Transport Measurement of Andreev Bound States in a Kondo-Correlated Quantum Dot

We report nonequilibrium transport measurements of gate-tunable Andreev bound states in a carbon nanotube quantum dot coupled to two superconducting leads. In particular, we observe clear features of two types of Kondo ridges, which can be understood in terms of the interplay between the Kondo effect and superconductivity. In the first type (type I), the coupling is strong and the Kondo effect is dominant. Levels of the Andreev bound states display anticrossing in the middle of the ridge. On the other hand, crossing of the two Andreev bound states is shown in the second type (type II) together with the 0-$\pi$ transition of the Josephson junction. Our scenario is well understood in terms of only a single dimensionless parameter, $k_B{T}_K^{\min }/\Delta$, where $T_K^{\min }$ and $\Delta$ are the minimum Kondo temperature of a ridge and the superconducting order parameter, respectively. Our observation is consistent with measurements of the critical current, and is supported by numerical renormalization group calculations.

Figure 1

(a) Schematic of the measurement configuration. Highly doped Si wafer and ${\mathrm{SiO}}_2$ layer are used as a back gate and an insulating barrier, respectively. Channel length of the CNT is designed to be 300 nm. Diameter of the CNT $\sim 1.5\mathrm{nm}$ is measured by AFM. Differential conductance $dI/d{V}_{sd}$ plot as a function of $V_{sd}$ and $V_g$ is displayed (b) for normal (with magnetic field $B$ $\sim 1\mathrm{kG}$), and (c) for superconducting states at zero field, respectively. The letters $\mathrm{e}$ and $\mathrm{o}$ denote the even and odd number states, respectively. Bias voltages corresponding to $\Delta /e$ and $2\Delta /e$ (with $\Delta \sim 140\mu \mathrm{eV}$) are indicated by arrows in (c). Red curves in (b) correspond to $dI/d{V}_{sd}$ vs $V_{sd}$ in the middle of the Kondo ridges $A\mathrm{\text{−}}E$.

Figure 2

(a) Differential conductance plot for the two types of the Kondo ridge (magnified view of Fig. 1c for ridges $D$ and $E$). Red color represents the region of the negative differential conductance. Blue dashed lines ($V_{\mathrm{ABS}}$) are associated with the ABS-assisted transport (see the text). (b) Schematic diagram of the major transport process forming the Andreev bound states at finite bias. This mechanism is expected to be pronounced in highly asymmetric barriers, where the Andreev bound states are pinned to the lead with a stronger coupling. (c) The ABS level position ${\epsilon }_a$ (in units of $\Delta$) obtained from Eq. (1) is plotted as a function of $V_g$. (d) $k_B{T}_K/\Delta$ vs $V_g$ for ridges $D$ (type-I) and $E$ (type-II), respectively. Filled circles and solid lines correspond to the experimental data and the theoretical fit with the formula $k_B{T}_K=(U\Gamma /2{)}^{1/2}\exp [-\pi [|4{\epsilon }^2-U^2|]/8U\Gamma ]$, and $U/\Delta =21.4$. The dashed line at $k_B{T}_K/\Delta =0.788$ is the theoretically expected ‘0-$\pi$ transition’ line [13]. (e) $I_c$ vs $V_g$ for ridges $D$ and $E$, respectively. Solid lines provide a guide for the eye. $I_c\mathrm{\text{−}}{V}_g$ curves for ridge $E$ show a sharp drop around the 0-$\pi$ transition point denoted by the two vertical dashed-dotted lines.

Figure 3

NRG results for the two types of Kondo ridge: (a) ABS level position ${\epsilon }_a$ (in units of $\Delta$) and (b) the Josephson critical current $I_c$ as a function of the QD energy level ${\epsilon }_d$. Both in (a) and (b), coupling constants $\Gamma /\Delta =4.9$ (left panels) and $\Gamma /\Delta =3.0$ (right panels) are used, which correspond to the experimental values of ridges $D$ (type-I) and $E$ (type-II), respectively. Experimentally extracted value of $U/\Delta =21.4$ is used in both cases. $k_B{T}_K^{\min }/\Delta$ are 1.5 and 0.5 for ridges $D$ and $E$, respectively.

Figure 4

$I\mathrm{\text{−}}{V}_{sd}$ curves measured in current bias (red line), and voltage bias (blue line) modes, respectively, in the middle of Kondo ridge $D$ ($V_g=-8.255\mathrm{V}$). Two switching currents, $I_c$ and $I_a$, appear at zero voltage and at $V\approx 0.2\mathrm{mV}$, respectively. The slope of the supercurrent branch is mainly due to the resistance of low pass filters ($\sim 4\mathrm{k}\Omega$). Upper-left inset: differential conductance $G=dI/d{V}_{sd}$ vs $V_{sd}$ in voltage bias mode. Lower-right inset: magnified view of the $I\mathrm{\text{−}}V$ curve around zero bias.