Magnetic-field-induced transition in a quantum dot coupled to a superconductor

The magnetic moment of a quantum dot can be screened by its coupling to a superconducting reservoir, depending on the hierarchy of the superconducting gap and the relevant Kondo scale. This screening-unscreening transition can be driven by electrostatic gating, tunnel coupling, and, as we demonstrate here, a magnetic field. We perform high-resolution spectroscopy of subgap excitations near the screening-unscreening transition of asymmetric superconductor-quantum dot-superconductor (S–QD–S) junctions formed by the electromigration technique. Our measurements reveal a re-entrant phase boundary determined by the competition between Zeeman energy and gap reduction with magnetic field. We further track the evolution of the phase transition with increasing temperature, which is also evinced by thermal replicas of subgap states.

Figure 1

(a) Scanning electron micrograph of a nanoparticle-covered Al constriction after electromigration. (b) Normal state $dI/dV$ differential conductance map of device A at a base temperature of $T=100\mathrm{mK}$, for a magnetic field $B=60\mathrm{mT}$ suppressing superconductivity. (c) Schematics of the S–QD–S device, introducing the three capacitances and two tunnel couplings at play. (d) Experimental gate dependence of the Kondo energy scale $T_K$, determined from the temperature dependence of the linear conductance (red squares), as well as by a rescaling with a dividing factor 2.9 of the FWHM of the low-bias conductance peak (blue open diamonds).

Figure 2

(a) Differential conductance of device B measured with a lock-in ac oscillation of $2\mu \mathrm{V}$ in the superconducting state at fixed $V_G=0.5\mathrm{V}$, displaying very-sharply-resolved YSR resonances, with a FWHM = $9\mu \mathrm{V}$ (see inset). (b) Gate-dependent differential conductance of device A in the superconducting state at a base temperature of $T=80\mathrm{mK}$, revealing the dispersion of the YSR states. The minimal spacing of the two resonances, given by a voltage span $2{\mathrm{\Delta }}_{\mathrm{probe}}/e$ with ${\mathrm{\Delta }}_{\mathrm{probe}}=205\mu \mathrm{eV}$, is associated with the quantum ground-state transition, occurring at $V_G=0.71\mathrm{V}$. (c) Extracted bound state energy $E_B$ versus gate $V_G$ and dimensionless theoretical Kondo temperature $T_K^{\mathrm{th}}/\mathrm{\Delta }$, in comparison to theoretical predictions (lines). The dashed vertical line indicates the ground-state transition, found at $T_K^{\mathrm{th}}/\mathrm{\Delta }\approx 0.26$. (d) Evolution of the conductance peak intensities with $V_G$, showing a kink at the transition, consistent with a sharp unscreening transition.

Figure 3

(a) Sketch of level structure of YSR states at zero magnetic field (continuous lines) and finite (dashed lines) magnetic field, in vicinity of the ground-state transition. (b) Zoom on the zero-energy crossings of the bound-state dispersion for two magnetic-field values, $B=9$ and 15 mT (symbols). The solid lines are a spline interpolation of the data, from the crossing of which the value of the critical gate voltage $V_G^c$ is determined (arrows). (c) Phase diagram of the screening transition in ($V_G, B$) space, showing a re-entrant phase boundary. (d) Spectroscopic analysis of the YSR states as a function of bias and at a fixed gate voltage of $V_G=0.70\mathrm{V}$, close to the zero-field critical point at $V_G^c=0.71\mathrm{V}$. By increasing the magnetic field, a sharp kink is seen indicating the re-entrant phase boundary at $B=12\mathrm{mT}$.

Figure 4

(a) Differential conductance of device A in the superconducting state at the higher temperature $T=940\mathrm{mK}$, displaying the YSR resonances and their thermal replicas (black arrows). (b) Gate dependence of the YSR states at 940 mK, as found from the main conductance resonances (similar data are found from the thermal replicas). Owing to the temperature-driven reduction of the superconducting gap $\mathrm{\Delta }$, the ground-state transition has moved to the larger critical gate voltage $V_G^c\approx 0.74\mathrm{V}$ at the largest measured temperature.