Effect of dilute impurities on short graphene Josephson junctions

Despite the structural simplicity of graphene, its mechanical and electronic remarkable properties make this material a credible starting point for new technologies across a wide range of fields. The recent realizations of graphene-based hybrid systems, such as Josephson junctions, make graphene a promising a platform for new generations of devices for topological quantum computing and quantum sensing. To this aim, accurate control of the electronic properties of graphene Josephson junctions in the presence of disorder is essential. Here, we study the effect of a dilute homogeneous spatial distribution of non-magnetic impurities on the equilibrium supercurrent sustained by a ballistic graphene Josephson junction in the short junction limit. Within the Dirac-Bogoliubov-de Gennes approach and modeling impurities by the Anderson model we derive the supercurrent and its equilibrium power spectrum. We find a modification of the current-phase relation with a reduction of the skewness induced by disorder, and a nonmonotonic temperature dependence of the critical current. The potentialities of the supercurrent power spectrum for accurate spectroscopy of the hybridized Andreev bound states-impurities spectrum are highlighted. In the low temperature limit, the supercurrent zero frequency thermal noise directly probes the spectral function at the Fermi energy.