Josephson energy of superconducting junctions: amorphous versus crystalline tunnel barriers

Abstract

The Josephson energy $E_J$ is a key parameter governing the properties of transmon superconducting qubits. In Al/AlO$_x$/Al junctions, $E_J$ is set by electron tunneling through an ultrathin oxide barrier and therefore depends exponentially on the atomic structure of the barrier. We compute $E_J$ by first-principles device modeling based on the NEGF-DFT quantum-transport method, comparing a junction with a crystalline Al$_2$O$_3$ barrier against ten junctions with melt--quenched amorphous Al$_2$O$_3$ barriers of the same thickness. From the Fermi-level transmission and the Ambegaokar--Baratoff relation, we obtain a mean $E_J/h$ of $2.78$ GHz for the amorphous ensemble, with a standard deviation of $4.67$ GHz, compared with $0.73$ GHz for the crystalline reference; individual amorphous values span nearly two orders of magnitude. Scattering-state analysis shows that transport is quantum tunneling and that the variability originates from stoichiometric inhomogeneity of the amorphous oxide: Al-rich, low-barrier regions can connect into percolation-like tunneling pathways that strongly enhance the conductance. A realistic $200\times200$ nm$^2$ junction self-averages over more than $2\times10^4$ such microscopic regions. These results establish a quantitative atomistic route from oxide microstructure to the superconducting-circuit energy scale $E_J$.