Doping dependence of $d$-wave bond-charge excitations in electron-doped cuprates

Motivated by the recent experiments reporting the doping dependence of the short-range charge order (CO) in electron-doped cuprates, we study the resonant x-ray scattering spectrum from $d$-wave bond-charge fluctuations obtained in the two-dimensional $t\text{−}J$ model. We find that (i) the CO is short-range, (ii) the CO peak is pronounced at low temperature, (iii) the peak intensity increases with decreasing carrier doping $\delta$ down to $\delta \approx 0.10$ and is substantially suppressed below $\delta \approx 0.10$ due to strong damping, and (iv) the momentum of the CO decreases monotonically down to $\delta \approx 0.10$ and goes up below $\delta \approx 0.10$. These results reasonably capture the major features of the experimental data, and the observed short-range CO can be consistently explained in terms of bond-charge fluctuations with an internal $d$-wave symmetry.

Figure 1

(a) $\mathbf{q}$ dependence of ${\chi }_d\left(\mathbf{q}\right)$ for various choices of temperatures at $\delta =0.13$. (b) $S\left(\mathbf{q}\right)$ for the same parameters as in (a) and the cutoff energy is ${\omega }_c=\infty$.

Figure 2

Doping dependence of the damping $\mathrm{\Gamma }$ (solid line) and the phase boundary of the $d$-wave bond-charge order at $T=0$ (dashed line).

Figure 3

(a) $\mathbf{q}$ dependence of $S\left(\mathbf{q}\right)$ for various choices of temperatures at doping $\delta =0.13$. (b) $\mathrm{\Delta }{S}^{\mathrm{peak}}$ as a function of temperature for several choices of doping rates. The cutoff energy is ${\omega }_c=\infty$.

Figure 4

Peak intensity of $S\left(\mathbf{q}\right)$ with ${\omega }_c=\infty$ and the momenta $Q_{\mathrm{co}}$, $Q_{\mathrm{co}}^{\mathrm{cut}}$, and $Q_{\mathrm{edge}}$ as a function of $\delta$ at $T=0$. $Q_{\mathrm{co}}$ is the peak position of $S\left(\mathbf{q}\right)$ with ${\omega }_c=\infty$ whereas $Q_{\mathrm{co}}^{\mathrm{cut}}$ is that of $S\left(\mathbf{q}\right)$ with ${\omega }_c=0.05$; $Q_{\mathrm{edge}}$ is defined in the inset. Solid circles are the experimental data from Ref. [24].

Figure 5

(a) $\mathbf{q}$ dependence of $S\left(\mathbf{q}\right)$ for several choices of the cutoff energy at $\delta =0.13$ and $T=0$. (b) $\mathbf{q}$ dependence of $S\left(\mathbf{q}\right)$ with a cutoff energy ${\omega }_c=0.05$ for various choices of temperatures at $\delta =0.13$. The inset shows the temperature dependence of the peak intensity of $\mathrm{\Delta }S$ for several choices of doping rate after subtraction of intensity at $T=0.035$.

Figure 6

$S\left(\mathbf{q}\right)$ along the $(0,0)\text{−}(\pi ,\pi )$ direction for various choices of $\mathrm{\Gamma }$ at $T=0$ and $\delta =0.08$; the cutoff energy is ${\omega }_c=\infty$.