Superconducting phase coherence in the presence of a pseudogap: Relation to specific heat, tunneling, and vortex core spectroscopies

In this paper we demonstrate how, using a natural generalization of BCS theory, superconducting phase coherence manifests itself in phase-insensitive measurements—when there is a smooth evolution of the excitation gap $\Delta$ from above to below $T_c.$ In this context, we address the underdoped cuprates. Our premise is that just as Fermi-liquid theory fails above $T_c,$ BCS theory fails below. The order parameter ${\Delta }_{\mathrm{sc}}$ is different from the excitation gap $\Delta .$ Equivalently there is a (pseudo)gap in the excitation spectrum above $T_c$ which is also present in the underlying normal state of the superconducting phase. A central emphasis of our paper is that the latter gap is most directly inferred from specific heat and vortex core experiments. At the same time there are indications that fermionic quasiparticles exist below $T_c$ so that many features of BCS theory are clearly present. A natural reconciliation of these observations is to modify BCS theory slightly without abandoning it altogether. Here we review such a modification based on a BCS-like ground-state wave function. A central parameter of our extended BCS theory is ${\Delta }^2-{\Delta }_{\mathrm{sc}}^2$ which is a measure of the number of bosonic pair excitations which have a nonzero net momentum. These bosons are present in addition to the usual fermionic quasiparticles. Applying this theory we find that the Bose condensation of Cooper pairs, which is reflected in ${\Delta }_{\mathrm{sc}},$ leads to sharp peaks in the spectral function once $T<~T_c.$ These are manifested in angle-resolved photoemission spectra as well as in specific heat jumps, which become more like the behavior in a $\lambda$ transition as the pseudogap develops. We end with a discussion of tunneling experiments and condensation energy issues. The comparison between theoretical and experimental plots of $C_v,$ tunneling, vortex core spectroscopy measurements is good.