Imaging the impact on cuprate superconductivity of varying the interatomic distances within individual crystal unit cells

Many theoretical models of high-temperature superconductivity focus only on the doping dependence of the CuO₂-plane electronic structure. However, such models are manifestly insufficient to explain the strong variations in superconducting critical temperature, Tc, among cuprates that have identical hole density but are crystallographically different outside of the CuO₂ plane. A key challenge, therefore, has been to identify a predominant out-of-plane influence controlling the superconductivity, with much attention focusing on the distance d_A between the apical oxygen and the planar copper atom. Here we report direct determination of how variations in interatomic distances within individual crystalline unit cells affect the superconducting energy-gap maximum Δ of Bi₂Sr ₂CaCu₂O_8+δ. In this material, quasiperiodic variations of unit cell geometry occur in the form of a bulk crystalline "super-modulation." Within each supermodulation period, we find ≈9 ± 1% cosinusoidal variation in local Δ that is anticorrelated with the associated d_A variations. Furthermore, we show that phenomenological consistency would exist between these effects and the random Δ variations found near dopant atoms if the primary effect of the interstitial dopant atom is to displace the apical oxygen so as to diminish d_A or tilt the CuO₅ pyramid. Thus, we reveal a strong, nonrandom out-of-plane effect on cuprate superconductivity at atomic scale.