Spectrum of low-energy excitations in the vortex state: Comparison of the Doppler-shift method to a quasiclassical approach

We present a detailed comparison of numerical solutions of the quasiclassical Eilenberger equations with several approximation schemes for the density of states of s- and d-wave superconductors in the vortex state, which have been used recently. In particular, we critically examine the use of the Doppler-shift method, which has been claimed to give good results for d-wave superconductors. Studying the single-vortex case we show that there are important contributions coming from core states, which extend far from the vortex cores into the nodal directions and are not present in the Doppler-shift method, but significantly affect the density of states at low energies. This leads to sizable corrections to Volovik's law, which we expect to be sensitive to impurity scattering. For a vortex lattice we also show comparisons with the method due to Brandt, Pesch, and Tewordt and an approximate analytical method, generalizing a method due to Pesch. These are high-field approximations strictly valid close to the upper critical field ${B}_{c2}.$ At low energies the approximate analytical method turns out to give impressively good results over a broad field range and we recommend the use of this method for studies of the vortex state at not too low magnetic fields.