Tunneling Processes into Localized Subgap States in Superconductors

We combine scanning-tunneling-spectroscopy experiments probing magnetic impurities on a superconducting surface with a theoretical analysis of the tunneling processes between (superconducting) tip and substrate. We show that the current through impurity-induced Shiba bound states is carried by single-electron tunneling at large tip-substrate distances and Andreev reflections at smaller distances. The single-electron current requires relaxation processes, allowing us to extract information on quasiparticle transitions and lifetimes.

Figure 1

$dI/dV$ spectra measured on an isolated Mn adatom on Pb(111) for increasing tunneling strength from top to bottom (recorded with a lock-in modulation amplitude of $15{\mu \mathrm{V}}_{\mathrm{rms}}$ at a frequency of 912 Hz). Spectra are normalized to the “normal-state” conductance measured at 4 meV (i.e., well outside the superconducting gap) in units of $G_0=2e^2/h$, offset for clarity and scaled when indicated for better visibility. The distance to the closest neighboring Mn atom was larger than 5 nm. A spectrum acquired above the clean Pb(111) surface overlays the smallest-conductance trace (top curve) for comparison. The four peaks originating from the deepest Shiba level are marked by dashed lines at $e|V|=\pm (\mathrm{\Delta }\pm {\epsilon }_0)$.

Figure 2

Principal tunneling processes involving a Shiba state (solid line) within the superconducting gap (enclosed by BCS quasiparticle peaks). The chemical potential is represented by a dashed line. (a) Single-electron tunneling from tip to substrate [rate ${\mathrm{\Gamma }}_e(\omega )$] with subsequent relaxation from the Shiba state to the quasiparticle continuum (rate ${\mathrm{\Gamma }}_1$). (b) Andreev process transferring a Cooper pair to the substrate by electron and hole tunneling [with rates ${\mathrm{\Gamma }}_e(\omega )$ and ${\mathrm{\Gamma }}_h(\omega )$, respectively]. The processes in (a) and (b) both contribute near the threshold $eV=\mathrm{\Delta }+{\epsilon }_0$. (c) Single-electron tunneling from substrate to tip [with rate ${\mathrm{\Gamma }}_e(\omega )$] after occupation of the Shiba state by the relaxation of a thermal quasiparticle (with rate ${\mathrm{\Gamma }}_2$), contributing to the thermal peak at $eV=-(\mathrm{\Delta }-{\epsilon }_0)$. The current at the other two thresholds $eV=-(\mathrm{\Delta }+{\epsilon }_0)$ and $eV=\mathrm{\Delta }-{\epsilon }_0$ is carried by analogous hole processes (see the Supplemental Material [28]).

Figure 3

(a) Peak heights ${\alpha }_{\pm }$ and ${\beta }_{\pm }$ of the four resonances associated with the deepest Shiba level (marked by dashed lines in Fig. 1) as a function of normal-state conductance at $T=1.2\mathrm{K}$ (left panel) and $T=4.8\mathrm{K}$ (right panel). The full (dashed) lines are fits to Eqs. (1) and (2) for the main (thermal) peaks. The crossover points between single-electron and Andreev contributions to ${\alpha }_{\pm }$ are indicated by arrows. (b) Experimental $dI/dV$ traces of the Shiba peak near $eV=-(\mathrm{\Delta }+{\epsilon }_0)\simeq -1.57\mathrm{meV}$ at different junction conductances (solid lines). The spectra are normalized to the peak maximum. The dashed lines are theoretical fits of the Shiba peaks excluding multiple Andreev reflections.