A full quantum-mechanical treatment of the Bogoliubov-de Gennes equation for a single vortex in a d-wave superconductor is presented. First, we find low-energy states extended in four diagonal directions, which have no counterpart in a vortex of s-wave superconductors. The four-fold symmetry is due to 'quantum effect', which is enhanced when $p_{ F}\xi $ is small. Second, for $p_{ F}\xi \sim 1$, a peak with a large energy gap $E_{ 0}\sim \Delta $ is found in the density of states, which is due to the formation of the lowest bound states.
Categories: cond-mat.supr-con