Imaging the impact on cuprate superconductivity of varying the inter-atomic 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. But such models are manifestly insufficient to explain the strong variations in superconducting critical temperature Tc among cuprates which have identical hole-density but are crystallographically different outside 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 of inter-atomic distances within individual crystalline unit cells, affect the superconducting energy-gap maximum $\Delta$ of Bi2Sr2CaCu2O8. In this material, quasi-periodic variations of unit cell geometry occur in the form of a bulk crystalline 'supermodulation'. Within each supermodulation period, we find a $\sim 9\pm1%$ co-sinusoidal variation in local $\Delta$ that is anti-correlated with the associated $d_A$ variations. Furthermore, we show that phenomenological consistency would exist between these effects and the random $\Delta$ 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 non-random out-of-plane effect on cuprate superconductivity at atomic scale.