Controlling charge quantization with quantum fluctuations

Abstract

In 1909, Millikan showed that the charge of electrically isolated systems is quantized in units of the elementary electron charge e. Today, the persistence of charge quantization in small, weakly connected conductors allows for circuits in which single electrons are manipulated, with applications in, for example, metrology, detectors and thermometry1,2,3,4,5. However, as the connection strength is increased, the discreteness of charge is progressively reduced by quantum fluctuations. Here we report the full quantum control and characterization of charge quantization. By using semiconductor-based tunable elemental conduction channels to connect a micrometre-scale metallic island to a circuit, we explore the complete evolution of charge quantization while scanning the entire range of connection strengths, from a very weak (tunnel) to a perfect (ballistic) contact. We observe, when approaching the ballistic limit, that charge quantization is destroyed by quantum fluctuations, and scales as the square root of the residual probability for an electron to be reflected across the quantum channel; this scaling also applies beyond the different regimes of connection strength currently accessible to theory6,7,8. At increased temperatures, the thermal fluctuations result in an exponential suppression of charge quantization and in a universal square-root scaling, valid for all connection strengths, in agreement with expectations8. Besides being pertinent for the improvement of single-electron circuits and their applications, and for the metal–semiconductor hybrids relevant to topological quantum computing9, knowledge of the quantum laws of electricity will be essential for the quantum engineering of future nanoelectronic devices.

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Figure 1: Tunable quantum connection to a metallic island.
Figure 2: Charge quantization versus connection strength at T ≈ 17 mK.
Figure 3: Charge quantization scaling near the ballistic critical point.
Figure 4: Crossover to a universal charge quantization scaling as temperature is increased.

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Acknowledgements

This work was supported by the European Research Council (ERC-2010-StG-20091028, no. 259033), the French RENATECH network, the national French programme ‘Investissements d’Avenir’ (Labex NanoSaclay, ANR-10-LABX-0035), the US Department of Energy (DE-FG02-08ER46482) and the Swiss National Science Foundation.

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Authors

Contributions

S.J. and Z.I. performed the experiment with inputs from A.A. and F.P.; S.J., Z.I., A.A. and F.P. analysed the data; F.D.P. fabricated the sample and contributed to a preliminary experiment; U.G., A.C. and A.O. grew the 2DEG; I.P.L., E.I., E.V.S. and L.I.G. developed the strong thermal fluctuations theory; F.P. led the project and wrote the manuscript with inputs from all authors.

Corresponding author

Correspondence to F. Pierre.

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The authors declare no competing financial interests.

Additional information

Reviewer Information Nature thanks Y. Nazarov and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Extended data figures and tables

Extended Data Figure 1 Measurement schematic.

The signal VLR (VRR) is the voltage measured with amplification chain L (R) in response to the injected voltage VR. The trenches etched in the 2DEG, which can be seen in the form of a ‘Y’ through the metallic island, ensure that the only way from one QPC to the other is across the metallic island. The experiment is performed in the quantum Hall regime at filling factor ν = 2, where the current propagates along the edges in the direction indicated by arrows.

Extended Data Figure 2 Crosstalk compensation.

a, (Intrinsic) conductance across the characterization gate adjacent to QPCR versus gate voltage . In the experiment, the left and right switches are independently set to the open and closed positions with and , respectively (vertical arrows in c). b, QPCR differential conductance in the presence of a d.c. bias of 72 μV (‘72 μVdc’) versus QPC gate voltage . The red and blue lines are measured with the adjacent switch in the open and closed positions, respectively (see inset schematics). The voltage drop across QPCR is smaller with the switch open, owing to the added series resistance. Although this does not result in a large error, because depends weakly on voltage bias, this effect is minimized by extracting the crosstalk compensation at low . c, Symbols represent the crosstalk compensation , with respect to the gate voltage , versus . Lines are linear fits of the crosstalk compensation at (red, −2.8% relative compensation), (green, −1.1% relative compensation) and (blue, −1.4% relative compensation).

Extended Data Figure 3 Conductance measurements versus quantitative predictions.

Direct GSETVg) comparison at T ≈ 17 mK between data (symbols) and predictions (solid lines, grey areas correspond to the temperature uncertainty of ±4 mK) in the two limits addressed by theory (equation (3) for τL ≈ 0 (top panels), equation (6) for τL ≈ 1 (bottom panels)).

Extended Data Figure 4 Theoretical description of the experimental set-up in formalism (A) for strong thermal fluctuations.

We consider the regime of the quantum Hall effect, where only one spinless edge mode contributes to the transport. The corresponding edge states are described by four charge density operators, labelled by s {L, R} and α {1, 2}. These states are mixed (backscattered) at the two QPCs (red dashed lines) with amplitudes γL and γR (equations (14) and (15)). The edge densities enter into the interaction Hamiltonian (equation (12) through the total chargeI8 of the metallic island (equation (13)). The average current 〈I〉 is calculated through a cross-section immediately to the right of QPCR (vertical blue lines).

Extended Data Figure 5 Charge quantization based on conductance versus transmission probability values.

a, b, Schematics of the configurations, both with the same QPCL setting τL = 0.24. In the configuration shown in a, QPCR is set to an ‘intrinsic’ conductance , which decomposes into one ballistic channel and one channel of intrinsic transmission probability 0.5. In the configuration shown in b, QPCR is set to the same intrinsic conductance , which now decomposes into two non-ballistic channels of intrinsic transmission probabilities 0.7 and 0.8. c, Sweeps of the device conductance are plotted versus gate voltage for the two configurations (a, red triangles; b, black squares). Conductance oscillations are visible only in the configuration shown in b, in the absence of a ballistic channel connected to the island.

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Jezouin, S., Iftikhar, Z., Anthore, A. et al. Controlling charge quantization with quantum fluctuations. Nature 536, 58–62 (2016). https://doi.org/10.1038/nature19072

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