Plasmon excitations in layered high-$T_c$ cuprates

Motivated by the recent resonant inelastic x-ray scattering (RIXS) experiment for the electron-doped cuprates ${\mathrm{Nd}}_{2-x}{\mathrm{Ce}}_x{\mathrm{CuO}}_4$ with $x\approx 0.15$, we compute the density-density correlation function in the $t-J$ model on a square lattice by including interlayer hopping and the long-range Coulomb interaction. We find that collective charge excitations are realized not inside the particle-hole continuum, but above the continuum as plasmons. The plasmon mode has a rather flat dispersion near the in-plane momentum ${\mathbf{q}}_{\parallel }=(0,0)$ with a typical excitation energy of the order of the intralayer hopping $t$ when the out-of-plane momentum $q_z$ is zero. However, when $q_z$ becomes finite, the plasmon dispersion changes drastically near ${\mathbf{q}}_{\parallel }=(0,0)$, leading to a strong dispersive feature with an excitation gap scaled by the interlayer hopping $t_z$. We discuss the mode recently observed by RIXS near ${\mathbf{q}}_{\parallel }=(0,0)$ in terms of the plasmon mode with a finite $q_z$.

Figure 1

Spectral weight of the density-density correlation $\mathrm{Im}{\chi }^c(\mathbf{q},\omega )$ in the plane of energy $\omega$ and in-plane momentum ${\mathbf{q}}_{\parallel }$ along $(\pi ,\pi )-(0,0)-(\pi ,0)-(\pi ,\pi )$ for $q_z=0$ and $\pi$. The dotted line denotes the upper boundary of a particle-hole continuum for $q_z=0$. The spectral intensity of the plasmon is by far higher than the continuum and is cut at 8 to get a better contrast for the continuum.

Figure 2

$q_z$ dependence of the plasmon dispersion around ${\mathbf{q}}_{\parallel }=(0,0)$ along $(0.4\pi ,0.4\pi )-(0,0)-(0.4\pi ,0)$. $q_z$ is changed from zero to $\pi$ with a step of $\pi /15$. This value of $\pi /15$ comes from our resolution of $q_z$ since we take 30 layers in our model.

Figure 3

$t_z$ dependence of the plasmon energy at ${\mathbf{q}}_{\parallel }=(0,0)$ for $q_z=0,\pi /15$, and $\pi$.

Figure 4

Plasmon dispersions for $q_z=0$ (dashed line) and $\pi$ (solid line) for several choices of doping rate $\delta$ along $(\pi ,\pi )-(0,0)-(\pi ,0)$.

Figure 5

(a) Temperature dependence of $\mathrm{Im}{\chi }^c(\mathbf{q},\omega )$ at $\mathbf{q}=(0,0,\pi )$ for $\mathrm{\Gamma }=0.01t$. (b) $\mathrm{Im}{\chi }^c(\mathbf{q},\omega )$ at $T=0$ for several choices of $\mathrm{\Gamma }$.

Figure 6

Plasmon dispersions for $q_z=0$ (dashed line) and $\pi$ (solid line) for $\delta =0.15$ and $t_z=0.5t$ along $(\pi ,\pi )-(0,0)-(\pi ,0)$.

Figure 7

Plasmon energy at $\mathbf{q}=(0,0,0)$ (dashed line) and $(0,0,\pi )$ (solid line) as a function of doping.