Density of states in d-wave superconductors of finite size

We consider the effect of the finite size in the ab plane on the surface density of states (DOS) in clean d-wave superconductors. We demonstrate that the angle-resolved DOS consists of energy bands that are formed similarly to the Kronig-Penney model. In contrast to the gapless DOS on a surface of a bulk sample, finiteness of the superconductor in one dimension provides the energy gap for all directions of quasiparticle motion except for theta= 45° (theta is the angle between the trajectory and the surface normal). As a result, the angle-averaged DOS behaves linearly at low energies. At the same time, the transport DOS can still have a gap. In the special case of alpha= 0° (alpha is the angle between the a axis of the crystal and the surface normal), the spectrum is gapped for all trajectories theta; the angle-averaged DOS is also gapped. For alpha= 45°, the spectrum is gapless for all theta; the angle-averaged DOS is then large at low energies.