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We study optimally dopedBi$_{2}$Sr$_{2}$Ca$_{0.92}$Y$_{0.08}$Cu$_{2}$O$_{8+\delta}$ (Bi2212) usingangle-resolved two-photon photoemission spectroscopy. Three spectral featuresare resolved near 1.5, 2.7, and 3.6 eV above the Fermi level. By tuning thephoton energy, we determine that the 2.7-eV feature arises predominantly fromunoccupied states. The 1.5- and 3.6-eV features reflect unoccupied states whosespectral intensities are strongly modulated by the corresponding occupiedstates. These unoccupied states are consistent with the prediction from acluster perturbation theory based on the single-band Hubbard model. Throughthis comparison, a Coulomb interaction strength U of 2.7 eV is extracted. Ourstudy complements equilibrium photoemission spectroscopy and provides a directspectroscopic measurement of the unoccupied states in cuprates. The determinedCoulomb U indicates that the charge-transfer gap of optimally doped Bi2212 is1.1 eV.

Revealing the Coulomb interaction strength in a cuprate superconductor