A COMSOL-Based Study of the Electrical Transport Properties of Al/ α $\alpha$ - Al 2 O 3 ${\rm Al}_{2}{\rm O} _{3}$ /Al Josephson Junctions

Superconducting Josephson junctions, as integral components of quantum circuits, are vital for the production of high-quality, reproducible, and scalable quantum chips. The aluminum tunnel junctions is currently regarded as one of the most high-performing and well-established Josephson junctions for application in quantum devices. Nonetheless, the critical current of the junctions is highly sensitive to its thickness, which significantly influences both the tunneling effect and the electrical properties of the device. This study develops a numerical model of the 3D Al/$\alpha$-${\mathrm{Al}}_2{O}_3$/Al Josephson junction employing finite element analysis. At the nanometer scale of the junction, the Cooper pairs tunneling probability of the atomically resolved electrostatic potential as well as the WKB-equivalent barrier height of the device is calculated. The results of the model reveal a channel effect that influences the critical current of the Josephson junctions, and it is found that the probability of particles tunneling varies with the thickness of the alumina film. The probability current density distribution near the junction for various barrier thicknesses is further investigated, simultaneously employing two tunneling probability calculation methods. Moreover, a 1D square potential barrier model is also established to elucidate the quantum tunneling effect, in which the amplitude of the wave function for tunneled particles diminishes as the thickness of the potential barrier increases.