  A remarkable mystery of the copper oxide high-transition-temperature (Tc)
superconductors is the dependence of Tc on the number of CuO2 layers, n, in the
unit cell of a crystal. In a given family of these superconductors, Tc rises
with the number of layers, reaching a peak at n=3, and then declines: the
result is a bell-shaped curve. Despite the ubiquity of this phenomenon, it is
still poorly understood and attention has instead been mainly focused on the
properties of a single CuO2 plane. Here we show that the quantum tunnelling of
Cooper pairs between the layers simply and naturally explains the experimental
results, when combined with the recently quantified charge imbalance of the
layers and the latest notion of a competing order nucleated by this charge
imbalance that suppresses superconductivity. We calculate the bell-shaped curve
and show that, if materials can be engineered so as to minimize the charge
imbalance as n increases, Tc can be raised further.
