Quasi-particle spectra around a single vortex in a d-wave superconductor

Using the Bogoliubov-de Gennes equation, we study quasi-particle spectra around a single vortex in a d-wave superconductor, where a magnetic field is parallel to the c-axis. In the temperature region where the Ginzburg-Landau theory is valid, we find that the local density of states preserves a circular symmetry when the symmetry of the superconducting order parameter is {\it pure} d-wave. It, however, exhibits a four-fold symmetry when the mixing of a s-wave component occurs. A peak with a {\it large energy gap} is found in the local density of states at the center of the vortex, which corresponds to the lowest bound state. Our results are consistent with a recent scanning tunneling microscopy experiment in an YBa$_{2}$Cu$_{3}$O$_{7-{\delta}}$ (YBCO) monocrystal. The breakdown of the Eilenberger theory in YBCO in particular and in the high-T$_{c}$ superconductors in generalis discussed.