Plasmon dispersion in bilayer cuprate superconductors

The essential building blocks of cuprate superconductors are two-dimensional ${\mathrm{CuO}}_2$ sheets interspersed with charge reservoir layers. In bilayer cuprates, two closely spaced ${\mathrm{CuO}}_2$ sheets are separated by a larger distance from the subsequent pair in the next unit cell. In contrast to single-layer cuprates, prior theoretical work on bilayer systems has predicted two distinct acoustic plasmon bands for a given out-of-plane momentum transfer. Here we report random phase approximation (RPA) calculations for bilayer systems which corroborate the existence of two distinct plasmon bands. We find that the intensity of the lower-energy band is negligibly small in most parts of the Brillouin zone, whereas the higher-energy band carries significant spectral weight. We also present resonant inelastic x-ray scattering (RIXS) experiments at the O $K$-edge on the bilayer cuprate ${\mathrm{Y}}_{0.85}{\mathrm{Ca}}_{0.15}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ (Ca-YBCO), which show only one dispersive plasmon branch, in agreement with the RPA calculations. In addition, the RPA results indicate that the dispersion of the higher-energy plasmon band in Ca-YBCO is not strictly acoustic but exhibits a substantial energy gap of approximately 250 meV at the two-dimensional Brillouin zone center.

Figure 1

(a) Sketch of a system with equally spaced conducting sheets (red shaded planes). Hopping between the sheets is denoted by $t_z$. The black boxes represent unit cells with the lattice parameters $a$, $b$, and $c$. The distance between the planes is denoted by $d$. (b) Sketch of a bilayer system that contains two closely spaced sheets within a unit cell. The intrabilayer hopping is denoted by $t_z$ and the interbilayer hopping by $t_z^{\prime }$. The distance between the two closely spaced planes is denoted by $d$. (c) Schematic of the crystal structure of the bilayer cuprate Ca-YBCO. The red shaded planes are the ${\mathrm{CuO}}_2$ sheets.

Figure 2

Computed intensity maps of the RPA charge response for a generic bilayer system. (a-c) Intensity maps in the ${\mathbf{q}}_{\parallel }-\omega$ plane for $q_z=0$, $0.2\pi$, and $\pi$, respectively. Red dashed lines correspond to the zeros of the dielectric function (${\omega }_{+}$ and ${\omega }_{-}$). White dotted lines are guides to the eye marking the upper bound above which the spectral weight of the particle-hole continuum becomes insignificant. (d) Intensity map in the $q_z-\omega$ plane for $q_x=q_y=0.1\pi$.

Figure 3

(a) XAS of Ca-YBCO across the O $K$ edge measured with $\sigma$-polarized photons. The arrow indicates the incident photon energy (528.5 eV) used for the RIXS experiment. (b) RIXS spectra of Ca-YBCO measured at various momenta along the $K$ direction, while $H$ and $L$ were fixed to 0 and 0.9, respectively. The fit (black line) to the experimental data (blue symbols) includes the plasmon peak (orange peak profile) and other contributions (not shown here) that are described in Appendix pp2. Curves for different momenta are offset in the vertical direction for clarity.

Figure 4

Computed intensity maps of the RPA charge response using renormalized effective parameters optimized for the bilayer cuprate Ca-YBCO. The dispersion is along the $K$ direction for $q_z=0.2\pi$. The superimposed red symbols are the plasmon energies for Ca-YBCO extracted from fits to the RIXS data [see Fig. 3].

Figure 5

(a) XAS of the YBCO film across the O $K$ edge. The arrow indicates the incident photon energy (527.32 eV) used for the RIXS experiment. (b) RIXS spectra of the YBCO film measured at various momenta along the $H$ direction between $H=0.04$ and 0.16. Curves for different momenta are offset in the vertical direction for clarity.

Figure 6

Representative fits of the RIXS data. (a) Fit of the RIXS spectrum for momentum (0, 0.02, 0.9), with a model using a Gaussian for the elastic peak (dashed black line) and anti-symmetrized Lorentzians for the other contributions, convoluted with the energy resolution of 27 meV via Gaussian convolution. The individual contributions are described in the text. [(b) and (c)] Fit of the spectrum for momentum (0, 0.04, 0.9) and (0, 0.06, 0.9), respectively.

Figure 7

Computed intensity map in analogy to Fig. 4, but for bare parameters of Ca-YBCO.