Scaling of subgap excitations in a superconductor-semiconductor nanowire quantum dot

A quantum dot coupled to a superconducting contact provides a tunable artificial analog of a magnetic atom in a superconductor, a paradigmatic quantum impurity problem. We realize such a system with an InAs semiconductor nanowire contacted by an Al-based superconducting electrode. We use an additional normal-type contact as a weakly coupled tunnel probe to perform tunneling spectroscopy measurements of the elementary subgap excitations, known as Andreev bound states or Yu-Shiba-Rusinov states. We demonstrate that the energy of these states $\zeta$ scales with the ratio between the Kondo temperature $T_K$ and the superconducting gap $\mathrm{\Delta }$. $\zeta$ vanishes for $T_K/\mathrm{\Delta }\approx 0.6$, denoting a quantum phase transition between the spin singlet and doublet ground states. By further leveraging the gate control over the quantum dot parameters, we determine the singlet-doublet phase boundary in the stability diagram of the system. Our experimental results show remarkable quantitative agreement with numerical renormalization group calculations.

Figure 1

(a) Schematics of the studied dual-gate N-QD-S devices. The nanowires are contacted by N and S leads comprising Ti/Au and Ti/Al bilayers, respectively. A Ti/Au thin strip covered by ${\mathrm{HfO}}_2$ dielectrics acts as a local plunger gate (pg), whereas the degenerately doped substrate is employed as a global back gate (bg). (b) False color scanning electron micrograph of a typical device. (c) Qualitative phase diagram of the QD-S system in the wide-gap limit ($\mathrm{\Delta }\to \infty$). The horizontal (vertical) line underscores QPTs between the singlet and doublet states (circles) that occur upon varying the QD level position (QD-S coupling). (d) Schematics of the Andreev level spectroscopy transport cycle. Current is measured across the N-QD-S device when the chemical potential of the tunnel probe (${\mu }_N$) is aligned with an Andreev level at energies $\pm \left|\zeta \right|$. Transport occurs via Andreev reflection, whereby an injected electron (hole) is reflected back to N as a hole (electron), forming (breaking) a Cooper pair in S.

Figure 2

(a) Fitting of normal-state linear conductance at $V_{\text{bg}}=4.5$ V (black dots) with NRG model (red line). A reliable estimation of device parameters results from the fit (see text and Supplemental Material [34] for details). (b) Effect of $V_{\text{bg}}$ on the QD-S tunnel coupling, and (c) coupling asymmetry. The plots demonstrate a continuous back gate-induced tuning of ${\mathrm{\Gamma }}_S/U$. N behaves as a tunnel probe irrespective of $V_{\text{bg}}$, even if ${\mathrm{\Gamma }}_S/{\mathrm{\Gamma }}_N$ is also affected by the back gating.

Figure 3

(a) Series of $dI/dV$ vs $(V,V_{\text{pg}})$ plots depicting the impact of back gating on the energy of Andreev levels. $V_{\text{bg}}$ increases from $-4.5$ to 39 V. The horizontal lines in the top left panel highlight the positions of the superconducting gap, $eV=\mathrm{\Delta }$, and the Fermi level, $eV=0$. $|S\rangle$ and $|D\rangle$ refer to the singlet and doublet ground states, respectively. The doublet ground state region is gradually suppressed for increasing $V_{\text{bg}}$, suggesting a QPT induced by the electrical tuning of ${\mathrm{\Gamma }}_S$. Overlaid to the plots are the density of states spectra calculated by NRG (dashed lines). The DOS spectra, also shown in (b), were calculated using $U, {\mathrm{\Gamma }}_S$, and $\mathrm{\Delta }$ extracted from the experimental data.

Figure 4

Experimental phase diagram of the QD-S system. The parameter space is composed of the QD-S tunnel coupling ${\mathrm{\Gamma }}_S$ and the QD level position ${\epsilon }_0$ scaled by $U$. The open dots represent the phase boundaries extracted from the experimental data. The three data points located around ${\epsilon }_0/U=0$ were estimated from the $\zeta =0$ intercept in $\zeta ({\mathrm{\Gamma }}_S/U)$ traces. The error bars are the associated errors in these fits. The remaining points were obtained directly from the Andreev level tunnel spectra. The solid lines represent phase boundaries simulated by NRG with model parameters extracted from the normal-state conductance. The colored limits of the singlet and doublet states are guides to the eye.

Figure 5

$T_K/\mathrm{\Delta }$ scaling of the Andreev level energy $\zeta /\mathrm{\Delta }$. The QPT occurs at $\zeta =0$. Two data sets are shown: one corresponding to the device discussed in the main text (device 1, closed dots) and another of the device shown in Ref. [34] (device 2, open dots). The solid line is the scaling curve calculated by NRG for $B_{\perp }=0$, whereas the dashed lines take into account the Zeeman-related broadening of the Kondo resonances ($B_{\perp }=30$ mT, $g$ factors $\sim 3.5$ and $\sim 5.75$, as measured in similar devices [33, 40]).