Infrared divergence in d.c. Josephson current through charge density wave

Abstract

We calculate the d.c. Josephson current through charge density wave (CDW) region connected to two superconductors (S 1, S 2) by tunnel junctions. At T=0 and without impurities, we discover infrared divergence in the d.c. Josephson current caused by the phase mode in the long-junction limit: d/ v F≫ℏ/ Δ S, ℏ/ Δ CDW where T is temperature. These limits are expressed by using the energy gap in S region Δ S, the length d, the Fermi velocity v F, and the energy gap Δ CDW in CDW region. The critical current is proportional to ( Δ CDWℏ v F/ d) 1/2∝ Δ CDW( ξ CDW/ d) 1/2. At βΔ S( T), βΔ CDW( T)≫1 and βℏ v F/ d≪1, the T-dependence of the correction caused by the phase mode is T-linear where β=1/ k B T and k B is Boltzmann constant. At finite temperatures and with impurities, the correction caused by the phase mode becomes proportional to ln T . At finite temperatures and without impurities, the critical current is similar to the Ambegaokar–Baratoff formula which is proportional to Δ S( T) in the short-junction limit: d/ v F≪ℏ/ Δ S, ℏ/ Δ CDW.