Optical studies of gap, exchange, and hopping energies in the insulating cuprates

We have measured the insulating energy gap {Δ} and the exchange interaction J in a series of cuprate crystals, including T'-phase M₂{CuO}₄$ (M=Pr, Nd, Sm, Eu, and Gd), ${T}*$-phase La,Tb,${Sr}₂CuO₄, and T-phase La₂{CuO}₄$. We find that the energy gap scales predominantly with the in-plane Cu-O distance, scaling as {δ} log{Δ}/ {δ} logd{~}-6. Furthermore, contrary to simple expectations, the energy gap increases with decreasing Cu-O distance, suggesting that Coulomb and other repulsive energies dominate the effects of band hybridization. Using a three-band Hubbard-model expression, our studies of {Δ} and J in the cuprates allow us to estimate that the hopping energy t scales with Cu-O distance as {δ} logt/{δ} logd{~}-4.