Collective phase oscillation in two-dimensional d-wave superconductors

We theoretically investigate the Carlson-Goldman (CG) mode in two-dimensional d_x2-y2-wave superconductivity. In conventional s-wave superconductors, it is known that the CG mode is observed as a peak in the structure function of the pair field susceptibility S(q,ω) only just below the transition temperature T_c and only in dirty systems. On the other hand, in the d-wave state, we show that the peak structure can be observed down to T≁0.2T_c in clean systems. Furthermore, we also show that the peak splits into two when the direction of the momentum q deviates from the q_x or q_y axes, or nodal directions of the order parameter. These features originate from a singularity in the charge-density fluctuation due to the presence of the nodes of the d-wave order parameter: It enhances the screening of the long-range Coulomb interaction so that the CG mode can exist far below T_c. We also investigate a quasi-two-dimensional system, and show that the Josephson plasma can be observed in S(q,ω) when q is parallel to the c direction.