Nonequilibrium Transport through a Spinful Quantum Dot with Superconducting Leads

We study the nonlinear cotunneling current through a spinful quantum dot contacted by two superconducting leads. Applying a general nonequilibrium Green function formalism to an effective Kondo model, we study the rich variation in the $IV$ characteristics with varying asymmetry in the tunnel coupling to source and drain electrodes. The current is found to be carried, respectively, by multiple Andreev reflections in the symmetric limit, and by spin-induced Yu-Shiba-Rusinov bound states in the strongly asymmetric limit. The interplay between these two mechanisms leads to qualitatively different $IV$ characteristics in the crossover regime of intermediate symmetry, consistent with recent experimental observations of negative differential conductance and repositioned conductance peaks in subgap cotunneling spectroscopy.

Figure 1

The top figure shows the conductance versus gate ($V_{\mathrm{gate}}$) and bias ($V_{\mathrm{sd}}$) voltage for a carbon nanotube quantum dot coupled to superconducting leads revealing a qualitatively different subgap structure at fillings 3, 7, and 11. For more details on the experimental setup, see Ref. 14. The bottom figure shows calculated $IV$ characteristics through a S/QD/S cotunnel junction obtained within the present approach. For all curves $J_{LL}/\Delta =11.9$ and (top to bottom) $J_{RR}/\Delta =0.05$, 0.8, 3.2, 11.9. The curves are offset for clarity. Top inset: Sketch of the S/QD/S cotunnel junction consisting of a spinful quantum dot tunnel, coupled to the superconducting leads. Bottom inset: Second order self-energy diagram included in the calculation.

Figure 2

$IV$ curves for a S/QD/S cotunnel junction with symmetric couplings, $J_{LL}=J_{RR}=J_{LR}=J$. From top to bottom the curves correspond to $J/\Delta =22.6$, 17.3, 12.7, 8.8, 5.7. All curves are normalized by $G_N$ determined for $J/\Delta =22.6$. Inset: dot spectral function for $J/\Delta =12.7$ with $V=0$ (solid) and $V=\Delta /e$ (dashed).

Figure 3

$IV$ curves for a S/QD/S cotunnel junction with coupling asymmetry $J_{RR}/J_{LL}=0.004$, 0.007, 0.016, 0.062, 1.0 for lines 1–5, respectively. All curves have $G_N=0.013(2e^2/h)$, corresponding to ${\nu }_F{J}_{LL}\approx 0.744$, 0.511, 0.335, 0.170, 0.042. Inset: as in Fig. 2 for the case with $J_{RR}/J_{LL}=0.016$.