Abstract
Although the unit of charge in nature is a fundamental constant, the charge of individual quasiparticles in some low-dimensional systems may be fractionalized. Quantum one-dimensional (1D) systems, for instance, are theoretically predicted to carry charge in units smaller than the electron charge e. Unlike 2D systems, the charge of these excitations is not quantized and depends directly on the strength of the Coulomb interactions. For example, in a 1D system with momentum conservation, it is predicted that the charge of a unidirectional electron that is injected into the wire decomposes into right- and left-moving charge excitations carrying fractional charges f0e and (1−f0)e respectively1,2. f0 approaches unity for non-interacting electrons and is less than one for repulsive interactions. Here, we provide the first experimental evidence for charge fractionalization in one dimension. Unidirectional electrons are injected at the bulk of a wire and the imbalance in the currents detected at two drains on opposite sides of the injection region is used to determine f0. Our results elucidate further3,4 the collective nature of electrons in one dimension.
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Main
Charge fractionalization in one dimension is already predicted for the spinless Luttinger model1,2. The charge fraction f0 is given by
where gc is the Luttinger-liquid interaction parameter. For a galilean invariant system, gc=vF/vc, where vF is the bare Fermi velocity and vc is the velocity of charge excitations. Roughly, gc≈(1+U/2ɛF)−1/2, where U is the Coulomb interaction energy and ɛF is the Fermi energy. In spinfull one-dimensional (1D) systems, charge fractionalization occurs in addition to spin–charge separation, which has been recently confirmed by spectroscopy and tunnelling experiments4,5,6.
Observing interaction effects in 1D systems using transport experiments is a considerable challenge. For example, the d.c. two-terminal conductance with ideal contacts is universal and given by G=G0≡2e2/h, independent of interactions7,8,9,10,11. Shot-noise measurements have been useful in detecting fractional charge in 2D systems12,13,14. However, low-frequency shot-noise measurements in an ideal wire would only reveal the physics of the Fermi-liquid contacts, remaining insensitive to fractionalization15. Although both noise and conductance should reveal interaction effects at frequencies exceeding vF/gcL∼1010 Hz, where the excitation wavelength is shorter than the wire segment16,17,18, these frequencies are difficult to explore experimentally at low temperatures.
Initial experimental indication of electron fractionalization in one dimension is provided by angle-resolved photo-emission spectroscopy measurements on stripe-ordered cuprate materials5. Recent theoretical studies have proposed transport experiments aimed at detecting the same physics in quantum wires. Generally, these involve the realization of multi-terminal geometries, including: (1) local injection of electrons into a wire, where high-frequency noise correlations are expected18; (2) a four-probe geometry, measuring voltage shot noise due to an impurity19; (3) studying the d.c. I(V) curves in the presence of a bulk contact2; and (4) measuring the suppression of Aharonov–Bohm interference between two weakly coupled wires20.
Here, we realize another multi-terminal geometry: we use a parallel-wire system previously applied to the study of spin–charge separation3,4 and localization in one dimension21. Using momentum conservation in the tunnelling process between the two wires, we inject unidirectional electrons into the bulk of a wire, with fractionalization resulting in currents detected on both sides of the injection region. The current asymmetry AS≡(IR−IL)/(IR+IL), where IR(L) is the current detected on the right (left), depends on the extent of fractionalization, but also on details of the coupling at the drains. We evaluate the effect of drain coupling by measuring the two-terminal conductance G2T between the right and left drains, finding that G2T/G0=AS. We argue, on the basis of a theoretical model, that this observed equality proves that the tunnelling electron charge fractionalizes as predicted by theory.
The double quantum-wire sample is shown in Fig. 1a. Details about the sample and measurement technique are given in the Supplementary Information. The three-terminal geometry shown in the figure requires biasing two gates (G1, G2). The gates deplete the upper wire and leave the lower wire continuous, defining a finite central junction of length LS=10–40 μm, and a semi-infinite junction on each side. The short junction serves as the source and the long junctions 1 and 3 as drains. We measure the differential tunnelling conductance ∂ IT/∂ VSD between the source and each drain using standard lock-in techniques. Typically dVSD is 14 μV and the frequency is a few hertz. The measurements are carried out in a 3He refrigerator at T=0.25 K.
It is instructive to describe tunnelling between the wires in terms of their energy–momentum dispersions. Magnetic field B has the effect of shifting both dispersions relative to each other by ℏqB=e B d along the momentum axis, as seen in Fig. 2a, which depicts their relative positions for various magnetic fields. For VSD=0, both dispersions have the same electrochemical potential. Typically at B=0 the two 1D dispersions do not overlap (i), as the wires have different densities, and tunnelling is suppressed. Applying a field B− shifts the dispersions to overlap near one of the Fermi points (ii), enabling tunnelling between co-propagating electrons. The high field crossing of the two dispersions, where electrons tunnel between counter-propagating modes is marked as B+ (iv,v) where B±=ℏ|kFUW±kFLW| (with kFUW(LW) being the upper (lower) wire Fermi momenta). A key feature of this measurement geometry is that at B=B±, electrons with a well-defined momentum state, near the Fermi point, are added to the lower wire. Typically each wire is populated by several sub-bands, but tunnelling between different sub-bands is suppressed by near-orthogonality3, so that each sub-band j contributes a single pair of peaks, denoted Bj±.
In Fig. 2b, the right and left differential conductances ∂ IR/∂ VSD and ∂ IL/∂ VSD for LS=10 μm are plotted versus B. As B is scanned from zero to positive fields, small, sharp peaks corresponding to co-propagating electrons tunnelling between sub-bands j=1,2 are first encountered at B1,2−. The wide feature at 1 T<B<3 T results from tunnelling between the populated 2D electron gas (2DEG) in the upper quantum well and the lower wire (Fig. 2a(iii)). Further increasing the field, we finally encounter the B+ feature at B=6.5 T. To check whether the original directionality of the injected electrons is conserved, we now compare the conductance at both drains. Pronounced directional asymmetry emerges near the B+ feature: at B=+B+ left-moving electrons are injected into the lower wire (Fig. 2a(iv)), and the current detected at the left drain is indeed significantly stronger than on the right. Exactly opposite values appear at B=−B+, attesting to the geometric symmetry of the sample. A similar effect appears near the B− features, but is more difficult to observe as the differential conductance is small relative to a strong background signal.
The effect observed near −B+ is quantified by an asymmetry parameter AS≡(IR−IL)/(IR+IL); the deviation from perfect asymmetry cannot a priori be attributed to fractionalization as the wire is not perfect, and microscopic effects such as backscattering can suppress AS by distributing the charge evenly in the wire before it has the chance to be detected. Moreover, complicated processes associated with coupling of the interacting wire with the non-interacting leads also take place in the drains and may lead to a smaller overall asymmetry. To isolate fractionalization physics from such microscopic effects, we measure the two-terminal conductance G2T between the left and right drains. G2T, being independently sensitive to these microscopic processes, enables us to extract the extent of fractionalization from AS. To demonstrate this we use a model (the geometry of which is shown in Fig. 1b) assuming a single sub-band is occupied in the lower wire. The model is based on Luttinger-liquid theory as in our system gc≈0.4–0.5 (see the Supplementary Information for the method of evaluating gc and a discussion justifying the use of a single-sub-band model).
The chemical potentials V1,3 in the upper wire (subscript indicates junction) are set by the 2DEG that couples to both right- and left-moving upper-wire modes22. According to Luttinger-liquid theory, we may define separate chemical potentials for right- and left-moving lower-wire charge modes on each side of the central junction, denoted y1,3, w1,3 respectively. These are defined so that the current (right-moving) at any point is given by I=gcG0(y−w), whereas w+y is determined by the charge density, such that in equilibrium, where I=0, y and w are equal to the electron chemical potential V. With these definitions, we find that a right-moving charge excitation will affect the local value of y, but leave w unchanged, and vice versa. At the two end contacts, if the currents are small enough so that linear response applies, we must have y1=β V1+γ w1,w3=β V3+γ y3, where the parameters β and γ must satisfy β+γ=1 so that y=w=V in equilibrium. We have assumed here that the two junctions have identical parameters. We note without limiting the generality of the model, that β accounts for both backscattering occurring in the lower wire as well as any process associated with the coupling of the lower wire to the Fermi-liquid leads.
For the two-terminal conductance, no current enters the lower wire at the centre contact, so if there is no scattering there, we must have w1=w3 and y1=y3, and we compute G2T=I/(V3−V1). This yields G2T=gcG0β/(2−β). We note that the gc dependence is a consequence of the boundary conditions at equilibrium. We also note that theoretically β<1/(2−gc) and correspondingly G2T≤G0. For the three-terminal conductance, we set V1=V3=0, and require that IR+IL=IS, where IS is the injected current, IR=gcG0(y3−w3) is the net right-moving current in the right half of the wire and IL=gcG0(w1−y1) is the left-moving current in the left half of the wire.
At B=−B+, where the centre contact injects right-moving electrons, if we assume that an (unknown) fraction f of each electron charge travels to the right, this means that gcG0(y3−y1)=ISf and gcG0(w1−w3)=IS(1−f).
Solving for AS=(IR−IL)/(IR+IL), we obtain:
and hence the ratio:
at least for a single-mode wire. This is a fundamental result: it implies that regardless of the microscopic details, it is sufficient to measure G2T, AS and gc under the same conditions to extract f.
We therefore proceed to measure G2T between contacts O1 and O3 (Fig. 1a). This requires tuning the voltage of a single gate (G2 or G1 in the figure) to deplete the upper wire. As coupling to the lower wire is via tunnelling, G2T depends on B, and should be maximal when B=B+. G2T versus VG scans are shown in Fig. 3a, showing line scans taken at different magnetic fields close to B+. For VG=−1.58 V, conductance through the upper wire is suppressed. For −1.76<VG<−1.58 V, current flows through the lower wire, coupled by junctions 1, 3. For each line scan, G2T(B) is taken as the maximal value in this range and is shown as a circle in Fig. 3b. The result seems to fit a lorentzian, possibly related to variations of the density or of the tunnelling rate along the wire.
We study the ratio G2T/AS by comparing G2T(B+) and AS(B+) at different wire densities. Density is controlled either by biasing a side gate, evaporated over the cleave plane, or by shining infrared light on the sample. The results are plotted in Fig. 4. In the observed density range, G2T seems to depend linearly on nL, the line intercepting G2T(nL)=0 at a finite density of nL=20 μm−1. The mechanism underlying this dependence is not understood. Superimposing the asymmetry results on this plot yields the key experimental finding of this work: AS(nL) lies on top of the G2T(nL) line, showing that G2T=AS in all of these cases. Only few AS data points are presented because this measurement imposes a stringent requirement: the density distribution along the sample has to be very smooth, so that the density in the source junction would be identical to the densities in both drain junctions. This implies that the maximal tunnelling rate appearing at B=B+ will occur at the same magnetic field everywhere. The inset to Fig. 4 shows that G2T=AS even when the density distribution is not perfectly smooth: in these cases, the B+ feature appearing in the asymmetry scan reflects the local density at the source junction, marked BS+, and the B+ feature of the two-terminal measurement reflects density in larger regions at the drains. The AS result is therefore taken at BS+ and should be compared with G2T at the same field. This is done by superimposing a data point AS(BS+) on a plot of G2T(B), taken at the same conditions. To present compiled data from different scans, where the height and width of the G2T(B) lorentzian are different, we normalize each lorentzian to unity height and width, and apply the same transformation to the respective AS(BS+) data point, verifying that AS(BS+)=G2T(BS+).
The compiled measurements presented in Fig. 4 show that the result G2T=AS is robust: it holds for different samples, at different densities, and even when the sample has an uneven density distribution. Returning now to equation (1), we have (2f−1)=gc, confirming that the fractionalization factor is f=(gc+1)/2, as theoretically predicted by Luttinger-liquid theory1 for an ideal momentum-conserving directional injector. As 0.4<gc<0.5, we conclude that the fractionalization ratios are in the range of 0.7<f<0.75.
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Acknowledgements
We would like to acknowledge useful discussions with Y. Oreg. This work is partly supported by the Israeli US Bi-national Science Foundation and by the US National Science Foundation under contract No. DMR-0707484. H.S. is supported by a grant from the Israeli Ministry of Science.
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The experiments were carried out by H.S., G.B. and A.Y.; L.N.P. and K.W.W. provided the samples, and the theoretical model was developed by B.I.H. and K.L.
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Steinberg, H., Barak, G., Yacoby, A. et al. Charge fractionalization in quantum wires. Nature Phys 4, 116–119 (2008). https://doi.org/10.1038/nphys810
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DOI: https://doi.org/10.1038/nphys810
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