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C-Axis Tunneling Spectra in High-T$_c$ Superconductors in the Presence of a d Charge-Density Wave

The optimally doped and underdoped region of the $t-J$ model at large N (N is the number of spin components) is governed by the competition of d-wave superconductivity (SC) and a d Charge-Density Wave (d-CDW).The partial destruction of the Fermi surface by the d-CDW and the resulting density of states are discussed. Furthermore, c-axis conductances for incoherent and coherent tunneling are calculated, considering both an isotropic and an anisotropic in-plane momentum dependence of the hopping matrix element between the planes. The influence of self-energy effects on the conductances is also considered using a model where the electrons interact with a dispersionless, low-lying branch of bosons. We show that available tunneling spectra from break-junctions are best explained by assuming that they result from incoherent tunneling with a strongly anisotropic hopping matrix element of the form suggested by band structure calculations. The conductance spectra are then characterized by one single peak which evolves continuously from the superconducting to the d-CDW state with decreasing doping. The intrinsic c-axis tunneling spectra are, on the other hand, best explained by coherent tunneling. Calculated spectra show at low temperatures two peaks due to SC and d-CDW. With increasing temperature the BCS-like peak moves to zero voltage and vanishes at T$_c$,exactly as in experiment.Our results thus can explain why break junction and intrinsic tunneling spectra are different from each other. Moreover, they support a scenario of two competing order parameters in the underdoped region of high-T$_c$ superconductors.