Electronic Screening-Enhanced Hole Pairing in Two-Leg Spin Ladders Studied by High-Resolution Resonant Inelastic X-Ray Scattering at Cu $M$ Edges

We study the electronic screening mechanisms of the effective Coulomb on-site repulsion in hole-doped ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ compared to undoped ${\mathrm{La}}_6{\mathrm{Ca}}_8{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ using polarization dependent high-resolution resonant inelastic x-ray scattering at Cu $M$ edges. By measuring the energy of the effective Coulomb on-site repulsion and the spin excitations, we estimate superexchange and hopping matrix element energies along rungs and legs, respectively. Interestingly, hole doping locally screens the Coulomb on-site repulsion reducing it by as much as 25%. We suggest that the increased ratio of the electronic kinetic to the electronic correlation energy contributes to the local superexchange mediated pairing between holes.

Figure 1

X-ray absorption at (a) $\mathrm{Cu}\text{−}3p\to \mathrm{Cu}\text{−}3d$ and (b) $\mathrm{Cu}\text{−}3s\to \mathrm{Cu}\text{−}4p$ $M$-edge transitions and incident photon energies selected for resonance Raman scattering (open bullets) together with the “white-light” intensity distribution of FLASH (red lines). (c) Proposed electronic band structure of the spin-ladder compounds (SLCs) denoting the relevant energy scales for transitions from the $\mathrm{Cu}\text{−}3p$ and $\mathrm{Cu}\text{−}3s$ states as well as the correlation energies splitting the states close to the Fermi level. Occupied states are shaded in gray. Dashed lines denote the undoped case, with no accessible electronic states at the Fermi level ($E_F$). (d) The spin ladder with its different superexchange parallel to the leg (along $\widehat{c}$) and parallel to the rung (along $\widehat{a}$) of the ladder. (e) The excited final spin state representing the excitation of two magnons. (f) and (g) Intermediate states after absorption of the incident photon outlining the relevant electronic processes between neighboring Cu sites.

Figure 2

(a) Raman spectra (Stokes side is denoted by negative energies) taken at 79.3 and 122.0 eV from the hole-doped ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ (SCO) with the polarization along the rung ($E\parallel \widehat{a}$) and the elastic line from sandblasted Si for reference. (b) Contour plot of the incident photon energy versus excitation energy (Raman shift) of SCO for $E\parallel \widehat{a}$ showing the resonance behavior of the low-energy phonon excitations. The scale of contour plots is shown.

Figure 3

Contour plot and resonance profile of resonant inelastic x-ray scattering (RIXS) from undoped ${\mathrm{La}}_6{\mathrm{Ca}}_8{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ (LCCO) (a),(c) for $E\parallel \widehat{a}$ (along the rungs) and (b),(d) for $E\parallel \widehat{c}$ (along the leg). The inset of (a) enlarges the region of the two-magnon resonance energy. The maximum two-magnon energy is denoted as $E_{\mathrm{mag}}^{\max }$ of about 400 meV. The spectrum at 80 eV for ($E\parallel \widehat{c}$) (a gray dotted line) shows the anisotropy of the peak energy denoted as $\mathrm{\Delta }{E}_{\mathrm{mag}}^{\max }$ of about $80\pm 10\mathrm{meV}$. Contour plot and resonance profile of resonant inelastic x-ray scattering (RIXS) from doped ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ (SCO) (e),(g) for $E\parallel \widehat{a}$ polarization and (f),(h) for $E\parallel \widehat{c}$ polarization. In this case the $E_{\mathrm{mag}}^{\max }$ and $\mathrm{\Delta }{E}_{\mathrm{mag}}^{\max }$ are about 375 and $5\pm 10\mathrm{meV}$, respectively. For comparison, we replot the $\mathrm{E}\parallel \widehat{a}$ spectrum of LCCO and SCO in (d) and (h), respectively.