Electron attraction mediated by Coulomb repulsion

Journal name:
Nature
Volume:
535,
Pages:
395–400
Date published:
DOI:
doi:10.1038/nature18639
Received
Accepted
Published online

One of the defining properties of electrons is their mutual Coulomb repulsion. However, in solids this basic property may change; for example, in superconductors, the coupling of electrons to lattice vibrations makes the electrons attract one another, leading to the formation of bound pairs. Fifty years ago it was proposed1 that electrons can be made attractive even when all of the degrees of freedom in the solid are electronic, by exploiting their repulsion from other electrons. This attraction mechanism, termed ‘excitonic’, promised to achieve stronger and more exotic superconductivity2, 3, 4, 5, 6. Yet, despite an extensive search7, experimental evidence for excitonic attraction has yet to be found. Here we demonstrate this attraction by constructing, from the bottom up, the fundamental building block8 of the excitonic mechanism. Our experiments are based on quantum devices made from pristine carbon nanotubes, combined with cryogenic precision manipulation. Using this platform, we demonstrate that two electrons can be made to attract each other using an independent electronic system as the ‘glue’ that mediates attraction. Owing to its tunability, our system offers insights into the underlying physics, such as the dependence of the emergent attraction on the underlying repulsion, and the origin of the pairing energy. We also demonstrate transport signatures of excitonic pairing. This experimental demonstration of excitonic pairing paves the way for the design of exotic states of matter.

At a glance

Figures

  1. Model system and experimental realization of its fundamental building block.
    Figure 1: Model system and experimental realization of its fundamental building block.

    a, The organic-molecule model system proposed by Little1, consisting of two parts: the ‘system’, a one-dimensional conducting chain (individual lattice sites marked in green), and the ‘medium’, an array of side-chain ‘polarizers’ (purple) each having a single electron that can hop between a site close to, and a site further away from, the chain. The fundamental unit block that manifests attraction is a two-site system with one polarizer (dashed circle). b, In a bare electronic system (top) an electron creates a repulsive Coulomb potential (green). Embedding it in a medium that flips the sign of its potential (bottom) will make this electron attractive to other electrons. c, Implementation of the two components (‘polarizer’ and ‘system’) that make up the fundamental building block. These are fabricated as two separate devices, each having a pristine nanotube assembled on contacts (yellow) and suspended above an array of gates (blue), which set the potential landscape for the electrons (grey) (see Supplementary Information section S1 for dimensions). The polarizer device (top) has two gate voltages, VB and VT, that control the potentials of the bottom and top dots (purple), and is operated as an isolated dipole, whose sole degree of freedom is that of a single electron hopping between the sites. The system device (bottom) has two gate voltages, VL and VR, that control the potentials of the right and left dots (green). Here the central barrier is opaque and the side barriers are relatively transparent such that electrons enter the two dots from their corresponding leads. An additional dot on a side segment of the same nanotube (blue) serves as a single electron transistor charge detector: a voltage bias across it, VCD, leads to current, ICD, that is sensitive to the population of the system dots through weak electrostatic coupling. d, A custom-built scanning probe microscope (centre), operating inside a dilution refrigerator, brings the two oppositely facing devices into proximity (about 100–150 nm apart) such that the polarizer nanotube is perpendicular to the system nanotube (right) and such that one of its dots is directly above the system while the other is further away (left), creating a structure that is analogous to that of Little’s molecule.

  2. From repulsive to attractive electrons.
    Figure 2: From repulsive to attractive electrons.

    a, Measured charge stability diagram of the bare system: charge detector current (ICD, colour scale) plotted as a function of the voltage detuning between its right (R) and left (L) sites, δV = (VL − VR)/2, and the mean gate voltage, V = (VL + VR)/2. Steps in ICD (dashed grey lines) correspond to single electrons populating the L/R sites (green/white circles label an electron presence/absence). The middle vertical shift (black dashed line) is a direct measure of the Coulomb repulsion between the neighbouring electrons (see text). b, Similar charge stability diagram to a, but with the polarizer nanotube positioned nearby (approximately 125-nm separation between the nanotubes). The interaction vertex is now horizontal, reflecting an attraction between the electrons. Along this line (dashed black) the ground state is degenerate between the two even states (0, 0) and (1, 1) ((nL, nR) represents the number of electrons in the L and R dots), whereas the odd states (1, 0) and (0, 1) are the excited states, separated by a pairing gap Δ from the ground state. This gap is maximal at the centre of the horizontal line, and its magnitude there is equal to the length of this line (2Δ when normalized to energy units; see Supplementary Information section S4). Note that the charge detector is far away from the right dot and that when the polarizer is close their mutual capacitance is strongly screened and so the charging lines of this dot are harder to observe. c, In the bare system, an electron populating the L site generates a Coulomb potential (green) that is positive (repulsive) in both the L and R sites. d, Embedding the system in a medium based on continuum electrostatics (purple) can at best screen the potential to zero far away from the electron, but cannot flip its sign. e, The key element for sign inversion is charge discreteness: a single-electron dipole (right) and a similarly shaped metal (left) will screen an external field differently (grey arrows). In the latter the internal field is nulled, whereas in the former it is larger and of opposite sign to the external field (red arrows; over-screening). This behaviour is rooted in the fact that an electron does not repel itself (see text). f, With a nearby polarizer, an electron charging the system gets dressed by the polarization (grey ellipse). The electrostatic potential (green) of the dressed particle will be substantially different to that of the bare electron (calculated in Supplementary Information section S4). Note that all the measurements presented here were done with single holes instead of single electrons, but to avoid unnecessary confusion we presented the physics in the language of electrons.

  3. Dependence of pairing energy on the polarizer detuning and the origin of the pair binding energy.
    Figure 3: Dependence of pairing energy on the polarizer detuning and the origin of the pair binding energy.

    ac, Charge stability diagrams similar to those in Fig. 2b, measured for different energy detunings of the polarizer: δ = 0.39 meV (a), 1.01 meV (b) and 2.57 meV (c). The observed attraction increases linearly with δ (more data in Supplementary Information section S6). The insets show the polarizer potential wells for the different detuning values. d, e, Rationalizing this observation using a toy model of the chain: two spatially separated electrons in the chain are each dressed by the polarization of their two adjacent polarizers (d); when the electrons are nearest neighbours, they share the centre polarizer and thus need to polarize one fewer polarizer (e). The energy gain, which gives the pairing energy, is consequently equal to δ, as we observe in the experiments in ac. A similar polarizer-sharing argument holds for the two-site case, as is explained in Supplementary Information section S6.

  4. Transport measurements.
    Figure 4: Transport measurements.

    a, Experimental configuration. The polarizer current, Ipol, is measured with a finite bias on one lead, VSD. The barrier between this lead and its nearby dot is reduced such that the dot becomes strongly connected to the lead, effectively making the polarizer nanotube a single quantum dot device with two leads. Compared to the experiments presented in Figs 2 and 3, in which the polarization occurred internally between two dots, here the polarization is between the dot and its lead. b, Ipol measured as a function of the local gate voltage, VB, for four different population states of the system: blue, (0, 0); red, (1, 0); orange, (0, 1); green, (1, 1). In these measurements, the system charge remained fixed by staying far away from the charging lines of the system. This measurement was performed at a high bias, VSD = −1.3 mV, which widens the observed charging lines. The vertical dashed line indicates the gate voltage VB used in c. c, Ipol (colour scale) measured at VSD = 100 μV as a function of the voltage detuning between the L and R system sites, δV = (VL − VR)/2, and the mean voltage, V = (VL + VR)/2. Finite current (red) is observed for the odd system states, (1, 0) and (1, 0), and negligible current (blue) is measured for the even states, (0, 0) and (1, 1), except for a special peak of finite current (indicated by the white arrow) appearing along their degeneracy line. d, A theoretical master-equation calculation of Ipol for the parameters of the experiment (Supplementary Information section S7). The theoretical result features a finite current peak on the degeneracy line between the (0, 0) and (1, 1) states only when considering correlated processes that involve co-tunnelling of a pair of electrons into the system in concert with an electron tunnelling out of the polarizer. e, Ipol (colour scale) measured along a line cutting through the centre of the degeneracy line as a function αV and VSD (the independently measured lever-arm, α = 0.61, normalizes V to energy units). Three regimes appear at different bias ranges, differing in their turn-on slope, s = dVSD/(αV), the value of which (s ≈ 0, black line; s ≈ 1, blue line; s ≈ 2, red line) reflects the number of system electrons participating in the dominant transport process (0e, 1e or 2e, respectively) (see Supplementary Information section S7). f, Theoretical master-equation calculation of Ipol versus αV and VSD with only a 0e system process considered (left), with 0e + 1e processes (middle) and with 0e + 1e + 2e processes (left). The 2e processes, which involve a pair tunnelling in the system, are the dominant processes at low bias, as is observed experimentally, although their calculated amplitude is lower than in the experiment, probably because the theory considers them to only the lowest order in tunnelling (Supplementary Information section S7).

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Author information

  1. These authors contributed equally to this work.

    • A. Hamo,
    • A. Benyamini &
    • I. Shapir
  2. Present addresses: Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (J.W.); Department of Micro- and Nanotechnology, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark (K.K.).

    • J. Waissman &
    • K. Kaasbjerg

Affiliations

  1. Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel

    • A. Hamo,
    • A. Benyamini,
    • I. Shapir,
    • I. Khivrich,
    • J. Waissman,
    • K. Kaasbjerg,
    • Y. Oreg &
    • S. Ilani
  2. Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany

    • F. von Oppen

Contributions

A.H., A.B., I.S. and S.I. performed the experiments, analysed the data, contributed to its theoretical interpretation and wrote the paper. I.S. built the scanning probe microscope. I.K. built the custom measurement instrumentation for the experiment. J.W. designed and fabricated the devices. K.K., Y.O. and F.v.O. developed the theoretical model. K.K. performed the theoretical simulations.

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The authors declare no competing financial interests.

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Supplementary information

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  1. Supplementary Information (2.7 MB)

    This file contains Supplementary Text and Data, which includes Supplementary Methods, Supplementary Figures 1-10, a Supplementary Discussion and additional references.

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