Density of states in d-wave superconductors of finite size

TL;DR

This paper investigates how finite size effects in d-wave superconductors influence the surface density of states, revealing conditions under which an energy gap appears or disappears, with analytical and numerical confirmation.

Contribution

It provides a detailed analysis of the surface density of states in finite-sized d-wave superconductors, highlighting the impact of orientation and size on the energy spectrum.

Findings

Finite size induces an energy gap in the DoS except at 45 degrees.

At  degrees, the spectrum is fully gapped for all trajectories.

At 45 degrees, the spectrum remains gapless, leading to a large low-energy DoS.

Abstract

We consider the effect of the finite size in the ab-plane on the surface density of states (DoS) in clean d-wave superconductors. In the bulk, the DoS is gapless along the nodal directions, while the presence of a surface leads to formation of another type of the low-energy states, the midgap states with zero energy. We demonstrate that finiteness of the superconductor in one of dimensions provides the energy gap for all directions of quasiparticle motion except for \theta=45 degrees (\theta is the angle between the trajectory and the surface normal); then the angle-averaged DoS behaves linearly at small energies. This result is valid unless the crystal is 0- or 45-oriented (\alpha \ne 0 or 45 degrees, where \alpha is the angle between the a-axis and the surface normal). In the special case of \alpha=0, the spectrum is gapped for all trajectories \theta; the angle-averaged DoS is also gapped. In the special case of \alpha=45, the spectrum is gapless for all trajectories \theta; the angle-averaged DoS is then large at low energies. In all the cases, the angle-resolved DoS consists of energy bands that are formed similarly to the Kronig-Penney model. The analytical results are confirmed by a self-consistent numerical calculation.