Robustness of Yu-Shiba-Rusinov resonances in the presence of a complex superconducting order parameter

Robust quantum systems rely on having a protective environment with minimized relaxation channels. Superconducting gaps play an important role in the design of such environments. The interaction of localized single spins with a conventional superconductor generally leads to intrinsically extremely narrow Yu-Shiba-Rusinov (YSR) resonances protected inside the superconducting gap. However, this may not apply to superconductors with more complex, energy-dependent order parameters. Exploiting the Fe-doped two-band superconductor ${\mathrm{NbSe}}_2$, we show that due to the nontrivial relation between its complex-valued and energy-dependent order parameters, YSR states are no longer restricted to be inside the gap. They can appear outside the gap (i.e., inside the coherence peaks), where they can also acquire a substantial intrinsic lifetime broadening. T-matrix scattering calculations show excellent agreement with the experimental data and relate the intrinsic YSR state broadening to the imaginary part of the host's order parameters. Our results suggest that nonthermal relaxation mechanisms contribute to the finite lifetime of the YSR states, even within the superconducting gap, making them less protected against residual interactions than previously assumed. YSR states may serve as valuable probes for nontrivial order parameters promoting a judicious selection of protective superconductors.

Figure 1

Topography and differential conductance spectra of Fe in ${\mathrm{NbSe}}_2$: Topographies of the two most common Fe impurities having (a) a triangular shape ($\mathrm{\Delta }$) and (b) a $\mathsf{W}$ shape (W). (c) The triangular defect typically has a smaller exchange coupling, so that the YSR states appear within the coherence peaks. (d) The $\mathsf{W}$ defect typically has a higher exchange coupling, and the YSR states commonly appear inside the gap. An unperturbed reference spectrum with no Fe impurity in the vicinity is shown in red. The current set point for the topography was 20 pA at a bias voltage of 100 mV; the set point for the spectra was 200 pA at 4 mV.

Figure 2

Extracting the order parameters of Fe-doped ${\mathrm{NbSe}}_2$: (a) Fit of the interband-impurity model to an unperturbed conductance spectrum. The extracted fit parameters provide the input values for the subsequent analysis. (b) Calculated total density of states of the superconducting substrate from the extracted fit parameters (sum). The weighted densities of states for band 1 (b1) and band 2 (b2) are shown as dashed lines. Note that the gap edges of both bands are at the same energy. (c) Resulting order parameter ${\mathrm{\Delta }}_1\left(\omega \right)$ of the first band. (d) Resulting order parameter ${\mathrm{\Delta }}_2\left(\omega \right)$ of the second band. The blue shaded region indicates the gap.

Figure 3

Calculated YSR states: The solid lines correspond to the YSR spectra coupling to band 1 (YSR 1) and band 2 (YSR 2), while the dashed lines represent the unperturbed spectra for the first (bare 1) and second (bare 2) bands, respectively. The ratio between the spectra of the two bands corresponds to the typical ratio $\eta$ observed in the experiment. A small value for the Coulomb interaction $U_1^{\prime }=0.05$ and $U_2^{\prime }=0.01=U_1^{\prime }{n}_2/n_1$ was chosen to make the spectrum asymmetric. (a) YSR spectrum for weak exchange coupling. The YSR state is broad and within the region of the coherence peaks. (b) Stronger coupling than in (a), but still before the zero-bias crossing. The YSR state resides within the gap and has become extremely sharp. (c) and (d) YSR spectra vs exchange coupling in band 1 and band 2, respectively. The color scaling is adapted to show the weaker features; it is nonlinear for values larger than 3.5. The sharpening of the peak as it enters the gap is clearly visible. In (d) the YSR peak develops much slower due to the reduced density of states in band 2. For all panels, $J_1^{\prime }/J_2^{\prime }=n_1/n_2$.

Figure 4

Evolution of the YSR peak width: (a) YSR peak width vs peak location. The width becomes extremely narrow in the region of the superconducting gap. The color coding represents the strength of the exchange coupling. The red line is twice the imaginary part of the order parameter ${\mathrm{\Delta }}_1\left(\omega \right)$ of the first band. We see a clear correlation between the peak width and the imaginary part. (b) Differential conductance calculated from the YSR spectral function and the density of states of the superconducting V tip as well as finite energy resolution. The sharpening of the YSR state when entering the gap is clearly visible.

Figure 5

Comparison of experiment with theory: Peak positions and widths indicated for YSR states (a) outside the gap and (b) inside the gap, where an unperturbed reference spectrum was subtracted. (c) and (d) The corresponding calculated spectra. (e) Experimental YSR peak widths vs peak position in comparison with the extracted YSR peak widths from the calculated differential conductance spectra. The $\mathsf{W}$-shaped defect and the triangular defect ($\mathrm{\Delta }$) are color-coded in blue and red, respectively. The blue shaded region indicates the sample gap.