Magnetic excitations and phonons simultaneously studied by resonant inelastic x-ray scattering in optimally doped ${\mathrm{Bi}}_{1.5}{\mathrm{Pb}}_{0.55}{\mathrm{Sr}}_{1.6}{\mathrm{La}}_{0.4}{\mathrm{CuO}}_{6+\delta }$

Magnetic excitations in the optimally doped high-$T_{\mathrm{c}}$ superconductor ${\mathrm{Bi}}_{1.5}{\mathrm{Pb}}_{0.55}{\mathrm{Sr}}_{1.6}{\mathrm{La}}_{0.4}{\mathrm{CuO}}_{6+\delta }$ (OP-Bi2201, $T_{\mathrm{c}}\simeq 34$ K) are investigated by Cu $L_3$ edge resonant inelastic x-ray scattering (RIXS), below and above the pseudogap opening temperature. At both temperatures the broad spectral distribution disperses along the (1,0) direction up to $\sim 350$ meV at zone boundary, similar to other hole-doped cuprates. However, above $\sim 0.22$ reciprocal lattice units, we observe a concurrent intensity decrease for magnetic excitations and quasielastic signals with weak temperature dependence. This anomaly seems to indicate a coupling between magnetic, lattice, and charge modes in this compound. We also compare the magnetic excitation spectra near the antinodal zone boundary in the single layer OP-Bi2201 and in the bilayer optimally doped ${\mathrm{Bi}}_{1.5}{\mathrm{Pb}}_{0.6}{\mathrm{Sr}}_{1.54}{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ (OP-Bi2212, $T_{\mathrm{c}}\simeq 96$ K). The strong similarities in the paramagnon dispersion and in their energy at zone boundary indicate that the strength of the superexchange interaction and the short-range magnetic correlation cannot be directly related to $T_{\mathrm{c}}$, not even within the same family of cuprates.

Figure 1

(a) Experimental geometry. (b) Recipr-ocal-space image, the nuclear and magnetic first Brillouin zones are drawn with solid and dashed lines, respectively; the thick green line indicates the range covered by the experiments. (c) Representative RIXS spectra of OP-Bi2201 for various $\delta$ values in steps of ${25}^{\circ }$ as indicated by the black empty circles in panel (b), i.e., from ${15}^{\circ }$ grazing incidence to ${15}^{\circ }$ grazing emission, as measured with respect to the sample surface. Top: $\pi$ polarized incident x rays. Bottom: $\sigma$ polarized incident x rays. Data were collected at 50 K and 200 K. (d) Single spin-flip fraction (without orbital excitation) for $\pi$ or $\sigma$ polarizations according to RIXS cross section calculations for $2\theta ={130}^{\circ }$. The non-spin-flip channel can be spread over elastic, charge excitations, double-spin-flip excitations, and phonons, covering a broad spectral range from 0 to 1 eV energy loss. This explains why spin-flip channel, less intense but more concentrated in energy, is often the most easily recognizable spectral feature. (e) RIXS spectra measured at $Q_{\parallel }=-0.4$ r.l.u. with grazing-incidence geometry ($\delta =-{50}^{\circ }$) and measured at $Q_{\parallel }=0.4$ r.l.u. with grazing-emission geometry ($\delta =+{50}^{\circ }$) for $\pi$- or $\sigma$-polarized light. Data were collected at 50 K. All spectra are normalized to the integrated intensity of the $dd$ excitations.

Figure 2

RIXS spectra of OP-Bi2201 along (0,0)-(0.5,0) symmetry direction (black solid circles) at 50 K (left) and 200 K (right). The spectra are decomposed into the dispersive magnetic excitations (blue line), the elastic scattering (green line), the phonon scattering (orange line), and the charge- and $dd$-excitations background (gray line). The blue arrow indicates the paramagnon peak. The data were collected at grazing-emission using $\pi$ incident polarization.

Figure 3

Energy/momentum intensity false-color maps of RIXS spectra along (0,0)-(0.5,0) symmetry direction measured at (a) 50 K and (b) 200 K with $\pi$ polarization on OP-Bi2201. The white solid squares indicate the paramagnon peak positions as determined with the fitting procedure illustrated in Fig. 2. (c) RIXS spectra at 50 K (black) and 200 K (green) at selected $Q_{\parallel }$. (d) Experimental paramagnon dispersion and (e) the integrated intensity of paramagnon peak at 50 K (black) and 200 K (green) determined from the fitting procedure. Integrated inelastic intensity of optimally doped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ from Ref. [5] is superimposed for comparison (blue). (f) Intensity at $[-0.15,0.1]$ eV energy window for quasielastic signal at 50 K (black) and 200 K (green). Self-absorption correction has been applied to (e) and (f). The error bars represent the uncertainty in the fitting.

Figure 4

Energy/momentum intensity false-color maps of RIXS spectra along ($-0.5,0$)-(0.5,0) symmetry direction measured at (a) 50 K and (b) 200 K with $\sigma$ polarization on OP-Bi2201. (c) Polarization comparison of RIXS spectra measured at $Q_{\parallel }=0.22$ r.l.u. at 50 K and 200 K.

Figure 5

(a) Comparison of RIXS spectra of OP-Bi2201 and OP-Bi2212 at large $Q_{\parallel }$ measured with $\pi$ polarization. Two data sets were collected for OP-Bi2201, at 50 K and 40 K with energy resolution of 150 meV (black) and 300 meV (purple), respectively; OP-Bi2212 data were collected at 40 K with energy resolution of 300 meV (magenta). (b) Enlarged view of the low energy portion. (c) The paramagnon energies of OP-Bi2201 and OP-Bi2212 from the fitting of the spectra with experimental resolution of 300 meV are overlaid with paramagnon energy of OP-Bi2201 at 50 K reproduced from Fig. 3. The large error bars are due to the fitting uncertainty for the lower resolution spectra. (d) Integrated intensities of RIXS spectra with energy window $[-0.8,0]$ eV, normalized to the respective value at $Q_{\parallel }=0.22$ r.l.u. Self-absorption correction has been applied to the intensities. The error bars represent the uncertainty in determining the spectral weight.