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

TL;DR

This paper compares numerical and approximate methods for calculating the density of states in vortex states of superconductors, highlighting the importance of core states and providing recommendations for effective approaches.

Contribution

It critically evaluates the Doppler shift method against quasiclassical solutions and other approximations, revealing their limitations and proposing effective analytical methods.

Findings

Core states significantly influence low-energy density of states.

The Doppler shift method misses contributions from extended core states.

An approximate analytical method provides accurate results at moderate fields.

Abstract

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 sizeable 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 Bc2. 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.