Doppler shift on local density of states and local impurity scattering in the vortex state

The vortex state thermal and transport properties of the high-$T_c$ copper oxides can be understood in a d-wave gap model and are dominated by the extended quasiparticle states that exist along the nodal directions in momentum space. The Doppler shift on these states due to the circulating supercurrents around the vortex core, introduces van Hove ridges into the energy dependent local density of states as a function of distance in the region between cores. We emphasize the topology of these ridges and the effect on them of local static impurity scattering in Born and unitary limit. We treat possible orthorhombicity. Effective local scattering rates are also obtained.