Close inspection of plasmon excitations in cuprate superconductors

Recently resonant inelastic x-ray scattering experiments reported fine details of the charge excitations around the in-plane momentum ${\mathbf{q}}_{\parallel }=(0,0)$ for various doping rates in electron-doped cuprates ${\text{La}}_{2-x}{\text{Ce}}_x{\text{CuO}}_4$. We find that those new experimental data are well captured by acousticlike plasmon excitations in a microscopic study of the layered $t-J$ model with the long-range Coulomb interaction. The acousticlike plasmon is not a usual plasmon typical to the two-dimensional system, but has a small gap proportional to the interlayer hopping $t_z$.

Figure 1

Plasmon dispersion as a function of in-plane momentum along $(\pi ,\pi )$-(0,0)-$(\pi ,0)$ direction for (a) $q_z=0$, (b) $5\pi /15$, and (c) $\pi$. The doping rate is $\delta =0.17$. The broadening is taken as $\mathrm{\Gamma }={10}^{-2}t$ to make the dispersion sharper. Faint spectral weight below 0.8 eV corresponds to the particle-hole continuum and becomes broader with increasing $q$. The insets in (b) and (c) magnify a very low-energy region around ${\mathbf{q}}_{\parallel }=(0,0)$. Plasmon dispersion as a function of $q_z$ for ${\mathbf{q}}_{\parallel }=(0.05,0)\pi$ (d), $(0.2,0)\pi$ (e), and $(0.8,0)\pi$ (f).

Figure 2

Charge excitation spectra as a function of $\omega$ at ${\mathbf{q}}_{\parallel }=(0.12,0)\pi$ (a) and $(0.22,0)\pi$ (b) for $\delta =0.17$ and two different momenta $q_z$. (c) and (d) Corresponding experimental data from Figs. 2(c) and 2(d) in Ref. [11]. The arrow in (d) indicates excitations of charge origin. The peaks at $\omega \approx 0.3$ eV in (d) correspond to paramagnon excitations and the peaks at $\omega =0$ in (c) and (d) are elastic; these features are out of the scope of the present work. While the in-plane momenta are set exactly the same as the experimental values, the values of $q_z$ are slightly different between our values and the experimental ones because we have a finite number of planes. For instance, $l=1.68$ in the experiment corresponds to $q_z=0.32\pi$ which is close to $q_z=5\pi /15$ in the present calculations for 30 planes.

Figure 3

(a) Charge excitation spectra as a function of $\omega$ at $\mathbf{q}=(0.12,0,5/15)\pi$ for different doping rates. The broadening $\mathrm{\Gamma }$ is assumed to decrease with increasing doping. (b) The corresponding experimental results from Fig. 4(a) in Ref. [11]. Although $q_z$ changes a little by changing doping in the experiment (see Fig. 4(g) in Ref. [11]), such a change is very small and our value of $q_z=5\pi /15$ is appropriate for all doping.

Figure 4

(a) Excitation spectra at $\mathbf{q}=(0.12,0,5/15)\pi$ and $(0.08,0.08,5/15)\pi$ for $\delta =0.18$. (b) The corresponding experimental data from Fig. 4(d) in Ref. [11]. (c) Doping dependence of the peak energy at $\mathbf{q}=(0.08,0.08,5/15)\pi$ (blue line), $(0.12,0,5/15)\pi$ (red line), and $(0.22,0,5/15)\pi$ (black line). (d) The peak width at $\mathbf{q}=(0.12,0,5/15)\pi$ as a function of doping. The symbols (triangles and circles) in (c) and (d) are the experimental data from Figs. 4(e) and 4(f) in Ref. [11], respectively.