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We investigate the electron-boson spectral density function,$I^2\chi(\omega,T)$, of CuO$_2$ plane in underdopedBi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi-2212) and underdopedYBa$_2$Cu$_3$O$_{6.50}$ (Y-123) using the Eliashberg formalism. We apply a new(in-plane) pseudogap model to extract the electron-boson spectral function. Forextracting the spectral function we assume that the spectral density functionconsists of two components: a sharp mode and the broad Millis-Monien-Pines(MMP) mode. We observe that both the resulting spectral density function andthe intensity of the pseudogap show strong temperature dependences: the sharpmode takes most spectral weight of the function and the peak position of thesharp mode shifts to lower frequency and the depth of pseudogap,$1-\tilde{N}(0,T)$, is getting deeper as temperature decreases. We observe alsothat the total spectral weight of the electron-boson density and the massenhancement coefficient increase as temperature decreases. We estimatefictitious (maximum) superconducting transition temperatures, $T_c(T)$, fromthe extracted spectral functions at various temperatures using a generalizedMcMillan formula. The estimated (maximum) $T_c$ also shows a strong temperaturedependence; it is higher than the actual $T_c$ at all measured temperatures anddecreases with temperature lowering. Since as lowering temperature thepseudogap is getting stronger and the maximum $T_c$ is getting lower we proposethat the pseudogap may suppress the superconductivity in cuprates.

The electron-boson spectral density function of underdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ and YBa$_2$Cu$_3$O