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

TL;DR

This work analyzes how a magnetic impurity creating Yu‑Shiba‑Rusinov (YSR) states affects Josephson tunneling in a tip–impurity–substrate setup. Using a mean-field Anderson impurity coupled to BdG superconducting leads, the authors compute the YSR spectrum, self-consistently determine the local superconducting order parameter, and evaluate the dc Josephson current through the impurity via Green's functions, including multiple tunneling channels. They find that both the current π-shift and the local order-parameter π-shift originate from the YSR state but do not strongly influence each other; notably, the π-shift in the order parameter does not induce a π-shift in the Josephson current, and the self-consistent Δ only modestly shifts the QPT coupling $t_C$ and slightly modifies $I_C$, with the current's spatial profile governed mainly by the YSR state's extent. The results imply that JSTM is not a reliable probe of the impurity-induced local order-parameter π-shift, and experimental interpretation must consider multiple transport channels and the YSR state's spatial spread.

Abstract

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.

Figures (4)

• Figure 1: Schematic setup. The magnetic impurity, responsible for creating the YSR states is depicted in red and is defined by the onsite energy level $\epsilon_i$ and spin-dependent energy splitting level interaction $J$. The impurity is coupled to the STM tip (top) through a hopping term $t_t$ and to the substrate by $t_s$ to the lattice site $x_0$. Additionally, there are other possible transmission channels between the tip and substrate with couplings $t_\textbf{x}$.
• Figure 2: (a) Energy of the unoccupied YSR state as a function of the coupling rate between the substrate and impurity $t_{s}$ for different temperatures $T$ relative to the critical temperature of the superconductor $T_C$. (b) Local order parameter at $\textbf{x}_0$ in the substrate. (c) Critical Josephson current $I_C$. Solid lines indicate self-consistent $\Delta_{\textbf{i}}$, dashed line the constant $\Delta_{\textbf{i}} =\Delta_0$ approximation. Dotted vertical lines indicate the critical coupling $t_s = t_C$ at the QPT.
• Figure 3: Critical Josephson current $I_C$ for tunneling to the substrate (a) and local order parameter $\Delta_{\textbf{i}}$ (b) along the $[11]$-axis intersecting $\textbf{x}_0$. (c) Critical Josephson current for tunneling into a substrate without the impurity present, but with the local order parameter distribution as in (b). The blue (orange) lines represent two different coupling strengths between the substrate and the impurity, close to the QPT in the weak (strong) coupling regime. Solid lines represent self-consistent order parameter solutions, dashed lines the constant order parameter approximation $\Delta_{\textbf{i}} =\Delta_0$ at zero temperature $T=0$ in (a). Note that the solid and dashed lines in (a) are very close and often overlap completely overlap.
• Figure 4: Critical Josephson tunneling current directly to the substrate close to the impurity as a function of the impurity substrate coupling $t_s$ for the same temperatures as in Fig. \ref{['fig:spectrum_delta_Josephsoncurrent']}. (a) Substrate site directly under the impurity $\textbf{x}_0$, (b) nearest neighbor site to $\textbf{x}_0$, and (c) next nearest neighbor site. Solid lines represent self-consistent order parameter solutions, dashed lines the constant order parameter approximation. Horizontal dotted lines indicate the critical coupling $t_C$ at the QPT for the respective temperatures.