The emergence of global phase coherence from local pairing in underdoped cuprates

In conventional metal superconductors such as aluminum, the large number of weakly bounded Cooper pairs become phase coherent as soon as they start to form. The cuprate high critical temperature ($T_c$) superconductors, in contrast, belong to a distinctively different category. To account for the high $T_c$, the attractive pairing interaction is expected to be strong and the coherence length is short. Being doped Mott insulators, the cuprates are known to have low superfluid density, thus are susceptible to phase fluctuations. It has been proposed that pairing and phase coherence may occur separately in cuprates, and $T_c$ corresponds to the phase coherence temperature controlled by the superfluid density. To elucidate the microscopic processes of pairing and phase ordering in cuprates, here we use scanning tunneling microscopy to image the evolution of electronic states in underdoped $\rm Bi_2La_xSr_{2-x}CuO_{6+{\delta}}$. Even in the insulating sample, we observe a smooth crossover from the Mott insulator to superconductor-type spectra on small islands with chequerboard order and emerging quasiparticle interference patterns following the octet model. Each chequerboard plaquette contains approximately two holes, and exhibits a stripy internal structure that has strong influence on the superconducting features. Across the insulator to superconductor boundary, the local spectra remain qualitatively the same while the quasiparticle interferences become long-ranged. These results suggest that the chequerboard plaquette with internal stripes plays a crucial role on local pairing in cuprates, and the global phase coherence is established once its spatial occupation exceeds a threshold.