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.

The junction between a superconductor and a quantum dot displays discrete subgap energy levels called Andreev bound states (ABSs) or, more specifically, Yu-Shiba-Rusinov (YSR) states when the highest occupied level hosts a single spin [1][2][3]. When the antiferromagnetic exchange interaction between this spin and the leads prevails over the superconducting gap , the localized spin is effectively Kondo screened, and a quasiparticle is bound at the interface. In contrast, if the exchange coupling is weaker, the superconducting condensate is marginally perturbed and the spin remains unscreened. The change between these two distinct ground states occurs via a sharp level crossing, which constitutes a simple realization of a quantum phase transition. In recent years, detailed investigations of this screening-unscreening transition have been performed, using as a control knob the variation of the level depth [4][5][6][7][8][9][10][11], the superconducting phase difference [12], or the tunnel coupling, which effectively modifies the exchange coupling strength [13][14][15][16]. The variation of an external magnetic field may provide an additional parameter with twofold consequences: increasing the magnetic field suppresses superconducting correlations (hence favoring a screened state), while the Zeeman effect enhances polarization towards one of the magnetic YSR states [9,10,17,18]. Thus, the two effects associated with a magnetic field lead to a shifting of level crossing in opposite directions.
Experimentally, it is challenging to explore the effect of the magnetic field on the screening-unscreening transition, precisely for the reason that superconductivity is usually quenched before the quantum phase transition can be accessed. Hence, the quantum dot systems needs to be tuned close to the critical point using another control parameter, here the back-gate voltage allowed in our transistor geometry. Furthermore, the detection of tiny level shifts between subgap states requires exquisite energy resolution in the μV range, which can be achieved only with superconducting leads.
In this Rapid Communication, we report on the observation of the magnetic field-controlled screened-unscreened ground-state transition of a quantum dot strongly coupled to one superconducting lead. An asymmetrically coupled superconductor-quantum dot-superconductor (S-QD-S) device combines the gate tunability of single-electron transistors with the unprecedented spectroscopic resolution of the subgap states. Monitoring the dispersion of the subgap states allows tracking the transition between the screened and unscreened spin ground states of the quantum dot as a function of the bare level of the dot, temperature, and, most importantly, magnetic field. We use the Anderson impurity as the main framework for the modelization of our data. A general phase diagram is drawn, which demonstrates a striking re-entrant behavior of the phase boundary, due to the previously mentioned competition between Zeeman splitting and superconducting gap closing. In addition, thermal replicas of YSR states are found to emerge at finite temperature, providing an alternative yet consistent picture of the subgap spectrum.
The device fabrication process relies on controlled electromigration of an on-chip all-metallic aluminum device presenting a constriction [19]. This technique produces nanometer sized gaps and was successfully applied for connecting single molecules [20][21][22][23]. Electrostatic gate control is provided through a local metallic back gate isolated by a 18-nm-thick ZrO 2 dielectric layer. Using aluminum as the constriction material, gated S-QD-S devices can thereby be formed [24][25][26]. Our quantum dots are colloidal gold nanoparticles of 5 nm diameter. Electromigration is performed at 4.2 K in cryogenic vacuum in a dilution cryostat. A scanning electron micrograph of an Al constriction after electromigration (without nanoparticles, for better visibility) is shown in Fig. 1(a). Samples showing stable gate-dependent conductance features are further investigated at temperatures down to T = 80 mK. The differential conductance G(V, V G ) = dI/dV is measured by using the lock-in technique, as a function of bias V and gate voltage V G . We show here data mostly from one sample, labeled A. Data from a second and similar sample (B) can be found in the Supplemental Material [27].
The normal-state differential conductance map (obtained at a magnetic field of 60 mT) is shown in Fig. 1(b), around the only experimentally accessible degeneracy point at V 0 G ≈ 0.40 V. The Kondo resonance is unaffected by magnetic up to fields of 120 mT (see Supplemental Material, Fig. S4). To the left, the linear conductance is suppressed, owing to a Coulomb-blockaded state with an even electron occupation number N. As the gate voltage V G is increased, a zero-bias resonance indicates the onset of Kondo correlations associated with the spin-1/2 degeneracy of the oddly occupied N + 1 electron state. The electrical model of the quantum dot junction is displayed in Fig. 1(c). The tunnel couplings to both leads are strongly asymmetric ( γ ), as evidenced by a nonunitary linear conductance limit, G(T → 0)/(2e 2 /h) = 4 γ /( + γ ) 2 ≈ 0.013 (see Supplemental Material [27]). This implies notably that the Kondo resonance builds between the QD and the drain electrode at experimentally accessible temperatures, the source contact acting as a tunnel probe, as is usually the case for the tip in a scanning tunneling microscopy experiment.
The following values of hybridization = 1.4 meV (implying γ = 5 μeV) and Coulomb repulsion U = 12.6 meV in the quantum dot are found from fits of the gate-dependent zero-bias conductance to numerical renormalization group (NRG) calculations (see Supplemental Material [27]). The ratio U/ 9 shows that the quantum dot is in the strongly correlated regime, with some deviations from Kondo scaling.
The full width at half maximum (FWHM) of the conduction resonance at the Fermi level, as shown in Fig. 1(b), is often taken as an approximate measure of the Kondo temperature T K , that we note T FW K (here and later we set k B = 1, identifying temperature and energy scales). Another direct and precise determination of T K is achieved considering the measured temperature dependence of the linear conductance G(T ). The latter can be fit by NRG calculations, or for a lower computational cost, by an empirical expression [42,43], leading to the gate-dependent Kondo temperature denoted T G K shown in Fig. 1(d) (see also Supplemental Material [27]). We find that these estimates agree closely within a scaling factor, such that T G K = T FW K /2.9. Therefore, it is seen that the quantum dot junction behaves like a single spin-1/2 Kondo impurity, with a gate-tunable T K that can be brought to the same order of magnitude as the superconducting order parameter of the leads, leading to a standard gate control of the screening transition [44,45].
We now turn to the study of the S-QD-S transistor at zero magnetic field. In presence of superconductivity in both leads, a transport gap of total width 2( + probe ) ≈ 900 μeV opens in the transport map, see Fig. 2(b). The Kondo peak is suppressed and two sharp symmetric resonances appear at a certain biasing voltage V so that 0 < |eV | − probe < . We take care to differentiate the gap ≈ 245 μeV of the strongly coupled lead, which governs the physical effects at play, from the gap probe ≈ 205 μeV of the weakly coupled electrode, which offsets essentially the conductance onset thresholds by ± probe . Thermal excitations at 940 mK, which will be discussed further below, provide unambiguous evidence of the probe's gap size [see Fig. 4(a)].
The presence of the YSR states is reflected by extremely sharp subgap resonances [ Fig. 2(a)] at |eV | = |E B | + probe ; that is, when the probe's chemical potential allows for driving the dot to its excited state by either adding or removing an electron. The transport mechanisms leading to a dc current here are essentially based on Andreev processes [46]. From the experimental gate dependence of the bound-state spectrum E B , the singlet-doublet ground-state transition, occurring for E B = 0, is readily seen to occur near V G = 0.71 V. Note that, owing to the very sharply defined gap edge of aluminum [25], but also low experimental temperatures and careful shielding of the experiment, we can achieve a spectroscopic resolution down to a FWHM of less than 10 μeV [ Fig. 2(a)], way below previously reported linewidths [13,[47][48][49]. The latter have indeed been discussed as a lifetime limiting factor in possible subgap state-based qubits [6,50].
Combining our knowledge of the superconducting-and normal-state properties, we can now plot the bound-state dispersion E B as a function of gate voltage V G , which we express as a function of the dimensionless ratio T G K / [ Fig. 2(c)]. We find the transition to the unscreened ground state for the critical value (T G K / ) c 0.7 consistent with Ref. [51] or with the value (T FW K /2 ) c 1.0 in Ref. [52]. Theoretical calculations [44] predict a critical value (T th K / ) c 0.30, using the scaling formula T th K 0.28 √ U exp[π 0 ( 0 + U )/(2 )]. While we can rescale our data to T th K (for instance at the value at the center of the diamond), which gives a reasonable value (T th K / ) c 0.26, we emphasize that our device is not strictly in the scaling regime where these predictions apply quantitatively [53]. In addition, a calculation using renormalized ABS theory [54] allows us to obtain a full gate-dispersion of the bound state in good agreement with the experimental observation; see Fig. 2(c) and the Supplemental Material [27] for details. Furthermore, the intensities of the conductance peaks, which reflect the weights carried by bound states, also follow the expected behavior [44] across the ground-state transition, as shown in Fig. 2(d).
Having understood in detail the zero-field properties of the QD-S hybrid, we now move to the main result of this work, in which we evince the competition of two magnetic effects on the ground-state transition of the quantum dot. A magnetic field B is expected to Zeeman split the two spin projections of the doublet state by E Z = ±gμ B B/2, with g being the gyromagnetic factor and μ B the Bohr magneton. The effect of the Zeeman splitting on the doublet state has been observed in superconductor-quantum dot junctions formed in semiconducting nanowires, owing to the large g ≈ 20 in these materials [9,10,17]. In these works, the subgap resonances are Zeeman split at the singlet ground-state phase because two excited states are accessible. In contrast, when the singlet is the excited state, no splitting was seen, because the only possible transition (at low temperature T < E Z ) is from the lower-energy spin-polarized state to the singlet.
Beyond the mere spectroscopic effect of the Zeeman splitting of the doublet excited state at a given dot level depth, we now consider the magnetic-field effect on the ground-state transition itself. Indeed, because one of the spin projections of the unscreened spin state has a lower energy, the screened ground-state phase space gradually shrinks, which is translated here into a critical value of V G moving to lower values. This is sketched in Fig. 3(a) and precisely observed in Fig. 3(c), where we plot the critical gate value V c G associated with the ground-state transition as a function of magnetic field. The latter is determined as previously from the kink (crossing) in the YSR dispersion, for each applied magnetic field as shown in Fig. 3(b). For small fields B < 10 mT, there is a clear downward trend of V c G , indicating a Zeeman-driven reduction of the parameter space associated with the singlet ground state. As the magnetic field is further increased, the reduction of the superconducting gap starts coming into play, with a quadratic magnetic field dependence of the gap to lowest order [55]. Intuitively, the gradual weakening of superconductivity favors Kondo screening of the spin in the dot, and thereby favors the singlet ground state, enhancing again the critical V c G [ Fig. 3(c)]. This re-entrance of the phase-boundary is confirmed when sweeping the magnetic field at a fixed gate voltage V G = 0.70 V [ Fig. 3(d)]. The transition of the ground-state parity induced near a field of 12 mT is accompanied by an abrupt change in the YSR spectra, which move to higher energies and acquire a broader lineshape (see Supplemental Material for details [27]).
For completeness, we finally focus on the effect of higher temperatures for the tunnelling spectroscopies as well as the ground-state transition. First, at higher temperatures, the nonzero probability of finding the dot in its excited state allows for new conductance resonances in tunneling spectroscopies emerging at |eV | = probe − |E B |, which are commonly referred to as thermal replicas of the YSR resonances [7,13]. This is readily seen as a pair of new peaks at low voltages in Fig. 4(a). The corresponding values of the bound-state energy E B can now also be deduced from the related supplementary threshold conditions, in excellent agreement with the bound-state energies deduced from the main resonances, leading to the data shown in Fig. 4(b). The singlet-doublet transition can thus be equally observed from the thermal replicas. Second, the thermal weakening of the superconducting gap provides another method for tuning the singlet-doublet ground-state transition. At T = 940 mK, we indeed find that the transition has moved to a larger gate value, about V c G 0.74 V (or equivalently at a lower T K than for the base temperature), in agreement with expectations. Obviously, no re-entrance is observed in the temperature dependence of the transition.
In conclusion, we have demonstrated a magnetic-field tuning of the screening-unscreening transition of a quantum dot coupled to superconductors in a transistor geometry. A novel phase diagram was established, demonstrating that the magnetic field leads to a re-entrant transition due to the competition between Zeeman stabilization of the lowest spin-polarized orbital and weakening of the superconducting gap. A complementary finite-temperature phase diagram was drawn, which reflects the sole thermal weakening on the superconducting gap, while signatures of the ground-state transition were also observed in thermally excited replicas. These results demonstrate that quantum dots constitute a rich model system for the controlled exploration of strong correlations effects in nanostructures. Further developments will address the influence of the screening-unscreening transition on operational properties of single-electron turnstiles [26]. This work was funded by the joint ANR-DFG grant JOSPEC (ANR-17-CE30-0030) and the Labex LANEF programme (ANR-10-LABX-51-01). Samples were fabricated at the Nanofab facility at Institut Néel-CNRS and PTA-CEA. We thank D. Basko, J. Pekola, N. Hatter, and B. Heinrich for useful discussions.