Abstract
Manybody correlations and macroscopic quantum behaviours are fascinating condensed matter problems. A powerful testbed for the manybody concepts and methods is the Kondo effect^{1,2}, which entails the coupling of a quantum impurity to a continuum of states. It is central in highly correlated systems^{3,4,5} and can be explored with tunable nanostructures^{6,7,8,9}. Although Kondo physics is usually associated with the hybridization of itinerant electrons with microscopic magnetic moments^{10}, theory predicts that it can arise whenever degenerate quantum states are coupled to a continuum^{4,11,12,13,14}. Here we demonstrate the previously elusive ‘charge’ Kondo effect in a hybrid metal–semiconductor implementation of a singleelectron transistor, with a quantum pseudospin of 1/2 constituted by two degenerate macroscopic charge states of a metallic island^{11,15,16,17,18,19,20}. In contrast to other Kondo nanostructures, each conduction channel connecting the island to an electrode constitutes a distinct and fully tunable Kondo channel^{11}, thereby providing unprecedented access to the twochannel Kondo effect and a clear path to multichannel Kondo physics^{1,4,21,22}. Using a weakly coupled probe, we find the renormalization flow, as temperature is reduced, of two Kondo channels competing to screen the charge pseudospin. This provides a direct view of how the predicted quantum phase transition develops across the symmetric quantum critical point^{4,21}. Detuning the pseudospin away from degeneracy, we demonstrate, on a fully characterized device, quantitative agreement with the predictions for the finitetemperature crossover from quantum criticality^{17}.
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Acknowledgements
This work was supported by the ERC (ERC2010StG20091028, no. 259033) and the French RENATECH network. We acknowledge E. Boulat, J. von Delft, S. De Franceschi, L. Glazman, D. GoldhaberGordon, K. Le Hur, A. Keller, K. Matveev, L. Peeters, P. Simon and G. Zaránd for critical reading of our manuscript and discussions.
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Z.I. and F.P. performed the experiment. Z.I., A.A. and F.P. analysed the data. F.D.P. fabricated the sample. U.G. and A.C. grew the 2DEG. S.J. contributed to a preliminary experiment. F.P. led the project and wrote the manuscript with input from Z.I., A.A. and U.G.
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Extended data figures and tables
Extended Data Figure 1 Measurement schematic.
Schematic of the measurement setup, showing explicitly the nine different and simultaneously measured signals. V_{ij} (i, j ∈ {1, 2, p}) is the voltage measured with amplification chain i in response to the injected voltage V_{j}. Trenches etched in the 2DEG in the form of a Y can be seen through the metallic island.
Extended Data Figure 2 QPC characterization.
a, b, Main panels, ‘intrinsic’ transmission probability across QPC_{1} (τ_{1}; a) and QPC_{2} (τ_{2}; b) measured at 11.5 mK (in the linear regime, without dc bias) by opening the QPC lateral characterization gate (see equivalent schematic in top left insets), and plotted versus the voltage applied to the split gate tuning the QPC (V_{qpc1,2}). The experimental transmission set points in the main text are indicated by symbols. Right insets, relative variation of the transmission probability with dc bias voltage, shifted vertically for clarity, for τ_{1,2} ≈ {0.06, 0.47, 0.93} from bottom to top, respectively. The larger noise in the inset of a (mostly visible for τ_{1} ≈ 0.06) is from the amplification chain. c, ‘Intrinsic’ conductance across one lateral characterization gate in units of e^{2}/h (τ_{lcg}, here adjacent to QPC1) plotted versus lateral gate voltage V_{lcg}. Increasing V_{lcg} results in the successive full opening of two electronic channels, as schematically illustrated. In practice, we close (open) the lateral characterization gates, corresponding to τ_{lcg} = 0 (τ_{lcg} = 2), by applying V_{lcg} ≈ −0.4 V (V_{lcg} = 0 V). Grey shaded areas correspond to the partial opening of one of the channels, a configuration not used in the experiment. d, Conductance of the QPCs measured at T = 22 mK versus dc voltage (continuous lines) with the adjacent lateral gate closed (τ_{lcg} = 0) and the lateral characterization gate opposite to the metallic island set to full transmission (τ_{lcg} = 2), which corresponds to the displayed schematic circuit. The low bias dips result from conductance suppression by the dynamical Coulomb blockade, while the high bias plateaus correspond to the ‘intrinsic’ transmission probabilities τ_{1,2} (horizontal dashed lines). e, ‘Intrinsic’ transmission probabilities τ_{1,2} at the experimental set points used in the main text, together with their relative increase Δτ_{1,2}/τ_{1,2} between V_{dc1,2} = 0 and V_{dc1,2} = 50 µV. This increase is the main experimental factor of uncertainty in the determination of τ_{1,2}.
Extended Data Figure 3 Coulomb diamonds.
The conductance G_{SET} (colour coded; brighter for larger G_{SET}) is displayed versus gate and dc voltages (δV_{g} and V_{dc}, respectively), with both QPCs set to a low transmission probability. The Coulomb diamonds (darker) correspond to a charging energy E_{C} = e^{2}/2C ≈ k_{B} × 290 mK.
Extended Data Figure 4 Reproducibility of conductance oscillations.
Shown is conductance across the hybrid SET (G_{SET}) when sweeping the gate voltage (V_{g}) across several periods for the symmetric configuration τ_{1} ≈ τ_{2} ≈ 0.93 and at base temperature T ≈ 11.5 mK (one period shown in Fig. 1c). The symbols display the measurements, the continuous line is the quantitative prediction of equation (9) in Methods.
Extended Data Figure 5 Observation of ‘in situ’ conductances that overstep the standard quantum limit e^{2}/h.
The displayed Coulomb peaks were measured at T ≈ 14 mK for the asymmetric QPCs configuration (τ_{1} ≈ 0.77, τ_{2} ≈ 0.93). Two sweeps (V_{g} increasing and decreasing) are shown for each measurement. a–c, Symbols are the normalized transmitted signal v_{i,j}/2, with i ≠ j (equation (5) in Methods) versus gate voltage. Each panel displays the two reciprocal signals v_{i,j} and v_{j,i}. The vertical lines in a are visual markers used in e. d, Symbols are the normalized reflected signal at the probe QPC_{p} (2(1 − v_{pp}), corresponding to G_{p}h/e^{2} in the limit G_{p} ≪ G_{1,2}). e, Symbols are the in situ conductance ratio G_{1}/G_{2}, measured from both v_{1p}/v_{2p} and v_{p1}/v_{p2}. For each measurement, the two sweeps (V_{g} increasing and decreasing) are shown with different symbols. The black line is an average at a given δV_{g}. The red line shows the value below which G_{2} > e^{2}/h near charge degeneracy (δV_{g} ≈ 0). The error bars shown in a–d represent the statistical uncertainties (s.e.m.) calculated from 10 successive measurements.
Extended Data Figure 6 Robustness to experimental conditions of ‘in situ’ conductances overstepping e^{2}/h.
Symbols display the ‘in situ’ conductance G_{2} measured at T ≈ 14 mK for the QPC setting τ_{1} ≈ 0.77 and τ_{2} ≈ 0.93, and under different experimental conditions. We repeatedly find G_{2} > e^{2}/h. a, The influence on G_{2} of the ac injection voltages V_{1,2,p}. The three lowest V_{ac} correspond to V_{1} = V_{2} = V_{p} = V_{ac}, whereas the fourth data point corresponds to V_{p} = V_{ac} with V_{1} = V_{2} = 1.15 µV_{rms}. b, Exploration of the influence on G_{2} of the coupling strength of QPC_{p}, characterized by G_{p}. The error bars represent the statistical uncertainties (s.e.m.) calculated from 20 or more different measurements.
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Iftikhar, Z., Jezouin, S., Anthore, A. et al. Twochannel Kondo effect and renormalization flow with macroscopic quantum charge states. Nature 526, 233–236 (2015). https://doi.org/10.1038/nature15384
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DOI: https://doi.org/10.1038/nature15384
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