Bulk and surface nanoscale hole density inhomogeneity in HgBa${}_2$CuO${}_{4+\delta }$ and Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ cuprates

It is well established that the hole density in the prototypical superconductor La${}_{2-x}$Sr${}_x$CuO${}_4$ is very inhomogeneous due to Sr-dopant induced disorder. On the other hand, the hole distribution in HgBa${}_2$CuO${}_{4+\delta }$ and Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ doped by interstitial oxygen is believed to be much more uniform. Recent nuclear magnetic resonance measurements indicate, however, that the charge inhomogeneity in HgBa${}_2$CuO${}_{4+\delta }$ is close to that in La${}_{2-x}$Sr${}_x$CuO${}_4$. Calculations performed in the present paper confirm this observation. We also show that the charge inhomogeneity is most pronounced at the surface layer that can be probed by scanning tunneling microscope. Our simulations demonstrate that, despite having similar amplitudes of charge inhomogeneity, the hole mean free path in HgBa${}_2$CuO${}_{4+\delta }$ is substantially longer than that in La${}_{2-x}$Sr${}_x$CuO${}_4$. The screening of the Coulomb repulsion in HgBa${}_2$CuO${}_{4+\delta }$ is also stronger. These two reasons might explain the difference in the superconducting critical temperatures between these two compounds.

Figure 1

Dispersion minima of the spinless fermion generated by Hamiltonian (2).

Figure 2

The HF model of La${}_{2-x}$Sr${}_x$CuO${}_4$ with four ($m=1,2,3,4$) layers of Coulomb defects (i.e., doped Sr ions) and with two “metallic” sheets.

Figure 3

The charge density of mobile holes in La${}_{2-x}$Sr${}_x$CuO${}_4$ calculated at zero temperature and different doping levels.

Figure 4

Theoretical NQR spectra for in-plane ${}^{63}\mathrm{Cu}$ in La${}_{2-x}$Sr${}_x$CuO${}_4$ at $T=0$ and $T=600$ K and for different doping levels $p$.

Figure 5

The HF model of HgBa${}_2$CuO${}_{4+\delta }$ with two layers of Coulomb defects (interstitial oxygen ions) and two “metallic” sheets.

Figure 6

The charge density of mobile holes in HgBa${}_2$CuO${}_{4+\delta }$ at zero temperature and at different doping levels.

Figure 7

Theoretical NQR spectra for in-plane ${}^{63}\mathrm{Cu}$ in HgBa${}_2$CuO${}_{4+\delta }$ at zero temperature and for different doping levels $p$.

Figure 8

(a) DOS and (b) charge density profile in HgBa${}_2$CuO${}_{4+\delta }$ at optimal doping $p=0.16$. Shown in the inset of panel (a) is the DOS of the homogeneous model with effective scattering rate $\eta =0.014$.

Figure 9

(a) DOS and (b) charge density profile in La${}_{2-x}$Sr${}_x$CuO${}_4$ at $p=0.16$. The inset shows the DOS of the homogeneous model with scattering rate $\eta =0.021$.

Figure 10

The HF model for Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ surface. Blue dots indicate the interstitial oxygen dopants.

Figure 11

(a, d) The zero temperature hole density plots and (b, e) the hole density maps for the surface layer of Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ for the average hole density $p=0.08$ (left panels) and $0.16$ (right panels). The lower panels (c) and (f) show positions of the oxygen dopants for the corresponding realizations of disorder. It is evident that the dopant oxygens locally increase the hole density.

Figure 12

The zero temperature hole density distribution in the surface layer of Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ at various doping levels $p$. Each curve is averaged over 20 realizations of disorder.

Figure 13

The correlation function Eq. (27) between the interstitial oxygen position and the local density of mobile holes for different values of doping $p$.