Resonant Inelastic X-Ray Scattering Study of Electron-Exciton Coupling in High-$T_c$ Cuprates

Explaining the mechanism of superconductivity in the high-$T_c$ cuprates requires an understanding of what causes electrons to form Cooper pairs. Pairing can be mediated by phonons, the screened Coulomb force, spin or charge fluctuations, excitons, or by a combination of these. An excitonic pairing mechanism has been postulated, but experimental evidence for coupling between conduction electrons and excitons in the cuprates is sporadic. Here we use resonant inelastic x-ray scattering to monitor the temperature dependence of the $\underset{\_}{d}d$ exciton spectrum of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8-x}$ crystals with different charge carrier concentrations. We observe a significant change of the $\underset{\_}{d}d$ exciton spectra when the materials pass from the normal state into the superconductor state. Our observations show that the $\underset{\_}{d}d$ excitons start to shift up (down) in the overdoped (underdoped) sample when the material enters the superconducting phase. We attribute the superconductivity-induced effect and its sign reversal from underdoped to overdoped to the exchange coupling of the site of the $\underset{\_}{d}d$ exciton to the surrounding copper spins.

Popular Summary

Explaining the mechanism underlying high-temperature superconductivity in the copper oxide materials known as cuprates requires an understanding of what causes electrons to form Cooper pairs. Pairing can be mediated by phonons, the screened Coulomb force, spin or charge fluctuations, or a combination of these. A pairing mechanism involving excitons—a bound state of an electron and a positively charged hole—has been postulated, but evidence for coupling between conduction electrons and excitons in the cuprates has been sporadic. Here, we provide clear experimental evidence for electron-exciton coupling in high-temperature superconductors.

We use resonant inelastic x-ray scattering to monitor the temperature dependence of the spectrum of $dd$ excitons—formed by localized electrons and holes in the $d$ orbitals—of the superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8-x}$. We observe a significant change of these excitons when the material passes from the normal state into the superconducting state. We observe that the superconductivity-induced modifications of the $dd$ excitons are dramatically influenced by the doping, which we ascribe to the underlying local spin ordering affecting the $dd$ excitons.

The coupling between electronic excitations and magnetism in strongly correlated materials is the key ingredient for exotic effects such as high-temperature superconductivity, charge or orbital ordering, and colossal magnetoresistance. Understanding, and even better controlling, such interplay will allow for the discovery of new functionalities. The electron-exciton coupling can lead toward an active manipulation in and out of the superconducting phase upon driving $dd$ excitons by fast laser pulses.

Figure 1

(a) Experimental RIXS spectra for the three different dopings and for different momentum transfers. To avoid clutter the spectra have been given vertical offsets grouped according to the momentum values. Vertical bars correspond to spin flip (SF, red), $\underset{\_}{d}d$ excitons (blue), and charge transfer (CT) excitations (orange) obtained from a cluster calculation (see Appendix pp2). (b) Energy-momentum dispersion of the fitted peak positions. Error bars are smaller than the symbol size and therefore not indicated. Momentum values are indicated on top.

Figure 2

Top panels: normalized RIXS spectra. The spectra for different temperatures overlap except near the maximum of the $\underset{\_}{d}d$ peak. Bottom panels: normalized RIXS spectra for the same temperatures as the top panels from which the spectra at $T_c$ have been subtracted. (a),(c) Underdoped Bi-2212 ($T_c=70\mathrm{K}$). (b),(e) Optimally doped Bi-2212 ($T_c=91\mathrm{K}$). (c),(f) Overdoped Bi-2212 ($T_c=70\mathrm{K}$). (a)–(c) Momentum value (0.36, 0). (d)–(f) Momentum value (0.25, 0.25).

Figure 3

Temperature dependence of the first moment of the $\underset{\_}{d}d$ exciton peak and the magnon peak, as extracted from the experimental spectra (considering, respectively, the energy intervals 1–4 and 0.1–0.6 eV). The dashed lines are fits to a curve parametrized as Eq. (2). The error bars represent the uncertainty on the elastic peak position computed as explained in Appendix pp1. The vertical dotted lines indicate $T_c$. Top panels: momentum value (0.36, 0). Bottom panels: momentum value (0.25, 0.25). (a),(d) Underdoped Bi-2212 ($T_c=70\mathrm{K}$). (b),(e) Optimally doped Bi-2212 ($T_c=91\mathrm{K}$). (c),(f) Overdoped Bi-2212 ($T_c=70\mathrm{K}$).

Figure 4

Temperature dependence of the integrated RIXS intensity $I_j(T)$ where the index $j=1$, 2 refers to the two energy ranges 1–2 and 2–3.5 eV, respectively. The dashed lines indicate $T_c$. Top panels: momentum value (0.36, 0). Bottom panels: momentum value (0.25, 0.25). (a),(d) Underdoped Bi-2212 ($T_c=70\mathrm{K}$). (b),(e) Optimally doped Bi-2212 ($T_c=91\mathrm{K}$). (c),(f) Overdoped Bi-2212 ($T_c=70\mathrm{K}$).

Figure 5

Doping dependence of the superconductivity-induced change of the average $\underset{\_}{d}d$ exciton and magnon energies. The momentum values are (0.36, 0) (green circles) and (0.25, 0.25) (blue squares). The error bars are extracted from the fit procedure.

Figure 6

(a) Phase diagram of the cuprates indicating the antiferromagnetic phase (beige), the normal phase (orange for underdoped and light blue for overdoped), and the superconducting phase (green) with the corresponding short-range spin correlations and $\underset{\_}{d}d$ exciton energies. (b),(c) Sketch of the temperature dependence of the $\underset{\_}{d}d$ exciton energy in the underdoped and overdoped cases.

Figure 7

(a) Fitted spectrum for the OD 70 K $\mathbf{q}=(0.36,0)$ at 30 K; the shaded area indicates the Poissonian noise. The shaded Voigt peaks represent the decomposition of the spectrum. (b) Distribution of the elastic peak position extracted by fitting ${10}^4$ spectra with random Poissonian noise.

Figure 8

Average energy and of the $\underset{\_}{d}d$ exciton peak of overdoped Bi-2212 ($p=0.21$,$T_c=70\mathrm{K}$) for momentum (0.36, 0) measured with 3 different detectors.

Figure 9

Integral function $G(\omega )$ defined in Eq. (a2) (solid line) and the spectrum $I(\omega ,T)$ (dashed line) normalized to their maximum, for the OD 70 K sample and $\mathbf{q}=(0.36,0)$.

Figure 10

Temperature dependence of the $\underset{\_}{d}d$ exciton average energy, from the spectral function computed in Eq. (c5) for the two directions in momentum space $\mathbf{q}=(0.36,0)$ (green circles) and $\mathbf{q}=(0.25,0.25)$ (blue squares).