Kondo Universal Scaling for a Quantum Dot Coupled to Superconducting Leads

We study competition between the Kondo effect and superconductivity in a single self-assembled InAs quantum dot contacted with Al lateral electrodes. Because of Kondo enhancement of Andreev reflections, the zero-bias anomaly develops side peaks, separated by the superconducting gap energy $\Delta$. For ten valleys of different Kondo temperature $T_K$ we tune the gap $\Delta$ with an external magnetic field. We find that the zero-bias conductance in each case collapses onto a single curve with $\Delta /k_B{T}_K$ as the only relevant energy scale, providing experimental evidence for universal scaling in this system.

Figure 1

(a) Differential conductance $dI/d{V}_{\mathrm{SD}}$ as a function of $V_G$ and $V_{\mathrm{SD}}$, with the leads driven normal by a field $B=100\mathrm{mT}$. $T=35\mathrm{mK}$. (b) $dI/d{V}_{\mathrm{SD}}$ at $V_{\mathrm{SD}}=0\mathrm{mV}$ in units of $e^2/h$. We show KV5 to KV7 with a $T_K$ of 1.04, 1.65, and 1.18 K, respectively. (c) Scanning electron micrograph of a similar device as used in these studies, with 100 nm scale bar. An InAs dot bridges the source and drain electrodes. (d) Temperature dependence of the conductance at KV5, with a best fit to scaling curve 1.

Figure 2

(a) $dI/d{V}_{\mathrm{SD}}$ as a function of $V_G$ and $V_{\mathrm{SD}}$, with the leads in the SC state ($B=0\mathrm{T}$). (b) Scan of the gap structure in the CB regime, as indicated by the square in upper figure. Conductance is suppressed for $V_{\mathrm{SD}}<2\Delta$. (c) Color-scale plot of $dI/d{V}_{\mathrm{SD}}$, showing the field dependence of the Kondo resonance (KV2, $T_K=1.23\mathrm{K}$). Horizontal lines denote the SC-$N$ transition in the leads. (d) The Kondo resonance out of equilibrium for three different Kondo valleys (KV 10,2,4 from upper to lower) with the leads in the SC state. The curves are normalized by their $N$-state conductance $G_0^N$. Offset for clarity. (e) Schematic depiction of the Kondo enhanced Andreev Reflection, with a split Kondo resonance in the dot DOS. (f) Zero-bias Kondo conductance (KV2) as a function of magnetic field, with $N$-state leads.

Figure 3

Semilog plot of $G^{\mathrm{SC}}/G_0^N$ for KV1–10 at various fields. Plotted against $\Delta /k_B{T}_K$, the data collapse on a single curve. The field dependent value of $\Delta$ was 135, 140, 120, 100, and $60\mu \mathrm{eV}$ for 0, 20, 40, 60, and 80 mT, respectively. The measurement uncertainty is given by the bars. Inset shows the same data plotted against $T_K$ only.