Coulomb disorder effects on angle-resolved photoemission and nuclear quadrupole resonance spectra in cuprates

The role of Coulomb disorder, either of extrinsic origin or introduced by dopant ions in undoped and lightly doped cuprates, is studied. We demonstrate that charged surface defects in an insulator lead to a Gaussian broadening of the angle-resolved photoemisson spectroscopy (ARPES) lines. The effect is due to the long-range nature of the Coulomb interaction. A tiny surface concentration of defects about a fraction of one percent is sufficient to explain the line broadening observed in ${\text{Sr}}_2{\text{CuO}}_2{\text{Cl}}_2$, ${\text{La}}_2{\text{CuO}}_4$, and ${\text{Ca}}_2{\text{CuO}}_2{\text{Cl}}_2$. Due to the Coulomb screening, the ARPES spectra evolve dramatically with doping, changing their shape from a broad Gaussian form to narrow Lorentzian ones. To understand the screening mechanism and the line-shape evolution in detail, we perform Hartree-Fock simulations with random positions of surface defects and dopant ions. To check validity of the model we calculate the nuclear quadrupole resonance (NQR) line shapes as a function of doping and reproduce the experimentally observed NQR spectra. Our study also indicates opening of a substantial Coulomb gap at the chemical potential. For a surface ${\text{CuO}}_2$ layer the value of the gap is on the order of 10 meV while in the bulk it is reduced to the value about a few meV.

Figure 1

Dispersion minima of the spinless fermion generated by Hamiltonian (19).

Figure 2

(Color online) ARPES spectral function, Eq. (24), for $36\times 36$ cluster averaged over 100 random realizations of Coulomb disorder. Concentration of defects is $C=0.6\%$. Values of momenta are $\mathit{k}=\left(\pi /2,\pi /2\right)$, $\left(0,\pi /2\right)$, and (0,0). Smooth lines correspond to energy dispersion $\omega =-{ϵ}_{\mathbf{k}}$ [with ${ϵ}_{\mathbf{k}}$ given by Eq. (20)], broadened according to Eqs. (5, 15, 16). In these equations, we set $L=36/\sqrt{\pi }$ and $a_d=1$.

Figure 3

Screening of in-plane Coulomb interaction by other layers. Dashed lines shows electric field of an in-plane charge bent due to a large polarizability of “other layers.”

Figure 4

The hole Sr binding energy as a function of the short-range cutoff $a_d$. We recall that we set $J=1$. The interlayer screening is taken into account according to Eq. (28).

Figure 5

(Color online) Plots of the hole density in a ${\text{CuO}}_2$ layer deep in the bulk, for a particular realization of random Sr positions, at doping $x=0.02$ and two different temperatures $T=0$ and 600 K. The length scale 5 nm $\left(\approx 13a_0\right)$ is shown.

Figure 6

(Color online) Plots of the hole density in a ${\text{CuO}}_2$ layer deep in the bulk, for a particular realization of random Sr positions, at doping $x=0.07$ and two different temperatures $T=0$ and 600 K.

Figure 7

(Color online) The calculated (upper and middle panels) and experimental (Ref. 12) (lower panel) ${}^{63}\text{C}\text{u}$ NQR spectra in ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ at different doping levels $x=0.02$, 0.04, 0.07, 0.115, and 0.16. The narrow line at $f=33.05\text{MHz}$ (Ref. 11) corresponds to the parent compound, $x=0$. Apart from broadening of the spectra, doping results also in a high-frequency structure similar to the B line observed in the experiment (Ref. 12). Note that linewidth at $T=600\text{K}$ is smaller than that at zero temperature.

Figure 8

The hole DOS in ${\text{CuO}}_2$ layer deep in the bulk at dopings $x=0.02$ and $x=0.07$. DOS vanishes at the chemical potential indicating the Coulomb gap which gradually decreases with doping.

Figure 9

Screening of in-plane Coulomb interaction in the top layer by other layers underneath. Dashed lines shows electric field of an in-plane charge bent due to polarization of “other layers.”

Figure 10

The hole DOS in the surface ${\text{CuO}}_2$ layer at dopings $x=0.02$ and $x=0.07$. DOS vanishes at the chemical potential indicating the Coulomb gap ${\Delta }_C$. The size of the gap, ${\Delta }_C\sim 10\text{meV}$, is larger than that in the bulk (see Fig. 8).

Figure 11

(Color online) The electron spectral function $A\left(\mathbf{k},\omega \right)$ at doping levels $x=0.01$, 0.03, 0.05, 0.07, 0.09, and 0.11 calculated for momenta ${\mathbf{p}}_0$, ${\mathbf{p}}_1$, ${\mathbf{p}}_2$, and ${\mathbf{p}}_3$ specified in Eq. (41). The Coulomb gap ${\Delta }_C$ is highlighted in the $x=0.11$ plot.