Full Counting Statistics of Yu-Shiba-Rusinov Bound States

TL;DR

This work develops a comprehensive full counting statistics (FCS) framework for spin-dependent transport in hybrid superconducting systems hosting Yu-Shiba-Rusinov (YSR) bound states. By combining the Floquet-Keldysh action with mean-field impurity models, it provides analytic and numerical tools to extract all current cumulants, classify tunneling processes, and predict signatures of resonant Andreev reflections, MARs, and direct YSR–YSR tunneling. The approach reproduces recent shot-noise measurements in NS configurations and delivers new predictions, including a universal Fano-factor minimum of 7/32 for direct YSR tunneling and detailed MAR structures in SS junctions. Overall, the FCS method opens a pathway to quantify subgap transport, YSR lifetimes, and spin-dependent effects in atomic-scale superconducting devices, with potential extensions to multiterminal and Majorana-related systems.

Abstract

With the help of scanning tunneling microscopy (STM) it has become possible to address single magnetic impurities on superconducting surfaces and to investigate the peculiar properties of the in-gap states known as Yu-Shiba-Rusinov (YSR) states. However, until very recently YSR states were only investigated with conventional tunneling spectroscopy, missing the crucial information contained in other transport properties such as shot noise. Here, we adapt the concept of full counting statistics (FCS) to provide a very deep insight into the spin-dependent transport in these hybrid systems. We illustrate the power of FCS by analyzing different situations in which YSR states show up including single-impurity junctions, as well as double-impurity systems where one can probe the tunneling between individual YSR states. The FCS concept allows us to unambiguously identify every tunneling process that plays a role in these situations. Moreover, FCS provides all the relevant transport properties, including current, shot noise and all the cumulants of the current distribution. Our approach can reproduce the experimental results recently reported on the shot noise of a single-impurity junction with a normal STM tip. We also predict the signatures of resonant (and non-resonant) multiple Andreev reflections in the shot noise of single-impurity junctions with two superconducting electrodes. In the case of double-impurity junctions we show that the direct tunneling between YSR states is characterized by a strong reduction of the Fano factor that reaches a minimum value of 7/32, a new fundamental result in quantum transport. The FCS approach presented here can be naturally extended to investigate the spin-dependent superconducting transport in a variety of situations, and it is also suitable to analyze multi-terminal superconducting junctions, irradiated contacts, and many other basic situations.

Figures (17)

• Figure 1: Schematic representation of a magnetic impurity coupled to a superconducting substrate and to an STM tip that can be either normal or superconducting. The tunneling rates $\Gamma_{\rm t}$ and $\Gamma_{\rm S}$ describe the strength of the coupling of the impurity to the tip and substrate, respectively. The superconducting gaps of the electrodes are denoted by $\Delta_{\rm t}$ and $\Delta_{\rm S}$.
• Figure 2: Tunneling processes in the case of an impurity coupled to a normal and to a superconducting lead. In these diagrams the left density of states (DOS) corresponds to a normal STM tip and the right one to the impurity coupled to the superconducting substrate. The red lines correspond to electron-like quasiparticles and the blue ones to hole-like. In all cases, we indicate the threshold voltage at which the process starts to contribute to the transport. (a) Single-quasiparticle tunneling in which a quasiparticle tunnels either into the continuum DOS of the superconducting electrode (i) or resonantly into the excited YSR state inside the gap (ii). (b) Standard non-resonant Andreev reflection in which an electron is reflected as a hole. (c) A resonant Andreev reflection in which the electron that is retro-reflected impinges at the energy of a YSR state.
• Figure 3: Differential conductance (a), shot noise (b), and Fano factor (c) as a function of the voltage for the case of a single impurity coupled to a normal tip and SC substrate for different values of the Dynes' parameter $\eta$ of the SC electrode, as indicated in the legend of panel (b). The parameters used are $\Gamma_{\mathrm{S}} = 100\Delta,\ J = 80\Delta,\ U = 60\Delta, \Gamma_{\mathrm{t}} = \Delta$, and $T=0$. With these parameters the junction has a normal-state conductance of $0.026G_0$ and the corresponding YSR energy is $\epsilon_{\rm YSR} = 0.41\Delta$.
• Figure 4: Charge-resolved differential conductance as a function of the voltage for the cases considered in Fig. \ref{['fig-NS1']}. Every panel corresponds to a value of the Dynes' parameter $\eta$ of the SC electrode: (a) $\eta = 0.1\Delta$, (b) $\eta = 0.01\Delta$, (c) $\eta = 0.001\Delta$, and (d) $\eta = 0.0001\Delta$. The red lines correspond to the contribution of single-quasiparticle tunneling ($G_1$) and the blue ones to the contribution of the Andreev reflection ($G_2$).
• Figure 5: Differential conductance (a), shot noise (b), and Fano factor (c) as a function of the voltage for the case of a single impurity coupled to a normal tip and SC substrate and for different values of the tip tunneling rate $\Gamma_{\rm t}$, as indicated in the legend of panel (b). The parameters used are $\Gamma_{\mathrm{S}} = 100\Delta, \ J = 80\Delta,\ U = 60\Delta, \eta = 0.001 \Delta$, and $T=0$.