Josephson tunneling through a Yu-Shiba-Rusinov state: Interplay of $π$-shifts in Josephson current and local superconducting order parameter

An impurity hosting a magnetic moment coupled to a conventional $s$-wave superconductor gives rise to so-called Yu-Shiba-Rusinov (YSR) states with energies inside the superconducting gap. Depending on the coupling between the impurity and the superconductor, the system can have two distinct quantum ground states separated by a quantum phase transition (QPT). We investigate the interplay of two effects observed at the QPT. First, the tunneling supercurrent through the impurity reverses its sign at the QPT, denoted as a $π$-shift in the current-phase relation. Secondly, the local superconducting order parameter at the impurity site is suppressed and becomes negative at the QPT, generally termed a $π$-shift in the local superconducting order parameter. We find that both these effects are governed by the presence of the YSR state, however, they do not significantly depend or influence each other. In particular, we establish that the $π$-shift in the superconducting order parameter does not induce a $π$-shift in the tunneling Josephson current, nor can the Josephson current and its spatial behavior be used to directly probe the impurity-induced changes in the local superconducting order parameter, which occur on a length scale substantially shorter than the superconducting coherence length.

💡 Research Summary

The authors investigate a magnetic impurity coupled to a conventional s‑wave superconductor, focusing on the Yu‑Shiba‑Rusinov (YSR) bound state that forms inside the superconducting gap. By varying the impurity‑substrate coupling, the system can be driven through a quantum phase transition (QPT) at which the YSR level crosses zero energy. Two distinct “π‑shifts” are known to occur at this transition: (i) a reversal of the sign of the Josephson current flowing through the impurity (the current‑phase relation acquires a π phase), and (ii) a local suppression and sign reversal of the superconducting order parameter Δ at the impurity site. The central question of the paper is whether these two phenomena are causally linked, i.e. whether the π‑shift of Δ induces the π‑shift of the Josephson current, and whether Josephson‑STM (JSTM) can be used to probe the local order‑parameter sign change.

The theoretical framework combines a mean‑field Anderson impurity model (spin‑½ impurity with on‑site energy εi=0 and exchange splitting J=1t) with Bogoliubov‑de Gennes (BdG) tight‑binding Hamiltonians for both the superconducting tip and substrate. The tip‑impurity hopping t_t is set to a weak value (0.05t) to mimic tunnelling, while the impurity‑substrate hopping t_s is varied to traverse the QPT. The substrate’s order parameter Δ_i is obtained self‑consistently: starting from a homogeneous Δ0, the BdG eigenstates are used to evaluate the gap equation Δ_i=−(g/2)∑_ν f_T(E_ν)