Momentum and Energy Dependence of the Anomalous High-Energy Dispersion in the Electronic Structure of High Temperature Superconductors

Using high-resolution angle-resolved photoemission spectroscopy we have studied the momentum and photon energy dependence of the anomalous high-energy dispersion, termed waterfalls, between the Fermi level and 1 eV binding energy in several high-$T_c$ superconductors. We observe strong changes of the dispersion between different Brillouin zones and a strong dependence on the photon energy around 75 eV, which we associate with the resonant photoemission at the $\mathrm{Cu}3p\to 3d_{x^2-y^2}$ edge. We conclude that the high-energy “waterfall” dispersion results from a strong suppression of the photoemission intensity at the center of the Brillouin zone due to matrix element effects and is, therefore, not an intrinsic feature of the spectral function. This indicates that the new high-energy scale in the electronic structure of cuprates derived from the waterfall-like dispersion may be incorrect.

Figure 1

(a)–(h) Typical snapshots of the one-particle excitation spectra of Bi-2212 (a)–(f) and Y-123 (g),(h) measured by angle-resolved photoemission with 100 eV photon energy. The spectra (a)–(d) are measured along the high-symmetry directions marked by the dashed lines on the Fermi surface map (e). Spectra (a) and (c) from the 1st BZ exhibit strong high-energy kinks (black arrows) and waterfalls, while the equivalent spectrum (b) from the 2nd BZ exhibits no pronounced high-energy scales. Additional spectral weight is clearly seen in panel (d), where the curve at the left of the panel shows the energy distribution curve at $(\pi ,0)$. (f) Constant-energy cut at 0.38 eV below the Fermi level in Bi-2212 showing spectral weight depletion along the 1st BZ diagonals. (g),(h) The respective constant-energy maps of Y-123. The 1st BZ on the constant-energy maps is confined by the dotted squares. (i),(j) and (k),(l) Pairs of equivalent spectra of Tb- and Pb-doped Bi-2212 taken in the 2nd BZ along the $(\pi ,0)$ and $(\pi ,\pi )$ directions with two different excitation energies as indicated on top of each panel. In both directions, the onset of the waterfalls-like behavior suddenly occurs at about 75 eV photon energy. The color scale in all panels represents photoelectron intensity. The spectra are normalized to the background above the Fermi level. The spectra in panels (k) and (l) are in addition multiplied by a linear function of momentum to enhance the right-hand part of the spectrum, which otherwise has much lower intensity than the left-hand part due to the experimental geometry. (m) Schematic representation of the experimentally accessible regions of momentum space showing different behaviors of the high-energy dispersion in different BZs immediately below 75 eV photon energy. Positive $k_x$ values correspond to the experimental geometry approaching normal incidence. (n) Energy distribution curves taken at the $\Gamma$ point from spectra shown in panels (i) and (k), showing a distinct bottom of the renormalized band at about 0.5 eV.