# Conventional superconductivity in quasicrystals

## TL;DR

This study investigates superconductivity in quasicrystals, revealing that despite their quasiperiodic structure, they tend to exhibit conventional BCS-like superconductivity, especially when the Fermi energy is not at a pseudogap.

## Contribution

The paper provides the first detailed theoretical analysis of superconductivity in quasicrystals using the Bogoliubov-de Gennes approach, highlighting the inhomogeneous nature and size dependence of the order parameter.

## Key findings

- Superconducting state is inhomogeneous but lacks superconductivity islands.
- Order parameter depends on system size when Fermi energy is at a pseudogap.
- Results align with experimental observations of superconductivity in Al-Zn-Mg quasicrystals.

## Abstract

Motivated by a recent experimental observation of superconductivity in the Al-Zn-Mg quasicrystal, we study the low-temperature behavior of electrons moving in the quasiperiodic potential of the Ammann-Beenker tiling in the presence of a local attraction. We employ the Bogoliubov-de Gennes approach for approximants of different sizes and determine the local pairing amplitude $\Delta_{i}$ as well its spatial average, $\Delta_{0}$, the superconducting order parameter. Due to the lack of periodicity of the octagonal tiling, the resulting superconducting state is inhomogeneous, but we find no evidence of the superconductivity islands, as observed in disordered systems, with $\Delta_{i}\rightarrow0$ at $T_{c}$ for all sites. In the weak-coupling regime, we find that the superconducting order parameter depends appreciably on the approximant size only if the Fermi energy sits at a pseudogap in the noninteracting density of states, with $\Delta_{0}$ decreasing as the system size increases. These results are in line with the experimental observations for the Al-Zn-Mg quasicrystal, and they suggest that, despite their electronic structure, quasicrystals are prone to display conventional BCS-like superconductivity.