We introduce a simple phenomenological form for the self-energy which allows us to extract important information from angle-resolved photoemission data on the high-$T_c$ superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$ for binding energies of order the spectral gap. First, we find a rapid suppression of the single particle scattering rate below $T_c$ for all doping levels. Second, we find that in the overdoped materials the gap $\Delta$ at all k points on the Fermi surface has significant temperature dependence and vanishes near $T_c.$ In contrast, in the underdoped samples such behavior is found only at k points close to the diagonal. Near $(\pi ,0)$, $\Delta$ is essentially $T$ independent in the underdoped samples. The filling-in of the pseudogap with increasing $T$ is described by a broadening proportional to $T-T_c,$ which is naturally explained by pairing correlations above $T_c.$

• Received 3 February 1998

DOI:https://doi.org/10.1103/PhysRevB.57.R11093