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.

• Received 13 February 2004

DOI:https://doi.org/10.1103/PhysRevB.70.212513