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Imaging the electronic Wigner crystal in one dimension

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Science  31 May 2019:
Vol. 364, Issue 6443, pp. 870-875
DOI: 10.1126/science.aat0905

Visualizing a tiny crystal

Electrons repel each other. When confined to a small space, electrons can form an ordered, crystalline state called the Wigner crystal. Observing this fragile crystal is tricky because it requires extreme conditions—low temperatures and densities—and very noninvasive probes. Shapir et al. created such conditions in a carbon nanotube, which housed the electrons, and a second nanotube that scanned the first nanotube, serving as a probe. The measured electronic densities were consistent with theoretical predictions for small Wigner crystals of up to six electrons in one dimension.

Science, this issue p. 870

Abstract

The quantum crystal of electrons, predicted more than 80 years ago by Eugene Wigner, remains one of the most elusive states of matter. In this study, we observed the 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 carbon nanotube, we used another nanotube as a scanning-charge perturbation. The images we obtained of a few electrons confined in one dimension match the theoretical predictions for strongly interacting crystals. The quantum nature of the crystal emerges in the observed collective tunneling through a potential barrier. These experiments provide the direct evidence for the formation of small Wigner crystals and open the way for studying other fragile interacting states by imaging their many-body density in real space.

Eugene Wigner predicted (1) that when long-range Coulomb interactions in a system of electrons dominate over kinetic energy and disorder, a crystalline ground state should emerge in which the electrons are kept apart irrespective of their internal quantum number (flavor). Experimental searches for this quantum crystal have been conducted in the cleanest available electronic systems, such as the surface of liquid helium (2) and low-dimensional semiconductor heterostructures. In semiconducting two-dimensional electronic systems, transport (3, 4), microwave (5, 6), nuclear magnetic resonance (7), optical (8, 9), tunneling (10), and bilayer correlation (11) measurements have indicated the existence of a crystal at high magnetic fields. In one dimension, thermal and quantum fluctuations destroy the long-range order, and a crystalline state in an infinite system is not expected. However, in finite systems, quasi-long-range order produces crystalline correlations, and this one-dimensional Wigner crystal state has been extensively studied theoretically (1218) and probed experimentally via transport measurements (19, 20). However, these experiments probed only macroscopic properties of this state.

The unambiguous fingerprint of a Wigner crystal lies in its real-space structure, which could in principle be observed with a suitable imaging tool. Scanning probe experiments (2126) have so far been able to image only the noninteracting state or have showed invasive gating by the probe. These measurements highlight the inherent difficulty of imaging interacting electrons with conventional scanning methods: To resolve individual electrons, a macroscopic, metallic, or dielectric tip should approach the electrons closer than their mutual separation, which inevitably screens their interactions, which are at the heart of the interacting state. Moreover, macroscopic tips generically carry uncontrollable charges that have strong gating effects on an interacting electron system, strongly distorting the state being studied. To image an interacting state, a different kind of scanning probe is therefore needed.

In this study, we use a scanning probe platform that utilizes a carbon nanotube (NT) as a highly sensitive, yet minimally invasive scanning probe for imaging the many-body density of strongly interacting electrons. Our platform comprises a custom-made scan probe microscope, operating at cryogenic temperatures (~10 mK), in which two oppositely facing NT devices can be brought in close proximity (~100 nm) (27) and scanned along each other (Fig. 1A). One device hosts the system NT, which is used as the one-dimensional system under study (Fig. 1A, bottom). The second device contains the probe NT, which is perpendicular to the system NT and can be scanned along it (Fig. 1A, top). The two devices are assembled using a nanoassembly technique (28) that yields pristine NTs suspended above an array of metallic gates. In the system NT, it is essential to maintain strong interactions and low disorder, both of which are crucial for obtaining a Wigner crystal; this is achieved by suspending the NT far above the metallic gates (400 nm). Using 10 electrically independent gates, we design a potential that confines the electrons between two barriers, ~1 μm apart, localizing them to the central part of a long suspended nanotube (length: 2.3 μm) (Fig. 1B), away from the contacts that produce undesirable distortions (e.g., screening, image charges, disorder, and band bending) [see section 1 in the supplementary materials (29)]. We use highly opaque barriers that prevent hybridization of the confined electron’s wave function with those of the electrons in the rest of the NT. Because transport in this situation is highly suppressed, we probe the confined electrons using a charge detector located on a separate segment of the same NT (Fig. 1B, purple). The addition of these electrons is detected as a small change in the detector’s current, ICD, flowing between the two outer contacts of the device (Fig. 1B, blue arrow) but not through the central segment of the system NT. The probe NT device has an almost identical structure, differing only by the NT suspension length (1.6 μm) and number of gates (three). Because the probe gates and contacts are perpendicular to the system NT, the potential they induce remains translationally invariant as the probe is scanned along the system, and the only moving perturbation comes from the moving NT itself [section 2 in (29)].

Fig. 1 Experimental platform for imaging strongly interacting electrons.

(A) Scanning probe setup consisting of two carbon nanotube (NT) devices—a system-NT device (bottom) that hosts the electrons to be imaged (green ellipse) and a probe-NT device (top) containing the probing electrons (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 the middle part of the suspended NT (green), away from the contacts. The addition of these electrons is detected using a charge detector—a separate quantum dot formed on a side segment of the same NT (purple). The detector is biased by a voltage, VCD, applied on an external contact, leading to a current, ICD, flowing only between the contacts of the charge detector (blue arrow), such that no current passes through the main part of the system NT.

To demonstrate the basic principle behind our imaging technique, which we term “scanning charge,” we start with the simplest experiments—imaging the charge distribution of a single electron confined to a one-dimensional box (Fig. 2A). By measuring the energetic response of the system to a scanned perturbation, we can directly determine the system’s density distribution. For simplicity, we assume first that the perturbation produced by the probe NT is highly localized at its position, xprobe, V(x) ≈ Vδ(xxprobe). To the lowest order, such a perturbation will shift the system’s energy as E1(xprobe)=ψ1|V(xxprobe)|ψ1ρ1(xprobe)(1)

Fig. 2 Real-space imaging of the density profile of a single confined electron.

(A) To image the density distribution of a single electron confined in a potential “box” (gray), we place a fixed charge in the probe NT and scan it across the system NT. This charge creates a local perturbation at the probe position xprobe (red), which shifts the ground state energy of the system electron, E1 (top panels), proportional to the local density at the probe position E1(xprobe) ~ ρ1(xprobe) (see text). By measuring the global gate voltage, Vg, needed to keep the charging of this single electron in resonance with the Fermi energy of the leads, EF, for varying xprobe (bottom panels), we effectively trace the profile of its charge distribution Vg(xprobe) ~ ρ1(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 first electron entering the system-NT potential well (in Figs. 2 and 3, the green and red 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 of the electron convolved with the point spread function of the probe. (Insets) Illustration of the system and probe devices for different measurement positions. (D) Same as in (C), but for different probe charges from qprobe = 0e to 3e. (E) The traces extracted from panel (D), plotted together.

where ρ1(x)=|ψ1(x)|2 (Fig. 2A, green) is the density distribution of the confined electron wave function, ψ1(x). Thus, by measuring E1 as a function of xprobe, the electron’s density profile can be directly resolved (30, 31). The energy E1 is measured by referencing it to the Fermi energy in the leads (EF ≡ 0) (Fig. 2A, bottom). Starting with the level in resonance with EF, a movement of the probe to xprobe will shift the energy level to E1(xprobe) and typically away from resonance with EF. We then bring the level back to resonance by applying a global gate voltage, Vg, formed by a proper linear combination of the ten gates, chosen to produce a rigid shift of the potential without changing its shape [section 1 in (29)]. The applied Vg, measured in energy units, is equal to the energy shift E1. Thus, by monitoring the gate voltage required to keep the level in resonance for varying xprobe, we directly image the charge distribution, Vg(xprobe) ~ ρ1(xprobe). Note that in reality, the perturbation produced by our probe is not a delta function but has a spatial extent determined by the separation between the two NTs and the spatial extent of the confined charge in the probe NT. The measured profile will therefore be a convolution of the corresponding point spread function and the imaged charge density distribution. Throughout this Report, to avoid confusion in describing the physics of the electronic Wigner crystal, we use the language of electrons, although, owing to technical advantages, the actual measurements were done with holes in the system and probe [section 3 in (29)].

The imaging of the charge distribution of a single electron is shown in Fig. 2, B and C. The population of the first confined electron in the system (qsystem = 1e) is identified by its sharp charging peak, observed when we measure the charge detector signal, dICD/dVg, a function of Vg [Fig. 2B and section 3 in (29)]. To image the charge distribution of this electron, we also place one electron in the probe NT (qprobe = 1e), scan it along the system NT, and monitor the corresponding shift of the charging peak in gate voltage. This measurement (Fig. 2C) reveals that the charging peak shifts smoothly when xprobe traverses between the two edges of the confinement well, peaking at its center. This trace directly yields the real-space charge distribution of the first confined electron, convolved with the point spread function of the probe [section 5 in (29)].

An essential test for the technique is to assess how the measured energy shifts scale with the strength of the scanned perturbation, which we can control down to the single-electron limit. Figure 2D depicts imaging measurements done with qprobe = 0e to 3e, showing that the energy shifts increase monotonically with increasing number of probing electrons. The overall shape of the imaged charge density remains similar, but for larger qprobe, the peak becomes slightly sharper, reflecting an increasing probe invasiveness. For qprobe = 1e, the probe causes a movement of the electron in the system that is approximately an order of magnitude smaller than the zero-point motion of this electron, indicating that the probe is in the noninvasive limit [section 4 in (29)]. In principle, our probe could have also had uncontrolled charges caused by localized states in the NT or imperfection in the metals, which can create an even larger scanning perturbation than the single charges that we place intentionally. However, our measurement with qprobe = 0e displays an energy shift an order of magnitude smaller than that with qprobe = 1e (Fig. 2E), demonstrating that in our experiment, the spurious charges are 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 the charging resonance of the Nth electron (N > 1) (Fig. 3A). This resonance occurs when the states with N and N − 1 electrons are energetically degenerate, EN = EN−1 (equivalently, μENEN1=0). The probe perturbation can modify both EN and EN−1, shifting the resonance in gate voltage as

Fig. 3 Imaging the differential density of many-electron states.

(A) In a charging transition from N – 1 to N electrons, the resonance occurs for EN = EN–1 and the gate voltage shift images the differential density Vg(xprobe) ~ ρN(xprobe) − ρN – 1(xprobe). (B) Illustration of the expected differential density of noninteracting (left) versus strongly interacting (right) electrons in a carbon NT. These sketches also include the finite resolution smearing [section 5 in (29)]. Noninteracting electrons occupy the particle-in-a-box wave functions, each being fourfold degenerate because of the spin and valley degeneracy (red and blue arrows). Consequently, the differential density of the first four electrons should be identical and single-peaked, and that of the next four should be double-peaked. For the strongly interacting case, the electrons separate in real space (bottom right), and each added electron will add one more peak to the differential density profile (top right). (C) Measurement of ICD as a function of Vg and xprobe, around the charging peaks of the first six electrons in the system. The curves directly trace the differential density of these many-electron states, showing that they are deep in the strongly interacting regime. (D) The differential density of the first six electrons, calculated with DMRG, which takes into account long-range electronic interactions [section 7 in (29)], as a function of the spatial coordinate x/ld and the effective strength of electronic interactions, r˜s, ranging from very weak (r˜s=0.01) to very strong (r˜s=100). Green stars mark the positions of the peaks measured in the experiment, and the green lines mark the calculated positions (with a single free parameter ld = 160 nm) [section 7 in (29)].

Vg(xprobe)=EN(xprobe)EN1(xprobe)ρN(xprobe)ρN1(xprobe)(2)

The trace Vg(xprobe) now images the density of the N electron state, ρN(x), minus that of the N − 1 electron state, ρN−1(x). This quantity, which we term the differential density, is intuitively understood in the single-particle picture—it is merely the density added by the last electron to enter the system. Note that Eq. 2 also holds for the 1e case, because in that case E0 ≡ 0, and thus Vg(xprobe) ∝ ρ1(xprobe).

When electrons have multiple internal “flavors” (in our case four, spin and valley), their differential density profiles should be markedly different in the noninteracting and the strongly interacting cases. In the absence of interactions, electrons populate the particle-in-a-box states of the potential, with a degeneracy given by the number of flavors. In NTs, this would be a fourfold degeneracy owing to the spin and valley degrees of freedom. Thus, the spatial distribution of the density added by each one of the first four electrons (their differential density) would be identical, given by that of the first particle-in-a-box state. The density profiles associated with the next four electrons will also look identical to one another, having the two spatial peaks of the second particle-in-a-box state, and so on (Fig. 3B, left). A completely different picture emerges in the presence of strong interactions: Because Coulomb interactions are flavor-independent, all electrons keep apart, independent of their flavor. Thus with any additional electron, one more peak will be added to the differential density profile (Fig. 3B, right). Imaging the differential density of “flavored” electrons should therefore make a clear distinction between these two regimes.

The six panels in Fig. 3C correspond, respectively, to the first six electrons added to the system NT. To keep the perturbation minimal, all these scans are performed with one electron in the probe NT. Unlike in Fig. 2C, here we plot the charge detector current, ICD, rather than its derivative, which shows a step rather than a peak when an electron is added to the system NT. A clear trend can be observed in the imaged differential density profiles—with every added electron, one more peak appears in the differential density. These profiles are clearly different than those predicted by single-particle physics, but they match those of a strongly interacting crystal. With an increasing number of electrons, we see that the electron spacing is reduced but also that their overall spread increases, signifying that the electrons are confined to a “box” with soft walls. The slight deviation from perfect periodicity stems from nonideality of the potential [section 6 in (29)]. These images give the direct real-space observation of electronic Wigner crystals.

To obtain a quantitative understanding of our measurements, we perform density matrix renormalization group (DMRG) calculations that include long-range Coulomb interactions (32). We use a quartic confinement potential, V(x)=14Ax4, that nicely approximates the experimental confinement potential (fig. S1B). Figure 3D shows 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 this potential), and interaction strength, r˜Sa/aB, which estimates the ratio of the average electron spacing and the Bohr radius [section 7 in (29)]. The measured electron positions (Fig. 3D, green stars) agree well with those predicted by DMRG, showing that r˜S ranges from 50 for two electrons to 20 for six electrons (fig. S12A), placing the observed crystals well within the strongly interacting regime. The comparison further shows that the zero-point motion of the electrons is ~2 to 3 times smaller than their distance (Lindemann ratio) and that the exchange at these large electronic spacings is considerably smaller 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 between the electrons in a crystal will cause the crystal to tunnel through a barrier collectively (13). We create a tunneling barrier at the center of the confining potential box using the gates, effectively creating a double-well potential (Fig. 4A). To understand the measurement, we start with the simple case of one electron in the double well. Figure 4B shows the charge stability diagram of the system, measured using the charge detector current, dICD/dε, as a function of the common voltage on the two wells, Vg, and their detuning voltage, ε. This is the familiar double-dot stability diagram, with lines separating different charge states of the double dot. Below, we focus on the vertical line (red), which marks a transition between two states that have the same total charge. Along this line, the states with an electron in the right well and in the left well are in resonance. Notably, in our measurement, this line is wider than the others, signifying that it is broadened by quantum mechanical tunneling rather than by temperature.

Fig. 4 Many-body tunneling of the few-electron state.

(A) Illustration of the potential landscape, which now includes a central barrier through which an electron can tunnel (red arrow). The detuning voltage, ε, changes the relative height of the bottom of each well. (B) Charge stability diagram for 1e as a function of Vg and ε, measured using dICD/dε (color bar). The states (N, M) denote the charge N (M) in the left (right) wells. The vertical, wider line corresponds to an internal tunneling, occurring when EN+1,M = EN,M+1. (C) Schematic of the expected tunneling differential density for one electron (red “dipole”, bottom), given by the difference between its density distribution before and after tunneling [ρ10(x) and ρ01(x)] convolved with the probe’s point spread function (PSF). (D) Measured charge detector signal as a function xprobe and the difference in detuning relative to the unperturbed state, Δε. The red trace shows the Δε(xprobe) necessary to keep the tunneling in resonance (shown schematically in inset), giving the tunneling differential density. (E) Same as (A), but for three electrons in the trap. (F) Two scenarios for the tunneling: (Left) Only the central electron moves in the tunneling event; Δε(xprobe) will show a single dipole, as in the one-electron case illustrated in (C). (Right) Many-body tunneling, in which the coordinates of all the electrons move coherently in the tunneling process; multiple dipoles are expected in the differential tunneling signal. (G) (Top inset) Charge-stability diagram of three electrons, with ICD/dε (a.u.) measured for −42 mV < ε < 10 mV, 170 mV < Vg < 130 mV. (Main panel) Measured Δε(xprobe) for three electrons, exhibiting multiple dipoles. (Bottom inset) DMRG-calculated many-body density distributions [section 7 in (29)] of three electrons that the electrons tunnel between (blue and red) and their difference, convolved with the probe’s PSF (blue, bottom), showing that all electrons move a fraction of their zero-point motion as part of the many-body tunneling.

We can image the redistribution of charge density in the tunneling process by probing the response of this internal-tunneling line to a scanning-probe perturbation. As the probe is scanned along the system, it changes the tunneling resonance condition, E10 = E01, by affecting either the energy of the initial state, E10, or the energy of the final state, E01. If we now monitor the detuning needed to keep the tunneling in resonance for every probe position, Δε(xprobe), then, similarly to the imaging in Fig. 2, we get a differential density, only here it is the difference in the density distribution of the two states between which the electron tunnels, ρ10(xprobe) − ρ01(xprobe), which we term the tunneling differential density. For one electron in a double well, this difference should have a simple dipolar structure (Fig. 4C), with a peak and a dip at the locations of the electron in the initial and final states. The measured Δε(xprobe) (Fig. 4D) indeed shows this structure.

The tunneling differential density becomes more interesting in a system with more than one electron, because then it can display direct fingerprints of collective motion. Consider a three-electron state, where the central electron tunnels through the barrier. In a naïve picture, if the change in the position of the central electron does not affect the positions of the other electrons, the tunneling differential density will look similar to that of the one-electron case, with a single “dipole” capturing the shift of this electron only (Fig. 4F, left). However, if there are correlations between the electrons, one would expect that as part of the tunneling process, as the position of one electron changes, the coordinates of the other electrons will change simultaneously. The collective nature of this tunneling should be captured by the tunneling differential density, as this quantity records how many electronic coordinates shift when tunneling between ground states. If several coordinates change simultaneously, the differential density should display multiple dipoles (Fig. 4F, right).

Figure 4G shows the measurement for three electrons. Here, too, the measured charge stability diagram exhibits a wider, tunneling-dominated, internal transition line (top inset). The imaged tunneling differential density, Δε(xprobe) (main panel), clearly exhibits three dipoles—a large central dipole and two smaller side dipoles, directly demonstrating the collective motion of all three electrons in the tunneling process. From the relative magnitude of the dipoles, we find that the central electron has moved ~500 nm and the side electrons have moved ~50 nm. The measurement is well captured by DMRG calculation of this many-body tunneling (Fig. 4G, bottom inset), showing that, in the collective tunneling, each electron indeed moves a fraction of its zero-point motion.

Given the ability to directly image the spatial ordering of interacting electrons, it should now be possible to address additional basic questions related to the quantum electronic crystal, for example, the nature of its magnetic ordering. More broadly, the scanning platform developed here should allow for the exploration of a much wider range of canonical interacting-electron states of matter, whose imaging was previously beyond reach.

Supplementary Materials

science.sciencemag.org/content/364/6443/870/suppl/DC1

Supplementary Text

Figs. S1 to S12

References (3538)

References and Notes

  1. See supplementary materials.
Acknowledgments: We thank A. Stern and E. Berg for the stimulating discussions, J. Waissman for fabricating the devices, and D. Mahalu for the e-beam writing. Funding: C.P.M. acknowledges support from the UEFISCDI Romanian Grant PN-III-P4-ID-PCE-2016-0032. O.L. and G.Z. acknowledge support from the NKFIH (grants K120569 and SNN118028) and the Hungarian Quantum Technology National Excellence 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 the experimental setup. I.S., A.H., S.P., and S.I. developed the measurement technique. C.P.M., O.L., and G.Z. developed and performed the theoretical simulations. I.S. and S.I. wrote the paper. Competing interests: The authors declare no competing interests. Data and materials availability: Data reported in this paper are archived online (33, 34).

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