Tuning Yu-Shiba-Rusinov states in a quantum dot

We present transport spectroscopy of subgap states in a bottom gated InAs nanowire coupled to a normal lead and a superconducting aluminium lead. The device shows clearly resolved subgap states which we can track as the coupling parameters of the system are tuned and as the gap is closed by means of a magnetic field. We systematically extract system parameters by using numerical renormalization-group theory fits as a level of the quantum dot is tuned through a quantum phase transition electrostatically and magnetically. We also give an intuitive description of subgap excitations.

Figure 1

Excitations between states in a quantum dot/superconductor system. (a) The states under consideration with arrows in circles representing electrons in the dot and arrows in open rectangles representing Bogoliubons. The dashed shape in the diagram for $\left|B\right.〉$ depicts a singlet correlation. The arrows annotated with +$e^{-}$ show dominant sequential tunneling processes for transport going from N to S at the level position marked $x$ in (b). (b) A schematic diagram showing subgap excitations in the system, with and without anticrossings induced by the coupling between the dot and superconductor. (c) NRG simulations of the lowest doublet to singlet transitions for different values of the coupling density of states, ${\mathrm{\Gamma }}_S$. For all curves in (c), we have $U=5\mathrm{\Delta }$. The curve going across the traces mark an excitation energy of zero. Traces have been offset for clarity as indicated on the right-hand axis.

Figure 2

(a) Artist impression of a $0.6\text{−}\mu \mathrm{m}\times 0.4\text{−}\mu \mathrm{m}$ cutout of the device, to scale. The model shows the surface of the ${\mathrm{SiO}}_2$ substrate, bottom gates, insulating ${\mathrm{HfO}}_2$ (shown in green), InAs nanowire, gold contact, and aluminium contact. Details are in Appendix pp1. We assign names to two of the gates as shown. (b) SEM micrograph of a lithographically similar device. Note that only the part of the device between the gold electrode, N, and the aluminium electrode, S, is used.

Figure 3

Differential conductance at 35 mK with and without an externally applied $150\text{−}\mathrm{mT}$ in-plane field. The field drives the aluminium contact normal in (b). Regions that show heavily tunnel-broadened Coulomb diamonds also show subgap excitations far inside the gap when the Al contact is superconducting.

Figure 4

Conductance at zero bias as a function of the tuning gate and the raw plunger gate potential, with and without a magnetic field driving the aluminium contact normal. The lines in these plots shows the cuts done by $a_0$–$a_6$ of Fig. 5. For certain configurations of the tuning gate, the ground state remains a singlet as a dot level is brought past the Fermi level with the plunger gate, and this is evident at the *.

Figure 5

Bias spectroscopy of the subgap states for different gate configurations. In every plot, the $y$ axis is the potential of the superconducting lead relative to the normal lead and ranges from $-0.2$ to $0.2\mathrm{mV}$. Note the different $dI/dV$ scales. In the plot $a_n$, the tuning gate is set to $(300+40n)\mathrm{mV}$. We adjust for cross capacitance as described in the text. Example NRG fits are overlaid plots $a_0$ and $a_6$. The red curves are made using method 1 and the black curves are made using method 2; see text for details.

Figure 6

Coupling strengths ${\mathrm{\Gamma }}_S$ and ${\mathrm{\Gamma }}_N$ of each level transition shown in Fig. 5. We extract ${\mathrm{\Gamma }}_S$ from our NRG fits, and use the conductance on resonance in the $B=150\text{−}\mathrm{mT}$ data of Fig. 5 to find ${\mathrm{\Gamma }}_N$. There is one trace in the plot for each level transition, and the roman numerals refer back to the labels in Fig. 5.

Figure 7

Evolution of the subgap states as the gap closes upon the application of an external field in plane with the substrate perpendicular to the nanowire. The tuning gate is set to $V_T=0.34\mathrm{V}$ in all these plots, and the bias range is $\pm 125\mu \mathrm{V}$. The roman numerals on odd charge states refer back to the labels in Fig. 5.

Figure 8

Magnetic field dependence of subgap transport in the center of a level transition for different values of the tuning gate potential. In the data sets $b_0$ through $b_5$ the field was applied in the plane of the sample perpendicular to the nanowire, and in the data sets $c_0$ through $c_5$ the field was held at $60\mathrm{mT}$ and rotated in the plane of the sample with a direction of 0 radians being perpendicular to the wire. The data has been corrected for a drifting zero bias across the device as detailed in the Supplemental Material. The plots also show data from an NRG simulation, specifically the allowed excitations from the ground states of the system (red lines). The black lines show a phenomenological model of the gap closing used as an input to the simulation. Additional input to the simulation includes the ${\mathrm{\Gamma }}_S$ values from Fig. 6 and a $g$ factor found by fitting the plot $c_1$ at 0 radians.

Figure 9

Electric potentials applied to each relevant bottom gate. The gates are numbered starting from the gold side of the device (left side in Fig. 2). Gate no. 2 we call the tuning gate and gate no. 5, which is more strongly coupled to the energy levels of the dot, we call the plunger gate.

Figure 10

This figure compares output from our NRG program to Fig. 1 in Ref. [39].