Local analysis of a single impurity on a graphene Josephson Junction

Scientific interest in two-dimensional (2D) materials has soared since the discovery of graphene in 2004, in the field of both classical (Nano Materials Science 6, 1 (2024), Applied and Computational Engineering 7, 171 (2023)) and quantum (Advanced Materials 35, 2109894 (2023); ACS Nano 13, 13573 (2019)). Because of its versatility and formidable properties, graphene represents one of the most interesting choices for JJ implementations. One of the most successful fabrication techniques, which allows one to realize and stabilize single-layer 2D materials, exploits the van der Waals (VdW) forces between single layer 2D material to create stacked structures, i.e. VdW heterostructures (ACS Nano 13, 13573 (2019); Encyclopedia of Condensed Matter Physics (Second Edition) pp.310-328). High-quality heterostructures with clean interfaces have been employed to realize graphene Josephson Junctions (GJJs), by encapsulating graphene inside hexagonal boron nitride (hBN), being capable of achieving ballistic transport of Cooper pairs over micrometer-scale lengths in graphene (Nature Nanotechnology 5, 722 (2010); Nano Letters 11, 2396 (2011); Science 342, 614 (2013)), allowing also to obtain gate-tunable supercurrents that persist at large parallel magnetic fields (Nature Nanotechnology 10, 761 (2015); Nature Physics 12, 318 (2016); Physical Review Letters 117, 237002 (2016)), and other phenomena produced by various features of two-dimensional Andreev physics (Nature Physics 12, 128 (2016); Science 352, 966 (2016); Nature Physics 13, 756 (2017)). The majority of theoretical frameworks developed for GJJs have been relying on the ultra-short junction regime, i.e. for junction length L much shorter than the coherence length, which allows analytical retrieval, other than the energy spectrum, the explicit form of the CPR, which results in a forwardly skewed shape with respect to the sinusoidal of the SIS JJs (Nature Physics 13, 756 (2017)), which has been recently measured through the use of SQUID devices (Nano Letters 17, 3396 (2017); Physical Review Letters 133, 106001 (2024)). Such properties and results make GJJs suitable to be employed in hybrid gatemon qubits (Nature Nanotechnology 13, 915 (2018); Nature Nanotechnology 14, 120 (2019)) and other devices which offer reduced dissipative losses, crosstalk, and compatibility with high magnetic fields (Nature Communications 9, 4069 (2018); Nature Communications 9, 4615 (2018)), paving the way for the realization of fault-tolerant topological qubits based on Majorana zero modes (Physical Review B 93, 155402 (2016); Review Modern Physics 80, 1083 (2008)). Recently, there has been interest in the spatial control properties of ballistic GJJs. For this reason, an experimental methodology has been developed to measure and control the real-space current density in two parallel ballistic GJJs, based on Fourier and Hilbert transformations of the magnetic field-induced modulation of the critical current (Physical Review Applied 20, 054049 (2023)). Tunneling spectroscopy measurements in GJJs revealed the possible presence of weakly coupled microscopic quantum dots to proximitized graphene, which act as energy filters in the tunneling process (Nature Nanotechnology 14, 120 (2019), Physical Review B 98, 121411 (2018)). A theoretical analysis has suggested that a homogeneous dilute distribution of impurities in a GJJ can alter the critical current and the skewness of the CPR (Communications Physics 5, 265 (2022)). In addition, defects in the nearby substrate can act as carrier traps and induce fluctuations in the carrier density of the graphene channel (Journal of Statistical Mechanics 2019, 094015 (2019); Applied Physics Letters 119, 223106 (2021)) and lead to critical current fluctuations with a spectrum 1/f (Communications Physics 3, 6 (2020); European Physical Journal Special Topics 230, 821 (2021), Review Modern Physics 86, 361 (2014)). The importance of these figures of merit led us to start an investigation focused on how the presence of impurities in graphene affects ABS, comparing them with the clean junction case. This Ph.D. thesis condenses the research work conducted investigating the effects of a single impurity inside the normal phase region of a short ballistic GJJ. All results have been published in the open-access journal Physical Review Research (Physical Review Research 7, 013189 (2025)). A detailed description is given to trace the steps to write the model Hamiltonian used to describe the junction, obtained within the Dirac-Bogoliubov de Gennes approach, and the spinorial representation used. The basic knowledge given in this chapter is employed to find the eigenenergy spectrum and the eigenfunctions of the system Hamiltonian, which leads to the Andreev bound states spectrum. These subgap states physically originate from multiple Andreev reflections at the interfaces of the normal phase region of the junction, which in the literature are commonly obtained analytically within the short junction limit, are here calculated numerically with a method that takes into account finite (but potentially arbitrary) lengths. A definition of the system Hamiltonian projected in the Andreev bound state subspace, is given, together with the introduction and operative definition of the most important quantity within this research work, which is the local density of states. The core results of the investigation carried out within this Ph.D. research work, is the study of the effects on the local density of states of a single, short-range, impurity placed in the normal phase region of the junction, here considered short but still outside the short junction limit. Taking into account the Anderson and Lifshitz impurity models, using the spinorial formalism already delineated to give the correct Hamiltonian description of the full system. Particular attention was given, for each model, to the magnetic and non-magnetic impurity cases. To evaluate the effect of the impurity - ABS interactions, the Green function approach has been employed, starting from the Dyson equation and the self-energies, getting to an exact closed expression of the Green function expansion, through which the local density of states can be calculated. In the results of the numerical calculations, the energy resolved local density of states, which exhibits a forbidden energy range called mini-gap inside the superconducting gap, presents two symmetric (asymmetric) impurity peaks for an Anderson non-magnetic (magnetic) impurity, while in the Lifshitz non-magnetic case no peak inside the mini-gap emerges regardless of the impurity coupling parameter, unless the magnetic term is introduced. For the nonmagnetic Lifshitz case, a semi-analytical proof of this last statement was produced and discussed. The thesis work closes bringing the results of the space-resolved local density of states. For this analysis, this quantity is plotted as a function of the position in the direction transverse to the junction, which presents periodic, decreasing, and oscillating patterns. The Fourier analysis to find the characteristic wavenumbers of these oscillating patterns was performed numerically, and the results obtained are then compared with the momenta of the high-transmissive channels in ballistic graphene.