Optical weights and waterfalls in doped charge-transfer insulators: A local density approximation and dynamical mean-field theory study of ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$

We use the local density approximation in combination with the dynamical mean-field theory to investigate intermediate energy properties of the copper oxides. We identify coherent and incoherent spectral features that result from doping a charge-transfer insulator, namely quasiparticles, Zhang-Rice singlet band, and the upper and lower Hubbard bands. Angle resolving these features, we identify a waterfall-like feature between the quasiparticle part and the incoherent part of the Zhang-Rice band. We investigate the asymmetry between particle and hole doping. On the hole-doped side, there is a very rapid transfer of spectral weight upon doping in the one particle spectra. The optical spectral weight increases superlinearly on the hole-doped side in agreement with experiments.

Figure 1

(Color online) Doping $\delta$ (left scale) versus the chemical potential $\mu$. The parent compound $\left(\delta =0\right)$ is a charge-transfer insulator with a gap ${\Delta }_{pd}\approx 1.8\text{eV}$ (green region). The number of electrons in the $d_{x^2-y^2}$ orbital $\left(n_d\right)$ and in the $p_{\sigma }$ orbitals $\left(n_p\right)$ is also shown (right scale).

Figure 2

(Color online) (a) Left panel: Spectral function $A\left(\mathbf{k},\omega \right)$ for the hole-doped compound $\left(\delta =0.24\right)$. The black lines correspond to the LDA bands. Right panel: The local spectral functions of the $d_{x^2-y^2}$ band (dashed line) and of the $p_{\sigma }$ band (full line). We marked the positions of the lower Hubbard band (L.H.B.), coherent part of the Zhang-Rice singlet called the quasiparticle peak (Q U.P), the incoherent part of the Zhang-Rice singlet band (Z.R.S.) and the upper Hubbard band (U.H.B.). (b) The low energy part of $A\left(\mathbf{k},\omega \right)$ from panel (a). The dashed line shows the slope of the quasiparticle band, which gives the Fermi velocity. The derivative was obtained by a quadratic expansion of $\Sigma \left(\omega \right)$ at the Fermi level. The Fermi velocity in $\text{LDA}+\text{DMFT}$ is $v_{\text{DMFT}}=2.15\left(5\right)\text{Å}\text{eV}$ in very favorable agreement with experiment [$v_{\text{exp}}=2.0\text{Å}\text{eV}$ at doping $\delta =0.24$ (Ref. 26)]. The ratio between the DMFT and the band velocity is thus $v_{\text{DMFT}}/v_{\text{LDA}}=0.49$. (c) Fermi surface and the path along which $A\left(\mathbf{k},\omega \right)$ is shown in (a) and (b). (d) The density of states in the electron-doped compound. A very sharp quasiparticle peak develops close to the upper Hubbard band in the electron-doped case.

Figure 3

(Color online) Optical conductivity $\sigma \left(\omega \right)$ for the parent compound $\left(\delta =0\right)$ and the hole-doped compound $\left(\delta =0.24\right)$. For the parent compound, we show optics in the Neel state. The optical gap in the parent compound is around 1.8 eV in both the paramagnetic and antiferromagnetic state. The structure around $\omega =8\text{eV}$ corresponds to transitions between the lower and the upper Hubbard band. Upon hole doping, an additional peak appears close to $\omega \approx 1\text{eV}$ for a large range of doping ($\delta =5–30\%$). It corresponds to transitions between the quasiparticle part and incoherent part of the Zhang-Rice singlet.

Figure 4

(Color online) (a) Optical spectral weight $N_{\text{eff}}$ obtained by Eq. (7) using various cutoffs $\Lambda =1.2,1.5,1.8\text{eV}$ for both the hole doping $\left(\delta >0\right)$ and electron doping $\left(\delta <0\right)$. The experimental data (red circles) are reproduced from Ref. 27. The long dashed line corresponds to the curve $N_{\text{eff}}\left(\delta \right)=\delta$. Note the asymmetry between the electron $\left(\delta <0\right)$ and hole-doped $\left(\delta >0\right)$ side. For completeness, we also present more recent experimental data obtained by infrared spectroscopy of LSCO (white circle). The data are reproduced from Ref. 30. (b) The quasiparticle weight $Z$ versus doping. Inset: double occupancy of the $d_{x^2-y^2}$ orbital versus doping. For the electron-doped compound, the double occupancy increases linearly with doping, whereas the hole doping has only a weak effect on double occupancy of the $d_{x^2-y^2}$ orbital. The continuous lines are guides for the eyes.