Revealing the Coulomb interaction strength in a cuprate superconductor

We study optimally doped ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{Ca}}_{0.92}{\mathrm{Y}}_{0.08}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ (Bi2212) using angle-resolved two-photon photoemission spectroscopy. Three spectral features are resolved near 1.5, 2.7, and 3.6 eV above the Fermi level. By tuning the photon energy, we determine that the 2.7-eV feature arises predominantly from unoccupied states. The 1.5- and 3.6-eV features reflect unoccupied states whose spectral intensities are strongly modulated by the corresponding occupied states. These unoccupied states are consistent with the prediction from a cluster perturbation theory based on the single-band Hubbard model. Through this comparison, a Coulomb interaction strength $U$ of 2.7 eV is extracted. Our study complements equilibrium photoemission spectroscopy and provides a direct spectroscopic measurement of the unoccupied states in cuprates. The determined Coulomb $U$ indicates that the charge-transfer gap of optimally doped Bi2212 is 1.1 eV.