Dispersive collective charge modes in an incommensurately modulated cuprate Mott insulator

We report measurement of collective charge modes of insulating ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ using inelastic resonant x-ray scattering over the complete Brillouin zone. Our results show that the intense excitation modes at the charge gap edge predominantly originate from ladder-containing planar substructures. The observed modes are found to be dispersive for momentum transfers along the “legs” $(\hslash \vec{Q}\Vert \widehat{c})$ but nearly localized along the “rungs” $(\hslash \vec{Q}\Vert \widehat{a})$. We show that the dispersion and peak width characteristics of the modes can be understood in the strong-coupling quantum limit (Hubbard $U⪢t_{\mathit{\text{ladder}}}>t_{\mathit{\text{chain}}}$, where $t$ is the hopping parameter). Quite generally, we demonstrate that the momentum tunability ($\vec{Q}$ resolution) of inelastic x-ray scattering can be utilized to resolve mode contributions in multicomponent incommensurate quantum electron systems.

Figure 1

(Color online) Lattice substructures and scattering geometries. (a) Ladder unit cell in ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$. (b) Scattering geometry configured with $E$ field aligned in the [010] ($b$-axis) direction. (c) Two distinct Cu-O substructural units. The bonding between two adjacent ladders is very weak.

Figure 2

(Color) Inelastic resonance profiles. (a) Energy dispersion curves for incident energy dependence of $\mathrm{Sr}\mathrm{Cu}{\mathrm{O}}_2$ at momentum transfer $Q=\pi$ along the chain direction, with incident photon energy $\left(h\nu \right)$ from 8.980 (top) to $8.996\mathrm{keV}$ (bottom). (b) Observed resonance energies $E_1$ and $E_2$ in the total fluorescence yield Cu $K$ edge absorption. (c) Energy dispersion curves shown in (a) plotted in color to display more clearly the two resonance energies that couple to valence band charge excitations. (d) Energy dispersion curves for ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ at $Q=2\pi ∕c_L$ with $h\nu$ from $8.982\text{to}8.996\mathrm{keV}$, showing two resonances, labeled on the plot of fluorescence in (e). (f) Curves shown in (d) are plotted in color. Inelastic charge response for ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ at $8.984\mathrm{keV}$ incident energy and $Q=2\pi$ along the (g) $\widehat{c}$ and (h) $\widehat{a}$ axes with polarization along the $\widehat{b}$ axis, compared with optical conductivity data (solid line) with (g) $\widehat{c}$- and (h) $\widehat{a}$-axis polarization. (Ref. 6) The onset of DOS is in good agreement with x-ray response for $\sim 2\pi ∕c_L$ [ladder Brillouin Zone (BZ) center].

Figure 3

(Color online) $Q$ dependence of charge modes. Energy dispersion curves are shown for the RIXS spectrum at incident energies $\left(h\nu \right)$ of (a) 8.984 and (b) $8.9845\mathrm{keV}$, with momentum transfer in the ladder direction from $Q=2\pi ∕c_L$ (top) to $\pi ∕c_L$ (bottom). The dispersion of low-energy features is traced with red dashed lines, and a fitting (described in the text) is shown for features in the bottom curve of (a). (c) Low-energy features (black arrows) have no measurable dispersion in the $\widehat{a}$ direction, shown from $Q=6\pi$ (top) to $5\pi$ with incident energy $8.9845\mathrm{keV}$. Intensity scale in (c) is smaller than in (b) by a factor of 2. (d) Low-energy excitation feature in single-chain $\mathrm{Sr}\mathrm{Cu}{\mathrm{O}}_2$, measured from $Q=2\pi$ to $3\pi$, exhibits a similar dispersion to the ladder peaks but has one peak.

Figure 4

(Color) $Q$ dependence and lattice dimensionality. Inelastic charge response spectral weight across the Brillouin zone for (a),(b) quasi-1D $\mathrm{Sr}\mathrm{Cu}{\mathrm{O}}_2$, (c) ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$, and (d),(e) quasi-2D ${\mathrm{Nd}}_2\mathrm{Cu}{\mathrm{O}}_4$. Spectra were measured from (a),(b) $Q=2\pi$ to $3\pi$, (c) $Q=\pi$ to $3.25\pi$, and (d), (e) $\pi$ to $2\pi$, and (a), (b) and (d), (e) show data symmetrized about $Q=2\pi$. Color images in (b) and (d) represent the curves displayed in (a) and (e), respectively. In (c), red ticks mark the positions in the chain BZ and black ticks mark positions in the ladder BZ. Low-energy $(\sim 2–4.5\mathrm{eV})$ spectral weight is found to be periodic with the ladder.

Figure 5

(Color online) Two-particle charge spectrum. (a) Renormalized band structure of the ladder. Splitting is due to rung coupling. (b) Energy range of all possible two-particle excitations in the ladder based on (a). Regions with spectral weight from more than one band pair (continuum) are darker. Experimental peak positions determined from Fig. 3 are overlaid [red (gray) circles] for comparison. Hatched regions $(2.3–2.7\mathrm{eV})$ mark the experimental boundaries for the two-particle spectrum of the chain layer, which is expected to overlap with the ladder continuum, based on Ref. 14.