  Pairing of electrons in conventional superconductors occurs at the
superconducting transition temperature Tc, creating an energy gap D in the
electronic density of states (DOS). In the high-Tc superconductors, a partial
gap in the DOS exists for a range of temperatures above Tc. A key question is
whether the gap in the DOS above Tc is associated with pairing, and what
determines the temperature at which incoherent pairs form. Here we report the
first spatially resolved measurements of gap formation in a high-Tc
superconductor, measured on Bi2Sr2CaCu2O8+d samples with different Tc values
(hole concentration of 0.12 to 0.22) using scanning tunnelling microscopy. Over
a wide range of doping from 0.16 to 0.22 we find that pairing gaps nucleate in
nanoscale regions above Tc. These regions proliferate as the temperature is
lowered, resulting in a spatial distribution of gap sizes in the
superconducting state. Despite the inhomogeneity, we find that every pairing
gap develops locally at a temperature Tp, following the relation 2D/kBTp = 8.
At very low doping (<0.14), systematic changes in the DOS indicate the presence
of another phenomenon, which is unrelated and perhaps competes with electron
pairing. Our observation of nanometre-sized pairing regions provides the
missing microscopic basis for understanding recent reports of fluctuating
superconducting response above Tc in hole-doped high-Tc copper oxide
superconductors.
