Dual structure in the charge excitation spectrum of electron-doped cuprates

Motivated by the recent resonant x-ray scattering (RXS) and resonant inelastic x-ray scattering (RIXS) experiments for electron-doped cuprates, we study the charge excitation spectrum in a layered $t\text{−}J$ model with the long-range Coulomb interaction. We show that the spectrum is not dominated by a specific type of charge excitations, but by different kinds of charge fluctuations, and is characterized by a dual structure in the energy space. Low-energy charge excitations correspond to various types of bond-charge fluctuations driven by the exchange term ($J$ term), whereas high-energy charge excitations are due to usual on-site charge fluctuations and correspond to plasmon excitations above the particle-hole continuum. The interlayer coupling, which is frequently neglected in many theoretical studies, is particularly important to the high-energy charge excitations.

Figure 1

${\mathbf{q}}_{\parallel }\text{−}\omega$ maps of each element of $\mathrm{Im}{D}_{ab}(\mathbf{q},\omega )$ along the symmetry axes $(\pi ,0)\text{−}(\pi ,\pi )\text{−}(0,0)\text{−}(\pi ,0)$; the out-of-plane momentum is $q_z=\pi$. The elements are placed in a $6\times 6$ matrix form and only the upper triangle of the matrix is shown because $D_{ab}(\mathbf{q},\omega )$ is symmetric. To clarify each sector of $2\times 2$, $4\times 4$, and $2\times 4$, a small space is inserted between each sector.

Figure 2

Zeros of the determinant of $D_{ab}^{-1}(\mathbf{q},\omega )$ along the symmetry axes $(\pi ,0)\text{−}(\pi ,\pi )\text{−}(0,0)\text{−}(\pi ,0)$ for $J=0.3$ and $J=0$ at $q_z=\pi$. These zeros yield collective charge excitations above the particle-hole continuum.

Figure 3

${\mathbf{q}}_{\parallel }\text{−}\omega$ map of $\mathrm{Im}{\chi }_c(\mathbf{q},\omega )$ for $q_z=8\pi /15$. ${\mathbf{q}}_{\parallel }$ is scanned along the symmetry axes $(\pi ,0)\text{−}(\pi ,\pi )\text{−}(0,0)\text{−}(\pi ,0)$.

Figure 4

${\mathbf{q}}_{\parallel }\text{−}\omega$ maps of the spectral weight for (a) ${\chi }_{d\mathrm{bond}}$, (b) ${\chi }_{s\mathrm{bond}}$, and (c) ${\chi }_{d\mathrm{CDW}}$ along the symmetry axes $(\pi ,0)\text{−}(\pi ,\pi )\text{−}(0,0)\text{−}(\pi ,0)$. The out-of-plane momentum is $q_z=\pi$.

Figure 5

${\mathbf{q}}_{\parallel }\text{−}\omega$ maps of the superposition of excitation spectra of ${\chi }_c$, ${\chi }_{d\mathrm{bond}}$, ${\chi }_{s\mathrm{bond}}$, and ${\chi }_{d\mathrm{CDW}}$ along the symmetry axes $(\pi ,0)\text{−}(\pi ,\pi )\text{−}(0,0)\text{−}(\pi ,0)$ for $q_z=\pi$ at three different doping rates: (a) $\delta =0.15$, (b) $\delta =0.20$, and (c) $\delta =0.25$.

Figure 6

$S\left(\mathbf{q}\right)$ at $\delta =0.15$ and $T=0$ for ${\chi }_{d\mathrm{bond}}$, ${\chi }_{s\mathrm{bond}}$, ${\chi }_{d\mathrm{CDW}}$, and ${\chi }_c$ along the symmetry axes $(\pi ,0)\text{−}(\pi ,\pi )\text{−}(0,0)\text{−}(\pi ,0)$ for (a) $q_z=\pi$, (b) $q_z=8\pi /15$, and (c) $q_z=0$.

Figure 7

Charge excitation spectra for (a) ${\chi }_{d\mathrm{bond}}^{2\mathrm{D}}$, (b) ${\chi }_{s\mathrm{bond}}^{2\mathrm{D}}$, and (c) ${\chi }_{d\mathrm{CDW}}^{2\mathrm{D}}$ in the 2D $t\text{−}J\text{−}V$ model along the symmetry axes $(\pi ,0)\text{−}(\pi ,\pi )\text{−}(0,0)\text{−}(\pi ,0)$. (d) Superposition of the spectra of ${\chi }_c^{2\mathrm{D}}$, ${\chi }_{d\mathrm{bond}}^{2\mathrm{D}}$, ${\chi }_{s\mathrm{bond}}^{2\mathrm{D}}$, and ${\chi }_{d\mathrm{CDW}}^{2\mathrm{D}}$.

Figure 8

${\mathbf{q}}_{\parallel }\text{−}\omega$ maps of each element of $\mathrm{Im}{D}_{ab}(\mathbf{q},\omega )$ along the symmetry axes $(\pi ,0)\text{−}(\pi ,\pi )\text{−}(0,0)\text{−}(\pi ,0)$. The out-of-plane momentum is $q_z=0$. The elements are placed in a $6\times 6$ matrix form and only the upper triangular matrix is shown because $D_{ab}(\mathbf{q},\omega )$ is symmetric. To clarify each sector of $2\times 2$, $4\times 4$, and $2\times 4$, a small space is inserted between each sector. In the elements $(2,2)$, $(2,5)$, and $(2,6)$, we choose a smaller $\mathrm{\Gamma }={10}^{-5}$ to increase the accuracy when computing near ${\mathbf{q}}_{\parallel }=(0,0)$. The inset (bottom left) is the element $(1,1)$ at $q_z=\pi /15$. To make a better contrast, the spectral intensity around ${\mathbf{q}}_{\parallel }=(0,0)$ is multiplied by 80 and the vertical axis line at ${\mathbf{q}}_{\parallel }=(0,0)$ is broken between $\omega =0.1$ and 0.75.

Figure 9

${\mathbf{q}}_{\parallel }\text{−}\omega$ maps of $\mathrm{Im}{\tilde{\chi }}_{d\mathrm{bond}}(\mathbf{q},\omega )$, $\mathrm{Im}{\tilde{\chi }}_{s\mathrm{bond}}(\mathbf{q},\omega )$, and $\mathrm{Im}{\tilde{\chi }}_{d\mathrm{CDW}}(\mathbf{q},\omega )$ along the symmetry axes $(\pi ,0)\text{−}(\pi ,\pi )\text{−}(0,0)\text{−}(\pi ,0)$; the out-of-plane momentum is $q_z=\pi$.