Dephasing of Andreev pairs entering a charge density wave

A Cooper pair from a s-wave superconductor (S) entering a conventional charge density wave (CDW) below the Peierls gap dephases on the Fermi wavelength while one particle states are localized on the CDW coherence length ξ_CDW. It is thus practically impossible to observe a Josephson current through a CDW. The paths following different sequences of impurities interfere destructively, due to the different electron and hole densities in the CDW. The same conclusion holds for averaging over the conduction channels in the ballistic system. We apply two microscopic approaches to this phenomenon: (i) a Blonder, Tinkham, Klapwijk (BTK) approach for a single highly transparent S-CDW interface; and (ii) the Hamiltonian approach for the Josephson effect in a clean CDW and a CDW with non magnetic disorder. The Josephson effect through a spin density wave (SDW) is limited by the coherence length ξ_SDW, not by the Fermi wave-length. A Josephson current through a SDW might be observed in a structure with contacts on a SDW separated by a distance ξ_SDW.