Observation of momentum-dependent charge excitations in hole-doped cuprates using resonant inelastic x-ray scattering at the oxygen $K$ edge

We investigate electronic excitations in ${\mathrm{La}}_{2-x}{(\mathrm{Br},\mathrm{Sr})}_x{\mathrm{CuO}}_4$ using resonant inelastic x-ray scattering (RIXS) at the oxygen $K$ edge. RIXS spectra of the hole-doped cuprates show clear momentum dependence below 1 eV. The broad spectral weight exhibits positive dispersion and shifts to higher energy with increasing hole concentration. Theoretical calculation of the dynamical charge structure factor on oxygen orbitals in a three-band Hubbard model is consistent with the experimental observation of the momentum and doping dependence, and therefore the momentum-dependent spectral weight is ascribed to intraband charge excitations which have been observed in electron-doped cuprates. Our results confirm that the momentum-dependent charge excitations exist on the order of the transfer energy ($t$), and the broad spectral line shape indicates damped and incoherent character of the charge excitations at the energy range in the doped Mott insulators.

Figure 1

(a) X-ray absorption spectra near the oxygen $K$ edge. X-ray polarization is parallel to the ${\mathrm{CuO}}_2$ plane. (b) O $K$-edge RIXS spectra of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ ($x=0.25$) plotted against emitted photon energy. From bottom to top, the incident photon energy is tuned to $\mathrm{LEP}-0.3\mathrm{eV}$, LEP, and $\mathrm{LEP}+0.3\mathrm{eV}$ of XAS. (c) The same spectra as in (b) plotted as a function of energy loss. (d–f) RIXS spectra of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$. The energy of the $\sigma$-polarized incident x rays is tuned to UHB for $x=0$ (d) and $x=0.075$ (e) and LEP for $x$ = 0.18 (f). Black lines are the raw spectra and gray lines in (e) and (f) are the spectral weight after subtraction of the elastic peak (dotted line) and high-energy tail (dashed line). The vertical bars indicate the peak position of the spectral weight after subtraction.

Figure 2

(a–e) RIXS spectra of ${\mathrm{La}}_{2-x}{(\mathrm{Br},\mathrm{Sr})}_x{\mathrm{CuO}}_4$. The energy of the $\sigma$-polarized incident x-rays is tuned to LEP of XAS, except for the upper three spectra in (a). Black lines are the raw spectra, and gray lines are the spectral weight after subtraction of the elastic peak (dotted line) and high-energy tail (dashed line). The peak position is taken to be the centroid within the region where the spectral intensity exceeds 90% of maximum in the subtracted spectrum and indicated by the vertical bar. (f–j) Dispersion relation of the dispersive mode. Peak positions are plotted by open circles and filled squares. Full widths at 90% of the maximum intensity are displayed by vertical bars. Results from the spectra in Figs. 1 and 1 are included. In (g–j), RIXS intensity of (b–e) is displayed as a color map after subtracting the elastic component. Peak positions of the charge excitations [14] and the fast-dispersive mode [15] in electron-doped ${\mathrm{Nd}}_{2-x}{\mathrm{Ce}}_x{\mathrm{CuO}}_4$ are also shown by triangles and solid lines, respectively.

Figure 3

Dynamical charge structure factor on $\mathrm{O}2p_y$ orbital in a $6\times 4$ cylindrical three-band Hubbard cluster with $T_{pd}=1\mathrm{eV}$, $T_{pp}=0.3\mathrm{eV}$, $\mathrm{\Delta }=3\mathrm{eV}$, $U_d=8\mathrm{eV}$, and $U_p=4\mathrm{eV}$. Black, red, and green solid lines represent spectra with hole concentration $x=1/12$, $1/6$, and $1/4$, respectively, at $\mathbf{q}=(1/6,1/10)$. Black broken line is for $x=1/12$ at $\mathbf{q}=(1/3,1/10)$. A Gaussian broadening width of 0.1 eV is used for the spectral weights. We note that small weights centered at $\omega =0\mathrm{eV}$ are due to less convergence in dynamical DMRG.

Figure 4

Oxygen $K$-edge RIXS spectra of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ ($x=0.125$) measured below (30 K) and above (100 K) the transition temperature of charge order ($T_{\mathrm{co}}=50\mathrm{K}$).

Figure 5

The dynamical two-magnon correlation function for (a) $A_1$ mode, (b) $B_1$ mode, and (c) dynamical charge structure factor at $\mathbf{q}=(0.2,0.1)$ (black solid line) and $(0.1,0.3)$ (red solid line) for the $\sqrt{20}\times \sqrt{20}$ periodic cluster of the $t\text{−}{t}^{\prime }\text{−}{t}^{″}\text{−}J$ model with three-site term: $t=1$, $t^{\prime }=-0.25$, $t^{″}=0.12$, and $J=0.4$. A Lorentzian broadening with a width of $0.2t$ is used for the spectral weights.

Figure 6

The dependence of $N_{2p_y}(\mathbf{q},\omega )$ for a given parameter set on the truncation number $m$ in the dynamical DMRG method. A $6\times 4=24$-unit cell ${\mathrm{CuO}}_2$ cluster with cylindrical boundary condition is used. $\mathbf{q}=(6/6,1/10)$. $T_{pp}=0.3\mathrm{eV}$, $\mathrm{\Delta }=3\mathrm{eV}$, $U_d=8\mathrm{eV}$, and $U_p=4\mathrm{eV}$, and hole concentration $x=1/12$. A Gaussian broadening with a width of $0.1\mathrm{eV}$ is used for the spectral weights.