Incident-energy-dependent spectral weight of resonant inelastic x-ray scattering in doped cuprates

We theoretically investigate the incident-photon energy ${\omega }_{\mathrm{i}}$ dependence of resonant inelastic x-ray scattering (RIXS) tuned for the Cu $L$ edge in cuprate superconductors by using the exact diagonalization technique for a single-band Hubbard model. Depending on the value of core-hole Coulomb interaction in the intermediate state, RIXS for non-spin-flip channel shows either a ${\omega }_{\mathrm{i}}$-dependent fluorescencelike or ${\omega }_{\mathrm{i}}$-independent Raman-like behavior for hole doping. An analysis of x-ray absorption suggests that the core-hole Coulomb interaction is larger than on-site Coulomb interaction in the Hubbard model, resulting in a fluorescencelike behavior in RIXS consistent with recent RIXS experiments. A shift on the high-energy side of the center of spectral distribution is also predicted for electron-doped systems though spectral weight is small. Main structures in the spin-flip channel exhibit a Raman-like behavior as expected, accompanied with a fluorescencelike behavior with small intensity.

Figure 1

The core-hole potential $U_{\mathrm{c}}$ dependence of XAS spectrum $I^{\mathrm{XAS}}\left(\omega \right)$ for the $\sqrt{18}\times \sqrt{18}$ periodic cluster of the Hubbard model. (a) Electron doping and (b) hole doping with $x=2/18\sim 0.11$. $U=10t$ and $t^{\prime }=-0.25t$. The value of $U_{\mathrm{c}}$ is changed in the range of $U/2\ge U_{\mathrm{c}}\ge 3U/2$. The solid lines are obtained by a Lorenzian broadening with $\mathrm{\Gamma }=t$ for $\delta$ functions shown by bars.

Figure 2

The incident-phonon energy ${\omega }_{\mathrm{i}}$ dependence of the non-spin-flip spectrum $I_{\mathbf{q}}^{\mathrm{\Delta }S=0}$ at $\mathbf{q}=(2\pi /3,0)$ for the hole-doped $\sqrt{18}\times \sqrt{18}$ Hubbard cluster with $x=2/18\sim 0.11$. (a) $U_{\mathrm{c}}=12t$ and (b) $U_{\mathrm{c}}=8t$. Parameters are $U=10t, t^{\prime }=-0.25t$, and $\mathrm{\Gamma }=t$. ${\omega }_{\mathrm{i}}$ at the bottom panel is set to the edge of the XAS spectrum and increases by $t$ from bottom to top. (c) The dynamical charge structure factor $N(\mathbf{q},\omega )$ at $\mathbf{q}=(2\pi /3,0)$. (d) The $\mathbf{q}$-dependent dynamical two-magnon correlation function $M(\mathbf{q},\omega )$ at $\mathbf{q}=(2\pi /3,0)$. Black (red) line, ${\mathrm{B}}_{1\mathrm{g}}$ (${\mathrm{A}}_{1\mathrm{g}}$) mode.

Figure 3

Same as Fig. 2 but $x=4/18\sim 0.22$.

Figure 4

(a) The incident-phonon energy ${\omega }_{\mathrm{i}}$ dependence of the non-spin-flip spectrum $I_{\mathbf{q}}^{\mathrm{\Delta }S=0}$ at $\mathbf{q}=(2\pi /3,0)$ for the electron-doped $\sqrt{18}\times \sqrt{18}$ Hubbard cluster with $x=2/18\sim 0.11$. Parameters are $U=10t, t^{\prime }=-0.25t, U_{\mathrm{c}}=12t$, and $\mathrm{\Gamma }=t$. ${\omega }_{\mathrm{i}}$ at the bottom panel is set to the edge of the XAS spectrum and increases by $t$ from bottom to top. (b) The dynamical charge structure factor $N(\mathbf{q},\omega )$ at $\mathbf{q}=(2\pi /3,0)$. (c) The $\mathbf{q}$-dependent dynamical two-magnon correlation function $M(\mathbf{q},\omega )$ at $\mathbf{q}=(2\pi /3,0)$. Black (red) line, ${\mathrm{B}}_{1\mathrm{g}}$ (${\mathrm{A}}_{1\mathrm{g}}$) mode.

Figure 5

The incident-phonon energy ${\omega }_{\mathrm{i}}$ dependence of the spin-flip spectrum $I_{\mathbf{q}}^{\mathrm{\Delta }S=1}$ at $\mathbf{q}=(2\pi /3,0)$ for the $\sqrt{18}\times \sqrt{18}$ Hubbard cluster with $x=2/18\sim 0.11$ for (a) electron doping and (c) hole doping. Parameters are $U=10t, t^{\prime }=-0.25t, U_{\mathrm{c}}=15t$, and $\mathrm{\Gamma }=t$. ${\omega }_{\mathrm{i}}$ at the bottom panel is set to the edge of the XAS spectrum and increases by $t$ from bottom to top. The dynamical spin structure factor $S(\mathbf{q},\omega )$ at $\mathbf{q}=(2\pi /3,0)$ for (b) electron doping and (d) hole doping.

Figure 6

The contour plot of incident-phonon energy ${\omega }_{\mathrm{i}}$ dependence of spin-flip spectra $I_{\mathbf{q}}^{\mathrm{\Delta }S=1}$ and non-spin-flip spectra $I_{\mathbf{q}}^{\mathrm{\Delta }S=0}$ at $\mathbf{q}=(2\pi /3,0)$ for the hole-doped $\sqrt{18}\times \sqrt{18}$ Hubbard cluster with parameters $U=10t, t^{\prime }=-0.25t, U_{\mathrm{c}}=12t$, and $\mathrm{\Gamma }=t$. (a) $x=0$, (b) $x=2/18\sim 0.11$, and (c) $x=4/18\sim 0.22$ for $I_{\mathbf{q}}^{\mathrm{\Delta }S=1}$. (d) $x=0$, (e) $x\sim 0.11$, and (f) $x\sim 0.22$ for $I_{\mathbf{q}}^{\mathrm{\Delta }S=0}$.