REPORT◥

MESOSCOPIC PHYSICS

Imaging the electronic Wigner crystalin one dimensionI. Shapir1*, A. Hamo1*†, S. Pecker1, C. P. Moca2,3, Ö. Legeza4, G. Zarand2, S. Ilani1‡

The quantum crystal of electrons, predicted more than 80 years ago by EugeneWigner, remains one of the most elusive states of matter. In this study, we observedthe one-dimensional Wigner crystal directly by imaging its charge density in real space.To image, with minimal invasiveness, the many-body electronic density of a carbonnanotube, we used another nanotube as a scanning-charge perturbation. The imageswe obtained of a few electrons confined in one dimension match the theoreticalpredictions for strongly interacting crystals. The quantum nature of the crystal emergesin the observed collective tunneling through a potential barrier. These experiments providethe direct evidence for the formation of small Wigner crystals and open the way forstudying other fragile interacting states by imaging their many-body density in real space.

EugeneWigner predicted (1) thatwhen long-range Coulomb interactions in a system ofelectrons dominate over kinetic energy anddisorder, a crystalline ground state shouldemerge inwhich the electrons are kept apart

irrespective of their internal quantum number(flavor). Experimental searches for this quantumcrystal have been conducted in the cleanestavailable electronic systems, such as the surfaceof liquid helium (2) and low-dimensional semi-conductor heterostructures. In semiconductingtwo-dimensional electronic systems, transport(3, 4), microwave (5, 6), nuclear magnetic reso-nance (7), optical (8, 9), tunneling (10), and bi-layer correlation (11)measurements have indicatedthe existence of a crystal at high magnetic fields.In one dimension, thermal and quantum fluctu-ations destroy the long-range order, and a crys-talline state in an infinite system is not expected.However, in finite systems, quasi-long-range orderproduces crystalline correlations, and this one-dimensional Wigner crystal state has been ex-tensively studied theoretically (12–18) and probedexperimentally via transportmeasurements (19, 20).However, these experiments probed only macro-scopic properties of this state.The unambiguous fingerprint of aWigner crys-

tal lies in its real-space structure, which couldin principle be observed with a suitable imagingtool. Scanning probe experiments (21–26) haveso far been able to image only the noninteractingstate or have showed invasive gating by the probe.

These measurements highlight the inherent dif-ficulty of imaging interacting electrons with con-ventional scanningmethods: To resolve individualelectrons, a macroscopic, metallic, or dielectric tipshould approach the electrons closer than theirmutual separation, which inevitably screens theirinteractions, which are at the heart of the inter-acting state. Moreover, macroscopic tips gener-ically carry uncontrollable charges that havestrong gating effects on an interacting electronsystem, strongly distorting the state being studied.To image an interacting state, a different kind ofscanning probe is therefore needed.In this study, we use a scanning probe plat-

form that utilizes a carbon nanotube (NT) as a

highly sensitive, yet minimally invasive scanningprobe for imaging the many-body density ofstrongly interacting electrons. Our platform com-prises a custom-made scan probe microscope,operating at cryogenic temperatures (~10 mK),in which two oppositely facingNT devices can bebrought in close proximity (~100 nm) (27) andscanned along each other (Fig. 1A). One devicehosts the system NT, which is used as the one-dimensional system under study (Fig. 1A, bot-tom). The second device contains the probe NT,which is perpendicular to the systemNT and canbe scanned along it (Fig. 1A, top). The two de-vices are assembled using a nanoassembly tech-nique (28) that yields pristine NTs suspendedabove an array of metallic gates. In the systemNT, it is essential to maintain strong interactionsand low disorder, both of which are crucial forobtaining a Wigner crystal; this is achieved bysuspending the NT far above the metallic gates(400 nm). Using 10 electrically independent gates,we design a potential that confines the electronsbetween two barriers, ~1 mm apart, localizingthem to the central part of a long suspendednanotube (length: 2.3 mm) (Fig. 1B), away fromthe contacts that produce undesirable distortions(e.g., screening, image charges, disorder, and bandbending) [see section 1 in the supplementarymaterials (29)]. We use highly opaque barriersthat prevent hybridization of the confined elec-tron’s wave function with those of the electronsin the rest of the NT. Because transport in thissituation is highly suppressed, we probe the con-fined electrons using a charge detector locatedon a separate segment of the same NT (Fig. 1B,purple). The addition of these electrons is de-tected as a small change in the detector’s current,ICD, flowing between the two outer contacts ofthe device (Fig. 1B, blue arrow) but not throughthe central segment of the system NT. The probe

RESEARCH

Shapir et al., Science 364, 870–875 (2019) 31 May 2019 1 of 5

1Department of Condensed Matter Physics, Weizmann Instituteof Science, Rehovot 76100, Israel. 2Institute of Physics,Budapest University of Technology and Economics, H-1521Budapest, Hungary. 3Department of Physics, University ofOradea, 410087 Oradea, Romania. 4Wigner Research Centre forPhysics, H-1525 Budapest, Hungary.*These authors contributed equally to this work.†Present address: Department of Physics, Harvard University,Cambridge, MA 02138, USA.‡Corresponding author. Email: shahal.ilani@weizmann.ac.il

Fig. 1. Experimental platform for imaging strongly interacting electrons. (A) Scanning probesetup consisting of two carbon nanotube (NT) devices—a system-NT device (bottom) thathosts the electrons to be imaged (green ellipse) and a probe-NTdevice (top) containing the probingelectrons (red). In the experiment, the probe NT is scanned along the system NT (black arrow).(B) The system NT is connected to contacts (yellow) and is suspended above 10 gates (blue)used to create a potential well (shown schematically in gray) that confines a few electrons to themiddle part of the suspended NT (green), away from the contacts. The addition of these electronsis detected using a charge detector—a separate quantum dot formed on a side segment of thesame NT (purple). The detector is biased by a voltage, VCD, applied on an external contact, leadingto a current, ICD, flowing only between the contacts of the charge detector (blue arrow), such thatno current passes through the main part of the system NT.

on Novem

ber 1, 2020

http://science.sciencemag.org/

Dow

nloaded from

NT device has an almost identical structure, dif-fering only by the NT suspension length (1.6 mm)and number of gates (three). Because the probegates and contacts are perpendicular to the sys-temNT, the potential they induce remains trans-lationally invariant as the probe is scanned alongthe system, and the only moving perturbationcomes from themovingNT itself [section 2 in (29)].To demonstrate the basic principle behind our

imaging technique, which we term “scanningcharge,”we start with the simplest experiments—imaging the charge distribution of a single elec-tron confined to a one-dimensional box (Fig. 2A).By measuring the energetic response of the sys-tem to a scanned perturbation, we can directlydetermine the system’s density distribution. Forsimplicity, we assume first that the perturbationproduced by the probeNT is highly localized at itsposition, xprobe, V(x) ≈ Vd(x − xprobe). To thelowest order, such a perturbation will shift thesystem’s energy as

E1ðxprobeÞ ¼hy1jV ðx � xprobeÞjy1iºr1ðxprobeÞ ð1Þ

where r1ðxÞ ¼ jy1ðxÞj2 (Fig. 2A, green) is thedensity distribution of the confined electron wavefunction, y1ðxÞ . Thus, by measuring E1 as afunction of xprobe, the electron’s density profilecan be directly resolved (30, 31). The energy E1

ismeasured by referencing it to the Fermi energyin the leads (EF ≡ 0) (Fig. 2A, bottom). Startingwith the level in resonance with EF, a movementof the probe to xprobe will shift the energy levelto E1(xprobe) and typically away from resonancewith EF. We then bring the level back to reso-nance by applying a global gate voltage,Vg, formedby a proper linear combination of the ten gates,chosen to produce a rigid shift of the potentialwithout changing its shape [section 1 in (29)].The applied Vg, measured in energy units, isequal to the energy shift E1. Thus, by monitoringthe gate voltage required to keep the level in reso-nance for varying xprobe, we directly image thecharge distribution, Vg(xprobe) ~ r1(xprobe). Notethat in reality, the perturbation produced by ourprobe is not a delta function but has a spatialextent determined by the separation betweenthe two NTs and the spatial extent of the con-fined charge in the probe NT. The measured

profile will therefore be a convolution of the cor-responding point spread function and the imagedcharge density distribution. Throughout this Re-port, to avoid confusion in describing the physicsof the electronic Wigner crystal, we use thelanguage of electrons, although, owing to tech-nical advantages, the actual measurements weredone with holes in the system and probe [sec-tion 3 in (29)].The imaging of the charge distribution of a

single electron is shown in Fig. 2, B and C. Thepopulation of the first confined electron in thesystem (qsystem = 1e) is identified by its sharpcharging peak, observed when we measure thecharge detector signal, dICD/dVg, a function ofVg [Fig. 2B and section 3 in (29)]. To image thecharge distribution of this electron, we also placeone electron in the probe NT (qprobe = 1e), scanit along the system NT, and monitor the corre-sponding shift of the charging peak in gate volt-age. This measurement (Fig. 2C) reveals that thecharging peak shifts smoothly when xprobe tra-verses between the two edges of the confinementwell, peaking at its center. This trace directlyyields the real-space charge distribution of the

Shapir et al., Science 364, 870–875 (2019) 31 May 2019 2 of 5

Fig. 2. Real-space imaging of the density profile of a single confinedelectron. (A) To image the density distribution of a single electron confined ina potential “box” (gray), we place a fixed charge in the probe NT and scanit across the system NT.This charge creates a local perturbation at the probeposition xprobe (red), which shifts the ground state energy of the systemelectron, E1 (top panels), proportional to the local density at the probeposition E1(xprobe) ~ r1(xprobe) (see text). By measuring the global gatevoltage, Vg, needed to keep the charging of this single electron in resonancewith the Fermi energy of the leads, EF, for varying xprobe (bottom panels),we effectively trace the profile of its charge distribution Vg(xprobe) ~ r1(xprobe).

(B) The derivative of the charge detector current, with respect to Vg dICD/dVg,measured as a function Vg. The sharp charging peak corresponds to the firstelectron entering the system-NTpotential well (in Figs. 2 and 3, the green andred labels indicate the number of electrons in the system and probe,respectively). a.u., arbitrary units. (C) dICD/dVg as a function of Vg and xprobe.The charging resonance traces a curve that gives the charge density ofthe electron convolved with the point spread function of the probe. (Insets)Illustration of the system and probe devices for different measurementpositions. (D) Same as in (C), but for different probe charges from qprobe = 0eto 3e. (E) The traces extracted from panel (D), plotted together.

RESEARCH | REPORTon N

ovember 1, 2020

http://science.sciencem

ag.org/D

ownloaded from

first confined electron, convolved with the pointspread function of the probe [section 5 in (29)].An essential test for the technique is to assess

how the measured energy shifts scale with thestrength of the scanned perturbation, which wecan control down to the single-electron limit.Figure 2D depicts imaging measurements donewith qprobe = 0e to 3e, showing that the energyshifts increase monotonically with increasingnumber of probing electrons. The overall shapeof the imaged charge density remains similar, butfor larger qprobe, the peak becomes slightly sharp-er, reflecting an increasing probe invasiveness. Forqprobe = 1e, the probe causes a movement of theelectron in the system that is approximately an

order of magnitude smaller than the zero-pointmotion of this electron, indicating that the probeis in the noninvasive limit [section 4 in (29)]. Inprinciple, our probe could have also had uncon-trolled charges caused by localized states in theNTor imperfection in themetals, which can createan even larger scanning perturbation than thesingle charges that we place intentionally. How-ever, our measurement with qprobe = 0e displaysan energy shift an order of magnitude smallerthan that with qprobe = 1e (Fig. 2E), demonstrat-ing that in our experiment, the spurious chargesare much less influential than one electron.Next, we image the interacting states of many

electrons. The measurement is similar to the

one described above but is now done around thecharging resonance of the Nth electron (N > 1)(Fig. 3A). This resonance occurs when the stateswith N and N − 1 electrons are energeticallydegenerate, EN = EN−1 (equivalently, m ≡ EN �EN�1 ¼ 0). The probe perturbation can modifyboth EN and EN−1, shifting the resonance in gatevoltage as

VgðxprobeÞ ¼EN ðxprobeÞ � EN�1ðxprobeÞºrN ðxprobeÞ � rN�1ðxprobeÞ ð2Þ

The trace Vg(xprobe) now images the densityof the N electron state, rN(x), minus that of the

Shapir et al., Science 364, 870–875 (2019) 31 May 2019 3 of 5

1e 2e

3e 4e

5e 6e

1e 2e 3e 4e 5e 6e

Fig. 3. Imaging the differential density of many-electron states.(A) In a charging transition from N – 1 to N electrons, the resonance occursfor EN = EN–1 and the gate voltage shift images the differential densityVg(xprobe) ~ rN(xprobe)− rN – 1(xprobe). (B) Illustration of the expected differentialdensity of noninteracting (left) versus strongly interacting (right) electronsin a carbon NT.These sketches also include the finite resolution smearing[section 5 in (29)]. Noninteracting electrons occupy the particle-in-a-boxwave functions, each being fourfold degenerate because of the spin andvalley degeneracy (red and blue arrows). Consequently, the differentialdensity of the first four electrons should be identical and single-peaked,and that of the next four should be double-peaked. For the stronglyinteracting case, the electrons separate in real space (bottom right), and

each added electron will add onemore peak to the differential density profile(top right). (C) Measurement of ICD as a function of Vg and xprobe, aroundthe charging peaks of the first six electrons in the system.The curvesdirectly trace the differential density of thesemany-electron states, showingthat they are deep in the strongly interacting regime. (D) The differentialdensity of the first six electrons, calculated with DMRG, which takes intoaccount long-range electronic interactions [section 7 in (29)], as a functionof the spatial coordinate x/ld and the effective strength of electronicinteractions,~rs, ranging from very weak ð~rs ¼ 0:01Þ to very strong ð~rs ¼ 100Þ.Green stars mark the positions of the peaks measured in the experiment,and the green lines mark the calculated positions (with a single freeparameter ld = 160 nm) [section 7 in (29)].

RESEARCH | REPORTon N

ovember 1, 2020

http://science.sciencem

ag.org/D

ownloaded from

N − 1 electron state, rN−1(x). This quantity, whichwe term the differential density, is intuitivelyunderstood in the single-particle picture—it ismerely the density added by the last electron toenter the system. Note that Eq. 2 also holds forthe 1e case, because in that case E0 ≡ 0, andthus Vg(xprobe) º r1(xprobe).When electrons havemultiple internal “flavors”

(in our case four, spin and valley), their differ-ential density profiles should be markedly dif-ferent in the noninteracting and the stronglyinteracting cases. In the absence of interactions,electrons populate the particle-in-a-box states ofthe potential, with a degeneracy given by thenumber of flavors. In NTs, this would be a four-fold degeneracy owing to the spin and valley de-grees of freedom. Thus, the spatial distribution ofthe density added by each one of the first fourelectrons (their differential density) would beidentical, given by that of the first particle-in-a-box state. The density profiles associated with

the next four electrons will also look identical toone another, having the two spatial peaks of thesecond particle-in-a-box state, and so on (Fig. 3B,left). A completely different picture emergesin the presence of strong interactions: BecauseCoulomb interactions are flavor-independent,all electrons keep apart, independent of theirflavor. Thus with any additional electron, onemore peak will be added to the differential den-sity profile (Fig. 3B, right). Imaging the differentialdensity of “flavored” electrons should thereforemake a clear distinction between these tworegimes.The six panels in Fig. 3C correspond, respec-

tively, to the first six electrons added to the sys-tem NT. To keep the perturbation minimal, allthese scans are performed with one electron inthe probe NT. Unlike in Fig. 2C, here we plot thecharge detector current, ICD, rather than its deriv-ative, which shows a step rather than a peakwhenan electron is added to the system NT. A clear

trend can be observed in the imaged differentialdensity profiles—with every added electron, onemore peak appears in the differential density.These profiles are clearly different than thosepredicted by single-particle physics, but theymatch those of a strongly interacting crystal. Withan increasing number of electrons, we see thatthe electron spacing is reduced but also thattheir overall spread increases, signifying that theelectrons are confined to a “box” with soft walls.The slight deviation fromperfect periodicity stemsfromnonideality of the potential [section6 in (29)].These images give the direct real-space observationof electronic Wigner crystals.To obtain a quantitative understanding of our

measurements, we perform density matrix renor-malization group (DMRG) calculations that in-clude long-range Coulomb interactions (32). Weuse a quartic confinement potential,V ðxÞ ¼ 1

4Ax4,

that nicely approximates the experimental con-finement potential (fig. S1B). Figure 3D shows

Shapir et al., Science 364, 870–875 (2019) 31 May 2019 4 of 5

Fig. 4. Many-body tunnelingof the few-electron state.(A) Illustration of the potentiallandscape, which now includesa central barrier through whichan electron can tunnel (redarrow). The detuning voltage,e, changes the relative heightof the bottom of each well.(B) Charge stability diagramfor 1e as a function of Vg ande, measured using dICD/de(color bar). The states (N, M)denote the charge N (M) inthe left (right) wells.The vertical,wider line corresponds to aninternal tunneling, occurringwhen EN+1,M = EN,M+1.(C) Schematic of the expectedtunneling differential densityfor one electron (red “dipole”,bottom), given by thedifference between its densitydistribution before and aftertunneling [r10(x) and r01(x)]convolved with the probe’spoint spread function (PSF).(D) Measured charge detectorsignal as a function xprobeand the difference in detuningrelative to the unperturbedstate, De. The red trace showsthe De(xprobe) necessary tokeep the tunneling inresonance (shown schemati-cally in inset), giving thetunneling differential density.(E) Same as (A), but for three electrons in the trap. (F) Two scenariosfor the tunneling: (Left) Only the central electron moves in the tunnelingevent; De(xprobe) will show a single dipole, as in the one-electron caseillustrated in (C). (Right) Many-body tunneling, in which the of all theelectrons coordinate move coherently in the tunneling process; multipledipoles are expected in the differential tunneling signal. (G) (Top inset)Charge-stability diagram of three electrons, with ICD/de (a.u.) measured

for −42 mV < e < 10 mV, 170 mV < Vg < 130 mV. (Main panel) MeasuredDe(xprobe) for three electrons, exhibiting multiple dipoles. (Bottom inset)DMRG-calculated many-body density distributions [section 7 in (29)] ofthree electrons that the electrons tunnel between (blue andred and their difference, convolved with the probe’s PSF (blue, bottom),showing that all electrons move a fraction of their zero-point motionas part of the many-body tunneling.

RESEARCH | REPORTon N

ovember 1, 2020

http://science.sciencem

ag.org/D

ownloaded from

the differential densities of the first six electrons,calculated as a function of the spatial coordinate,x/ld (where ld is the natural length scale in thispotential), and interaction strength, ~rS ≈ a=aB ,which estimates the ratio of the average elec-tron spacing and the Bohr radius [section 7 in(29)]. The measured electron positions (Fig. 3D,green stars) agree well with those predicted byDMRG, showing that ~rS ranges from 50 for twoelectrons to 20 for six electrons (fig. S12A),placing the observed crystals well within thestrongly interacting regime. The comparison fur-ther shows that the zero-point motion of theelectrons is ~2 to 3 times smaller than their dis-tance (Lindemann ratio) and that the exchangeat these large electronic spacings is considerablysmaller than our temperature [section 8 in (29)].To probe the quantum nature of the Wigner

crystal, we measure its tunneling characteristics,with the expectation that the correlations betweenthe electrons in a crystal will cause the crystalto tunnel through a barrier collectively (13). Wecreate a tunneling barrier at the center of theconfining potential box using the gates, effec-tively creating a double-well potential (Fig. 4A).To understand the measurement, we start withthe simple case of one electron in the doublewell.Figure 4B shows the charge stability diagram ofthe system, measured using the charge detectorcurrent, dICD/de, as a function of the commonvoltage on the two wells, Vg, and their detuningvoltage, e. This is the familiar double-dot stabilitydiagram, with lines separating different chargestates of the double dot. Below, we focus on thevertical line (red), which marks a transition be-tween two states that have the same total charge.Along this line, the states with an electron in theright well and in the left well are in resonance.Notably, in our measurement, this line is widerthan the others, signifying that it is broadenedby quantum mechanical tunneling rather thanby temperature.We can image the redistribution of charge

density in the tunneling process by probing theresponse of this internal-tunneling line to ascanning-probe perturbation. As the probe isscanned along the system, it changes the tun-neling resonance condition, E10 = E01, by af-fecting either the energy of the initial state, E10,or the energy of the final state, E01. If we nowmonitor the detuning needed to keep the tun-neling in resonance for every probe position,De(xprobe), then, similarly to the imaging inFig. 2, we get a differential density, only here itis the difference in the density distribution of thetwo states between which the electron tunnels,r10(xprobe) − r01(xprobe), which we term the tun-neling differential density. For one electron ina double well, this difference should have a

simple dipolar structure (Fig. 4C), with a peakand a dip at the locations of the electron in theinitial and final states. The measured De(xprobe)(Fig. 4D) indeed shows this structure.The tunneling differential density becomes

more interesting in a system with more than oneelectron, because then it can display direct fin-gerprints of collective motion. Consider a three-electron state, where the central electron tunnelsthrough the barrier. In a naïve picture, if thechange in the position of the central electrondoes not affect the positions of the other elec-trons, the tunneling differential density will looksimilar to that of the one-electron case, with asingle “dipole” capturing the shift of this electrononly (Fig. 4F, left). However, if there are corre-lations between the electrons, one would expectthat as part of the tunneling process, as the posi-tion of one electron changes, the coordinates ofthe other electrons will change simultaneously.The collective nature of this tunneling should becaptured by the tunneling differential density,as this quantity records how many electronic co-ordinates shift when tunneling between groundstates. If several coordinates change simultane-ously, the differential density should display mul-tiple dipoles (Fig. 4F, right).Figure 4G shows the measurement for three

electrons. Here, too, the measured charge stabil-ity diagram exhibits a wider, tunneling-dominated,internal transition line (top inset). The imagedtunneling differential density, De(xprobe) (mainpanel), clearly exhibits three dipoles—a largecentral dipole and two smaller side dipoles, di-rectly demonstrating the collective motion of allthree electrons in the tunneling process. Fromthe relativemagnitude of the dipoles, we find thatthe central electron has moved ~500 nm and theside electrons have moved ~50 nm. The measure-ment is well captured by DMRG calculation ofthis many-body tunneling (Fig. 4G, bottom in-set), showing that, in the collective tunneling,each electron indeedmoves a fraction of its zero-point motion.Given the ability to directly image the spatial

ordering of interacting electrons, it should nowbe possible to address additional basic questionsrelated to the quantum electronic crystal, forexample, the nature of its magnetic ordering.More broadly, the scanning platform devel-oped here should allow for the exploration ofa much wider range of canonical interacting-electron states of matter, whose imaging waspreviously beyond reach.

REFERENCES AND NOTES

1. E. Wigner, Phys. Rev. 46, 1002–1011 (1934).2. C. C. Grimes, G. Adams, Phys. Rev. Lett. 42, 795–798

(1979).

3. E. E. Mendez, M. Heiblum, L. L. Chang, L. Esaki, Phys. Rev. B28, 4886–4888 (1983).

4. E. Y. Andrei et al., Phys. Rev. Lett. 60, 2765–2768 (1988).5. Y. P. Chen et al., Nat. Phys. 2, 452–455 (2006).6. Y. Chen et al., Phys. Rev. Lett. 91, 016801 (2003).7. L. Tiemann, T. D. Rhone, N. Shibata, K. Muraki, Nat. Phys. 10,

648–652 (2014).8. H. Buhmann et al., Phys. Rev. Lett. 66, 926–929 (1991).9. E. M. Goldys et al., Phys. Rev. B 46, 7957–7960 (1992).10. J. Jang, B. M. Hunt, L. N. Pfeiffer, K. W. West, R. C. Ashoori,

Nat. Phys. 13, 340–344 (2016).11. H. Deng et al., Phys. Rev. Lett. 117, 096601 (2016).12. H. J. Schulz, Phys. Rev. Lett. 71, 1864–1867 (1993).13. L. I. Glazman, I. M. Ruzin, B. I. Shklovskii, Phys. Rev. B 45,

8454–8463 (1992).14. C. Kane, L. Balents, M. Fisher, Phys. Rev. Lett. 79, 5086–5089

(1997).15. A. Secchi, M. Rontani, Phys. Rev. B 82, 035417 (2010).16. M. Roy, P. A. Maksym, Europhys. Lett. 86, 37001 (2009).17. M. Roy, P. A. Maksym, Phys. Rev. B 85, 205432 (2012).18. B. Wunsch, Phys. Rev. B 79, 235408 (2009).19. V. V. Deshpande, M. Bockrath, Nat. Phys. 4, 314–318 (2008).20. S. Pecker et al., Nat. Phys. 9, 576–581 (2013).21. L. C. Venema et al., Science 283, 52–55 (1999).22. S. G. Lemay et al., Nature 412, 617–620 (2001).23. M. Ouyang, J.-L. Huang, C. M. Lieber, Phys. Rev. Lett. 88,

066804 (2002).24. M. T. Woodside, P. L. McEuen, Science 296, 1098–1101

(2002).25. B. J. LeRoy, S. G. Lemay, J. Kong, C. Dekker, Appl. Phys. Lett.

84, 4280–4282 (2004).26. A. C. Bleszynski et al., Nano Lett. 7, 2559–2562 (2007).27. A. Hamo et al., Nature 535, 395–400 (2016).28. J. Waissman et al., Nat. Nanotechnol. 8, 569–574 (2013).29. See supplementary materials.30. E. E. Boyd, R. M. Westervelt, Phys. Rev. B 84, 205308

(2011).31. J. Qian, B. I. Halperin, E. J. Heller, Phys. Rev. B 81, 125323 (2010).32. Ö. Legeza, J. Sólyom, Phys. Rev. B 68, 195116 (2003).33. I. Shapir, A. Hamo, S. Pecker, C. P. Moca, Ö. Legeza,

G. Zarand, S. Ilani, Replication Data for: “Imaging the ElectronicWigner Crystal in One Dimension,” Harvard Dataverse,Version 1 (2019).

34. Ö. Legeza, Wigner Crystal DMRG, Zenodo, Version 1 (2019);http://doi.org/10.5281/zenodo.2648439.

ACKNOWLEDGMENTS

We thank A. Stern and E. Berg for the stimulating discussions,J. Waissman for fabricating the devices, and D. Mahalu for thee-beam writing. Funding: C.P.M. acknowledges support fromthe UEFISCDI Romanian Grant PN-III-P4-ID-PCE-2016-0032. O.L.and G.Z. acknowledge support from the NKFIH (grants K120569and SNN118028) and the Hungarian Quantum Technology NationalExcellence Program (grant 2017-1.2.1-NKP-2017-00001). S.I.acknowledges financial support by the ERC Cog grant (See-1D-Qmatter, 647413). Author contributions: I.S. and A.H.performed the experiments. I.S. and S.P. developed theexperimental setup. I.S., A.H., S.P., and S.I. developed themeasurement technique. C.P.M., O.L., and G.Z. developed andperformed the theoretical simulations. I.S. and S.I. wrote thepaper. Competing interests: The authors declare no competinginterests. Data and materials availability: Data reported in thispaper are archived online (33, 34).

SUPPLEMENTARY MATERIALS

science.sciencemag.org/content/364/6443/870/suppl/DC1Supplementary TextFigs. S1 to S12References (35–38)

15 February 2018; resubmitted 5 August 2018Accepted 1 May 201910.1126/science.aat0905

Shapir et al., Science 364, 870–875 (2019) 31 May 2019 5 of 5

RESEARCH | REPORTon N

ovember 1, 2020

http://science.sciencem

ag.org/D

ownloaded from

Imaging the electronic Wigner crystal in one dimensionI. Shapir, A. Hamo, S. Pecker, C. P. Moca, Ö. Legeza, G. Zarand and S. Ilani

DOI: 10.1126/science.aat0905 (6443), 870-875.364Science 

, this issue p. 870Sciencedimension.electronic densities were consistent with theoretical predictions for small Wigner crystals of up to six electrons in onewhich housed the electrons, and a second nanotube that scanned the first nanotube, serving as a probe. The measured

created such conditions in a carbon nanotube,et al.and very noninvasive probes. Shapir −−temperatures and densities low−−called the Wigner crystal. Observing this fragile crystal is tricky because it requires extreme conditions

Electrons repel each other. When confined to a small space, electrons can form an ordered, crystalline stateVisualizing a tiny crystal

ARTICLE TOOLS http://science.sciencemag.org/content/364/6443/870

MATERIALSSUPPLEMENTARY http://science.sciencemag.org/content/suppl/2019/05/29/364.6443.870.DC1

REFERENCES

http://science.sciencemag.org/content/364/6443/870#BIBLThis article cites 37 articles, 2 of which you can access for free

PERMISSIONS http://www.sciencemag.org/help/reprints-and-permissions

Terms of ServiceUse of this article is subject to the

is a registered trademark of AAAS.ScienceScience, 1200 New York Avenue NW, Washington, DC 20005. The title (print ISSN 0036-8075; online ISSN 1095-9203) is published by the American Association for the Advancement ofScience

Science. No claim to original U.S. Government WorksCopyright © 2019 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of

on Novem

ber 1, 2020

http://science.sciencemag.org/

Dow

nloaded from