Role of Lattice Coupling in Establishing Electronic and Magnetic Properties in Quasi-One-Dimensional Cuprates

High resolution resonant inelastic x-ray scattering has been performed to reveal the role of lattice coupling in a family of quasi-1D insulating cuprates, ${\mathrm{Ca}}_{2+5x}{\mathrm{Y}}_{2-5x}{\mathrm{Cu}}_5{\mathrm{O}}_{10}$. Site-dependent low-energy excitations arising from progressive emissions of a 70 meV lattice vibrational mode are resolved for the first time, providing a direct measurement of electron-lattice coupling strength. We show that such electron-lattice coupling causes doping-dependent distortions of the Cu-O-Cu bond angle, which sets the intrachain spin exchange interactions. Our results indicate that the lattice degrees of freedom are fully integrated into the electronic behavior in low-dimensional systems.

Figure 1

(a) X-ray absorption spectrum near the oxygen $K$ edge for ${\mathrm{Ca}}_{2+5x}{\mathrm{Y}}_{2-5x}{\mathrm{Cu}}_5{\mathrm{O}}_{10}$ compounds of three different doping levels. “UHB” and “LEP” denote the absorption peaks at 529.5 and 528 eV, respectively. The average number of hole per Cu ion is denoted as $x$. (b) $|{\Psi }_e^n⟩$ and $|{\Psi }_e^{n+1}⟩$ denote the electronic ground state and the excited electronic intermediate state with an additional electron excited from the oxygen $1s$ core level. $|n_p⟩$ ($n=0,1,2,\dots$) denotes the number of phonon quanta excited in the lattice degree of freedom with a wave function superposition coefficient $\alpha$. For purposes of illustration, the wave function is expressed as product states of the diagonalized electron and lattice Hamiltonians. Here, the lattice vacuum state $|0_p⟩$ is referenced to the initial equilibrium phonon occupation and $|n_p⟩$ describes excitations from this state. In the RIXS intermediate state, an electron (green ball) is excited, leaving a core hole (empty circle) in the O $1s$ level. The excited electron can be closer to either Cu or O site by tuning the incident photon energy to match the UHB or LEP resonance, respectively. The change of the local charge density on Cu (shaded area around Cu) and O (shaded area around O) sites induces local relaxation of the distorted lattice, creating a distribution of $|n_p⟩$. In the final state, the excited electron can recombine with the core hole, leaving the lattice in excited states. These final states have nonzero overlap with the intermediate state, yielding harmonic phonon excitations in RIXS spectrum.

Figure 2

The RIXS spectrum for (a) $x=0$ and (b) $x=0.33$ samples taken at several incident photon energies, as indicated by the red bars in the XAS plots. The length of these red vertical bars reflects the relative elastic peak intensity of the corresponding RIXS spectrum. The darker shaded areas highlight the spectra with prominent additional spectral weight near the elastic peak. (c) High resolution data taken at UHB resonance for the $x=0$ sample (upper panel) and at LEP resonance for the $x=0.33$ sample (lower panel). Multipeak features are annotated with number identifiers. The black curves are the Gaussian functions used for the fit (red curves). Purple curves are the remaining spectrum after subtracting the fit. Background is plotted as black dotted lines. (d) A summary plot of the fitted peak positions and widths (inset) of the RIXS spectra taken at UHB ($x=0$) and LEP ($x=0.3$ and 0.33) resonances. All data were taken at 30 K.

Figure 3

(a) The calculated RIXS intensities for the $x=0$ (left) and $x=0.33$ (right) hole-doped systems, respectively, with $g=0.2\mathrm{eV}$. The incident photon energies are indicated in the calculated XAS spectra and the elastic line has been removed by excluding the ground state from the final state summation. The calculations were performed on ${\mathrm{Cu}}_3{\mathrm{O}}_8$ and ${\mathrm{Cu}}_3{\mathrm{O}}_6$ clusters (open and periodic boundary conditions) for the $x=0$ and $x=0.33$ cases, respectively. The arrows indicate the displacement pattern of the phonon eigenvector involving $c$-axis motion of the O atoms. (b) A comparison of the normalized phonon excitations at the UHB resonance for the $x=0$ case and different coupling strengths. (c) The normalized phonon excitation spectrum at the LEP and UHB resonance in a doped cluster ($x=0.33$). Spectra of two different values of $g_{\mathrm{O}}$ were calculated to demonstrate the site dependence of the RIXS process at the LEP and UHB resonances. (d) Experimental data of harmonic phonon excitations (elastic peak and background subtracted) normalized to the first phonon peak intensity at the UHB resonance for three different doping levels. Data of $x=0.33$ sample taken at LEP resonance are also superimposed, illustrating the site-dependent electron-phonon coupling. (e) The ratio of the model RIXS intensity $I(\#2)/I(\#1)$ at the first and second phonon as a function of dimensionless coupling constant $g_{\mathrm{Cu}}/\Omega$, where $\Omega$ is the phonon energy, 70 meV. The data points with error bars indicate the experimental value of this ratio.

Figure 4

(a) Calculated number of phonons entangled with the ground state wave function as a function of coupling strength for undoped and doped clusters. A larger number of phonon quanta in the ground state correspond to a larger lattice distortion. Note that RIXS process excites additional phonons, leaving the system in an excited lattice state. The inset shows the estimated doping induced changes in the lattice constant as a function of the mode coupling strength. (b) RIXS spectra taken at two temperatures across the antiferromagnetic phase transition for the $x=0$ compound. The inset shows the rising edge of the $d\mathrm{\text{−}}d$ excitations. The agreement of the elastic peak and the $d\mathrm{\text{−}}d$ edge provides an internal energy reference, confirming the shift of the phonon mode energy across the antiferromagnetic transition.