spectra for all the samples acquired at $H=-0.15$ r.l.u. and at the energy corresponding to the plane resonance. The elastic peak has been subtracted for clarity. The spectra have been normalized to the spectral weight between 1 and 7 eV, corresponding to the CT excitation range. (g) Sketch of the experimental geometry. The incoming and outgoing x rays are reported in yellow arrows, and the incident ($\theta$) and scattering ($2\theta$) angles are highlighted. The component of the transferred momentum parallel to the $\mathrm{Cu}{\mathrm{O}}_2$ planes is reported in light green.

Figure 2

XAS and phonon resonances in Bi2212 OP91 and NBCO OP92. (a), (b) XAS spectra (grey shade area), the spectral area of phonon (orange circles), and charge-transfer excitations (green diamonds) for Bi2212 OP91 and NBCO OP92, respectively. The red and blue dashed vertical lines indicate the chain and plane resonant energies in NBCO OP92, respectively. (c) RIXS spectra collected at the chain (red) and plane (blue) resonances in NBCO OP92 after subtracting the elastic peaks for comparison.

Figure 3

Detuning data on Bi2212 OP91 (left panels) and NBCO UD38 (right panels). Incident energies were chosen around the plane resonance for Bi2212 OP91 and of chain resonance for NBCO UD38 for the reasons discussed in the text. The momentum transfer was $(-0.15,0)$ r.l.u. for Bi2212 OP91 and $(-0.25,0)$ r.l.u. for NBCO UD38. Solid lines are the fitting curves, decomposed into the various phonon branches, obtained through the global fitting discussed in the main text. We have highlighted that, in the case of plane resonance of Bi2212, the phonon resonance is shifted with respect to the maximum of the XAS. A failure of the model is evident at large detuning energies.

Figure 4

(a) Fits of RIXS spectra after subtracting the high-energy excitations for Bi2212 OD82, taken at momenta transfer $H=-0.2$ r.l.u.. Elastic peak is fitted by a Gaussian function determined by the energy resolution. The phonon excitations are fitted with the Frank-Condon model considering two phonons, i.e., the buckling mode ($\sim 40$ meV) and stretching mode ($\sim 70$ meV). (b) Phonon components are shown after subtracting the elastic peak. (c)–(e) Doping dependence of phonon amplitude for Bi2212, which shows a maximum at $p_c=0.19$. The phonon amplitude of NBCO is also shown in (c) and divided a factor of 2 to be close with the data of Bi2212. (f)–(h) Doping dependence of the electron-phonon coupling strengths $M$ for both NBCO and Bi2212. The SC phase diagram [52] of cuprates (shaded area) is displayed in (f).

Figure 5

The momentum dependence of the fitting parameters in Bi2212 OP91 (a), (b) and NBCO OP92 (c), (d). Phonon amplitudes increase with larger q, while the $M$ values do not show a clear momentum dependence. The extracted $M$ values of two samples are close to one another.

Figure 6

(a) Resistance of the underdoped and optimally doped NBCO epitaxial films as a function of temperature. (b) Magnetization measurements of the Bi2212 single crystals as a function of temperature in a magnetic field of 1 Oe. The horizontal dashed line is the zero of magnetization.