Determining the electron-phonon coupling in superconducting cuprates by resonant inelastic x-ray scattering: Methods and results on ${\mathrm{Nd}}_{1+x}{\mathrm{Ba}}_{2-x}{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$

The coupling between lattice vibration quanta and valence electrons can induce charge-density modulations and decisively influence the transport properties of materials, e.g., leading to conventional superconductivity. In high-critical-temperature superconductors, where electronic correlation is the main actor, the actual role of electron-phonon coupling (EPC) is being intensely debated theoretically and investigated experimentally. We present an in-depth study of how the EPC strength can be obtained directly from resonant inelastic x-ray scattering (RIXS) data through the theoretical approach derived by Ament et al. [Europhys. Lett. 95, 27008 (2011)]. The role of the model parameters (e.g., phonon energy ${\omega }_0$, intermediate state lifetime $1/\mathrm{\Gamma }$, EPC matrix element $M$, and detuning energy $\mathrm{\Omega }$) is thoroughly analyzed, providing general relations among them that can be used to make quantitative estimates of the dimensionless EPC $g={(M/{\omega }_0)}^2$ without detailed microscopic modeling. We then apply these methods to very high-resolution Cu $L_3$-edge RIXS spectra of three ${\mathrm{Nd}}_{1+x}{\mathrm{Ba}}_{2-x}{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ films. For the insulating antiferromagnetic parent compound, the value of $M$ as a function of the in-plane momentum transfer is obtained for Cu-O bond-stretching (breathing) and bond-bending (buckling) phonon branches. For the underdoped and the nearly optimally doped samples, the effects of Coulomb screening and of charge-density-wave correlations on $M$ are assessed. In light of the anticipated further improvements of the RIXS experimental resolution, this work provides a solid framework for an exhaustive investigation of the EPC in cuprates and other quantum materials.

Figure 1

(a) The crystal structure of undoped antiferromagnetic NBCO. (b) The layout of the experiment. (c) An example RIXS spectrum of NBCO collected at ${\mathbf{Q}}_{\parallel }=(\frac{2\pi }{a}h,0)$ with $h=-0.4$ and $T=20$ K. A wide energy range covering the $dd$ excitations is displayed. The inset shows a closeup of the region of interest to this work, which clearly reveals a phonon feature at approximately 70 meV.

Figure 2

A sketch of the RIXS process leading to low-energy phonon excitations at the Cu $L_3$ edge. Upon absorption of a photon, a $2p_{3/2}$ electron is promoted into an empty $3d$ state and a core hole is left in the Cu ion. Due to the attraction between the core hole and the excited electron, the intermediate charge distribution has an excitonic character. When viewed from the oxygen ion, the core hole is well screened while the weakened Cu-O bond pushes the oxygen ions toward an equilibrium position at farther distance. In the intermediate state, several phonons are thus excited. When the core hole is filled by the electron that was initially promoted to the valence state and a photon is emitted (radiative decay), the system reaches its original electronic ground state, but one or more phonons are left behind in the sample.

Figure 3

The universal plots of the phonon excitation intensities based on rescaled dimensionless variables. $M$ is the absolute value of the EPC matrix element, $\mathrm{\Gamma }$ is the intrinsic half-width at half-maximum of the Cu $L_3$ resonance so that $1/\mathrm{\Gamma }$ is proportional to the core-hole lifetime, and $g={(M/{\omega }_0)}^2$ is a dimensionless coupling constant. Left axis: the intensity of the one-phonon excitation $I_1$ rescaled by ${({\omega }_0/\mathrm{\Gamma })}^2$ and plotted as a function of ${(M/\mathrm{\Gamma })}^2$. The different lines corresponding to different values of $g$ superimpose almost perfectly, so that the curve is universal. Right axis: the ratio between the two- and one-phonon excitation intensities as a function of ${(M/\mathrm{\Gamma })}^2$. The behavior is also universal for ${(M/\mathrm{\Gamma })}^2\lesssim 1.5$. The inset shows that in the limit of small coupling, the universal curves of $I_1$ (solid black line) and $I_2/I_1$ (dashed red line) share the same behavior as a function of ${(M/\mathrm{\Gamma })}^2$ apart from an overall factor.

Figure 4

A geometrical construction showing the relationship between the parameters. The left panel plots the universal curve obtained by inverting the function of Fig. 3, while the right panel has ordinates ${(M/\mathrm{\Gamma })}^2$ (in common with the left panel) and ${({\omega }_0/\mathrm{\Gamma })}^2$ as abscissa. The way of using the diagram depends on the choice of the input data. For instance, suppose we know the intensity ratio $I_2/I_1$, the phonon frequency ${\omega }_0$, and the core-hole lifetime $1/\mathrm{\Gamma }$. We enter the left panel with a vertical line at $I_2/I_1$ that intercepts the universal curve at point ${\mathrm{P}}_1$; this gives the value of ${(M/\mathrm{\Gamma })}^2$. The horizontal line through ${\mathrm{P}}_1$ intersects the vertical line through ${({\omega }_0/\mathrm{\Gamma })}^2$ in the right panel at point ${\mathrm{P}}_2$. The straight line from the origin to ${\mathrm{P}}_2$ has the slope $tan\alpha =M^2/{\omega }_0^2=g$.

Figure 5

Comparison between the RIXS spectrum of NBCO-AF measured at in-plane momentum transfer $h=-0.4$ r.l.u. at resonance (black squares) and energy detuned by $\mathrm{\Omega }=-0.5$ eV (red circles). The spectra are normalized to the $dd$ excitations. The decrease in intensity of the phonon and of the bimagnon excitations is evident.

Figure 6

Theoretical intensity of local phonon excitations upon detuning. The different panels correspond to specific values of $\mathrm{\Gamma }/{\omega }_0$. Within each panel, the curves are labeled with the dimensionless coupling constant $g$ (from left to right: $g=100$, 64, 32, 16, 8, 4, 2). The curves are normalized to the value at resonance.

Figure 7

(a) Width at half-maximum of the theoretical detuning curves as a function of $\sqrt{g}$ with $\mathrm{\Gamma }/{\omega }_0$ as a parameter. The width $W$ is plotted in units of ${\omega }_0$. Note the linear dependence of the width over a wide range of $\sqrt{g}$. The lines at small values of the parameter $\mathrm{\Gamma }/{\omega }_0$ are very close one to the other, suggesting an approximate scaling law. (b) An example of the use of the simplified detuning rule. Suppose that the red straight line approximates the system in the parameter range $2\le \mathrm{\Gamma }/{\omega }_0\le 8$. If the width $W$ of the detuning curve is measured, the intersection of the horizontal line with height $W$ with the red line approximates the value of $g$ without numerical calculations.

Figure 8

Experimental results on NBCO-AF and assignments of the phonon modes. (a) Stack of raw RIXS spectra with the elastic line subtracted (black solid lines). The spectra show the presence of two main features in the phonon energy range (shaded areas) whose intensities have opposite momentum dependence. (b) Experimental dispersion relation of the two phonon branches: the Cu-O bond-buckling branch (red circles) at an energy of $\approx 30$ meV and the Cu-O bond-stretching or breathing branch (blue squares) at an energy of $\approx 70$ meV. The lines are linear fits to the dispersion relations.

Figure 9

The application of the detuning approach to the $h=-0.4$ r.l.u. mode of the breathing branch in NBCO-AF. (a) Cu $L_3$-edge RIXS spectra of NBCO-AF as a function of detuning energy $\mathrm{\Omega }$. (b) Detuning dependence of the phonon intensity (squares) compared against the theoretical detuning curves. Note that the experimental values discriminate very well among the theoretical curves having different dimensionless coupling constant $g$.

Figure 10

Fitting and decomposition of the RIXS spectra. The complete presentation is given in Appendix pp2, and here we summarize the main points. The decomposition of the high-statistics RIXS spectrum of NBCO-AF at $h=-0.4$ r.l.u., which is used to recover $I_1$ and $I_2$ of the buckling and breathing modes. (a) The RIXS spectrum (black empty squares) is decomposed into an elastic contribution (orange solid line) and an inelastic signal (gray filled circles). A tiny tail arising from the magnetic excitations has been subtracted. (b) Decomposition of the inelastic signal into buckling (red dotted line) and breathing (blue dashed line) modes and their overtones (orange dotted and light blue dashed lines, respectively). The dark yellow line is the sum of the components. The residuals are plotted as a green solid line, vertically offset for clarity. The full decomposition procedure is detailed in Appendix pp2.

Figure 11

The final results of the data analysis of NBCO-AF. The absolute value of the matrix element $M$ of the EPC in units of $\mathrm{\Gamma }$ is plotted as a function of the momentum transfer for the buckling (red circles) and breathing (blue squares) branches. Dashed and dotted lines are a fitting with sine and cosine functions, respectively.

Figure 12

The upper panel qualitatively shows two situations that modify the RIXS phonon signal: in the absence of charge order a smooth decrease of the EPC with doping is expected due to the increased screening from the free carriers. In the presence of CDWs, instead, CDW-induced effects may prevail so that a nonmonotonic trend is found as a function of doping. Indeed, the signal from the CDWs is absent in the AF sample, is the strongest in the UD sample, and is weaker in the OP sample. The nonmonotonic trend is not a necessary condition but it is sufficient to demonstrate the prevalence of the CDWs. This is indeed the case, as shown by the experimental results of the bottom panel, where spectra of NBCO-AF (black circles), NBCO-UD (red squares), and NBCO-OP (blue diamonds) are compared. The spectra were collected at $h=-0.4$ r.l.u. and are measured with an energy resolution of 60 meV. Note that in the 0.6–0.8 eV energy range, the spectral intensity scales with doping.

Figure 13

Evidence for the increase of the phonon signal from the NBCO-AF to the NBCO-OP sample at large momentum transfer $h=-0.4$ r.l.u. [(a), (c)]. No difference is detected in the phonon energy range at small momentum transfer $h=-0.1$ r.l.u. [(b), (d)]. The bottom panels show a closeup of the inelastic region of the spectra. The RIXS spectrum of NBCO-AF is shown in black, the one of NBCO-OP in red. In the latter, the continuum has been subtracted. Despite the difficulty of the subtraction, the difference spectra (OP minus AF, blue solid lines) are very clear. The difference spectra are vertically offset by $-0.4$ for clarity.

Figure 14

The different behavior of the NBCO-AF and NBCO-OP samples upon detuning. The doped sample (red circles) is definitely more robust than the undoped one (black squares). The difference spectrum (blue diamonds) shows that both phonon branches detected by RIXS are more robust in the OP sample.

Figure 15

EPC strength $M$ (in units of $\mathrm{\Gamma }$) as a function of momentum transfer for the buckling (red circles) and breathing (blue squares) modes. The momentum dependence of the EPC is reported from Fig. 11. Lines are best fits to the data obtained from the local (thick) and itinerant (thin) electron models.

Figure 16

Detailed presentation of the decomposition of the RIXS spectrum of NBCO-AF, already presented more briefly in Fig. 10. See text of Appendix pp2 for details.