Tunneling spectroscopy of a single quantum dot coupled to a superconductor: From Kondo ridge to Andreev bound states

We performed tunneling spectroscopy of a carbon nanotube quantum dot (QD) coupled to a metallic reservoir either in the normal or in the superconducting state. We explore how the Kondo resonance, observed when the QD's occupancy is odd and the reservoir is normal, evolves towards Andreev bound states (ABS) in the superconducting state. Within this regime, the ABS spectrum observed is consistent with a quantum phase transition from a singlet to a degenerate magnetic doublet ground state, in quantitative agreement with a single-level Anderson model with superconducting leads.

Figure 1

(a) Color-enhanced scanning electron micrograph (scale bar: 500 nm) of a device fabricated for the tunneling spectroscopy of a carbon nanotube (CNT). The substrate consists of highly doped silicon serving as a back gate (BG), shown here in cyan, with a $1\mu \mathrm{m}$ surface oxide layer. A second gate electrode [side gate (SG)] is used to electrostatically influence the left part the CNT. A grounded aluminum fork (green) is well connected to the CNT. The measurement of $dI/dV$ through the tunnel probe (red) gives access to the DOS in the CNT. (b) Schematic representation of the system: the presence of the tunnel probe electrode likely acts as a scatterer splitting the CNT into two QDs.

Figure 2

(a) Density plot of the CNT DOS measured, in the normal state ($B\sim 0.05\mathrm{T}$), at zero probe voltage as a function of SG (vertical axis) and BG (bottom axis). Superimposed dotted thin lines are guides to the eyes showing the electronic levels of two isolated QDs. Orange (cyan) lines correspond to resonances of the left (right) QD. Between these dotted lines, each diamond is indexed by a pair of integer numbers, which denote the equilibrium charge on left and right dot. Here the labeling is such that $\ell$ and $r$ are even numbers. Avoided crossings reveal the interdot coupling. (b) Schematic of the experiment testing the DOS of a single QD (${\mathrm{QD}}_{\mathrm{L}}$): by following a trajectory in the (SG, BG) plane that keeps ${\mathrm{QD}}_{\mathrm{R}}$ in a “Coulomb blockaded” even state [dashed yellow line 1 in (a)], the tunnel current is forced to solely go through ${\mathrm{QD}}_{\mathrm{L}}$ giving access to its DOS. Dashed yellow line 1 indicates the trajectory followed to explore the DOS of a single QD in the normal and in the superconducting state shown in Figs. 3a and 3b respectively. A, B, and C correspond to the odd valleys analyzed.

Figure 3

(a) Density plot of CNT's tunneling DOS in the normal state as a function of voltage probe $V$ (vertical axis) and BG (horizontal axis) following line 1 in the plane (BG, SG) depicted in Fig. 2a, after correcting for the gating effect of the probe junction. Odd valleys A, B, C show the development of a Kondo ridge. From the size of the diamonds (delimited by dashed red thin lines) the charging energy can be estimated. (b) Same as (a) in the superconducting state. Around the Fermi level, the superconducting gap ($\Delta =150\pm 5\mu \text{eV}$) develops and ABS are observed. By sweeping gate voltages along line 1, ABS form loops of different sizes each time an odd valleys is crossed.

Figure 4

Schematic of the excitation spectrum of the S-QD for (a) $\Gamma /\Delta \lesssim 1$ and (d) $\Gamma /\Delta \gg 1$ : Andreev transitions correspond to the excitation between the GS and excited state by addition of an electron or a hole [respectively blue and green dashed lines in (b) and (e)] at an energy $\left|E_{\mathrm{ABS}}\right|$ relative to the GS. The singlet $(S=0)$ and spin degenerate doublet $(S=\frac{1}{2})$ regions are indicated (red and double blue lines respectively). When $E_{\mathrm{ABS}}=0$ a quantum phase transition takes place, accompanied by an abrupt change in the spectral weights of ABS [see (c)]. (c) and (f) Color plot of the DOS spectrum for $U/\Delta =13.3$ and $\Gamma /\Delta =0.9,5$ [(c) and (f), respectively] as a function of the energy (left axis) and the chemical potential of the QD (bottom axis). The sharp resonances observed within $\Delta$ are artificially broadened to make them visible.

Figure 5

Top: Color plot of CNT's tunneling DOS measured in the superconducting state as a function of energy (left axis) and BG voltage (top axis) in odd valley A, B, and C. By using the proportionality factor that converts BG into energy, BG can be expressed as the dot level position ${\varepsilon }_0$ (bottom axis) used in the model described in the text. Green and blue dashed lines are the Andreev transitions obtained from the NRG calculation shown at the bottom: DOS spectrum of the QD calculated by a NRG algorithm using $U$and $\Delta$ obtained from the experiments ($U/\Delta =13.3,13.6,12.7$for A, B, and C, respectively) and $\Gamma$ as a sole fitting parameter ($\Gamma /\Delta =1.6,0.86,1.7$ for A, B, and C, respectively). Andreev transitions were artificially broadened to make them visible.

Figure 6

(a) Color plot of tunneling DOS measured in the normal state as a function of energy (left axis) and BG voltage (bottom axis) in odd valley A. (b) Tunneling DOS at $\varepsilon /\Delta =-0.47$ (black), 0 (red), and $0.47$ (green) as a function of the level position ${\varepsilon }_0$. Solid lines: NRG calculation using the parameter $\Gamma /\Delta =0.86$ obtained when fitting the superconducting state. (c) Tunneling DOS at the electron-hole symmetry point (${\varepsilon }_0/U=-0.5$) as a function of the energy. Solid line: NRG calculation using the parameter $\Gamma /\Delta =0.86$ obtained when fitting the superconducting state.