Abstract
Onedimensional electronic fluids are peculiar conducting systems, where the fundamental role of interactions leads to exotic, emergent phenomena, such as spincharge (spinonholon) separation. The distinct lowenergy properties of these 1D metals are successfully described within the theory of linear Luttinger liquids, but the challenging task of describing their highenergy nonlinear properties has long remained elusive. Recently, novel theoretical approaches accounting for nonlinearity have been developed, yet the rich phenomenology that they predict remains barely explored experimentally. Here, we probe the nonlinear spectral characteristics of short GaAs quantum wires by tunnelling spectroscopy, using an advanced device consisting of 6000 wires. We find evidence for the existence of an inverted (spinon) shadow band in the main region of the particle sector, one of the central predictions of the new nonlinear theories. A (holon) band with reduced effective mass is clearly visible in the particle sector at high energies.
Introduction
The electronic properties of onedimensional (1D) electron systems are fundamentally different from those of their twodimensional (2D) and threedimensional (3D) counterparts, owing to the prominent role of interactions under 1D confinement^{1,2,3,4,5,6,7,8,9}. Fermiliquid theory, applicable to normal 2D and 3D electron systems, breaks down spectacularly in the 1D case, which is better described by the theory of Tomonaga–Luttinger liquids^{10,11}. While elementary excitations of Fermi liquids behave as quasifree fermions, those of interacting electrons in 1D systems are collective bosonic modes. Out of equilibrium, electrons confined to 1D lose their individual identity, separating into two collective excitation types (quantized waves of density): spin modes (spinons) that carry spin without charge, and charge modes (holons) that carry charge without spin^{2,12}. Spinons and holons travel at different velocities, resulting in socalled spincharge separation, revealed in photoemission and tunnelling experiments^{12,13,14,15}. The spectral function A(ħ k, ħω), describing the probability for an electron with momentum ħ k and energy ħω>0 (ħω<0) to tunnel into (out of) the system, theoretically displays narrow Lorentzian singularities for normal 3D and 2D systems, whereas it displays wide asymmetric powerlaw singularities for 1D systems^{12,16,17}.
The theory of Tomonaga–Luttinger liquids, based on a linearization of the dispersion relation around the right and left Fermi points^{10,11}, has been successfully used for a long time to describe 1D electron systems in the limit of low energies but, in order to understand their behaviour at higher energies, that is, away from the Fermi points, the curvature of the dispersion relation has to be taken into account. Recently, a new nonlinear theory of 1D quantum fluids beyond the lowenergy limit has been developed^{18,19,20,21,22,23,24,25,26,27,28}. The new theory represents a giant advance towards full understanding of the behaviour of 1D electron systems, but many of its specific predictions remain barely explored experimentally.
Here, we report on the spectra of elementary excitations in short GaAs quantum wires, probed by momentum and energyresolved tunnelling spectroscopy, focusing on the nonlinear highenergy regime. We use a 1D–2D vertical tunnelling device with a large number of wires in parallel, to boost the wire signal and so maximize the chances to observe weak features of the wire spectral function. Our tunnelling conductance maps reveal, for the first time, the existence of an inverted (spinon) shadow band in the main region of the particle sector, symmetrically replicating the dispersion of the main spinon hole band, as anticipated by the nonlinear theory. They also reveal a (holon) band with reduced effective mass in the particle sector at high energies. Holons appear to be longlived in the particle sector, but shortlived in the hole sector.
Results
Vertical tunnelling device
Our device (Fig. 1) consists of a double GaAs quantumwell structure separated by a thin AlGaAs barrier, and various surface gates, among them an array of 1 μmlong wire gates (WG) interconnected with air bridges (Fig. 2). We apply a negative voltage V_{wg} to the wire gates, which is strong enough to pinch off (fully deplete) the upperwell (UW) regions under the wire gates, but not strong enough to pinch off the lowerwell (LW) regions under the wire gates. Hence, in the lower well, electrons can move under the wire gates, but in the upper well they cannot. Electrons in the upper well are laterally confined to a network of narrow parallel channels (W regions in Fig. 1c) socalled wires, and perpendicular wide trenches (La, Lb, Lc and Ld regions in Fig. 1c, jointly denoted L) socalled leads. In the UWwire regions, electrons are strongly confined to a narrow width, thus becoming 1D. In contrast, in the UWlead regions, no dimension is very narrow, and so electrons remain 2D. In the lower well, electrons are not laterally confined, and so their dimensionality is 2D. The quantum wells are separately contacted using a surfacegate depletion technique (see Methods section for details). A fixed ac voltage V_{ac}=50 μV and a variable dcbias V_{dc} are applied between them. The bias is applied to the lower well, while the upper well is grounded. The current I, injected through one of the quantum wells, is forced to tunnel through the barrier that separates them, before leaving the device through the other quantum well. The differential tunnelling conductance G=dI/dV_{ac}, in a variable inplane magnetic field B perpendicular to the wires, is measured at ∼57 mK lattice temperature (higher electron temperature) in a dilution refrigerator, using standard lockin techniques. The W and L regions of the device contribute distinct signals in G(B, V_{dc}) maps. The W signal has 1D–2D nature, that is, it corresponds to tunnelling between the UW 1D wires and the LW 2D electron gas. The L signal has 2D–2D nature, that is, it corresponds to tunnelling between the UW 2D leads and the LW 2D electron gas.
Control of electron density in the wires
The electron density in the UW wires can be controlled by tuning the wire–gate voltage V_{wg}. This is shown in Figs 3 and 4. Figure 3 maps dG/dV_{wg} for conditions close to equilibrium (V_{dc}≈0). The map reveals the 1D wire subbands participating in 1D–2D tunnelling. The singlesubband regime is achieved with V_{wg} in the range −0.78 V≲V_{wg}≲−0.69 V. For V_{wg}>−0.69 V, more than one wire subband participates in 1D–2D tunnelling and, for V_{wg}<−0.78 V, the wires are completely depleted of carriers. Figure 3 shows the magneticfield values B_{W}^{−}, B_{W}^{+}, −B_{L}^{−} and B_{L}^{+}, at which tunnelling resonances cross the V_{dc}=0 axis in G(B, V_{dc}) maps (for example Fig. 5), as a function of the wire–gate voltage. There are two sets of crossing values. One, ±B_{W}^{±}, corresponds to tunnelling in the W regions of the device, and another, ±B_{L}^{±}, to tunnelling in the L regions. Note that B_{W}^{±} denote the values corresponding to the first UW wire subband. The crossing values are related to the Fermiwavevector components along the wire direction as follows:
Here, is the Fermiwavevector component in the W regions of the upper well (lower well), is the Fermiwavevector component in the L regions of the upper well (lower well), d is the distance between the centres of the UW and LW wavefunctions, and −e is the electron charge. It happens that , and so and . The Fermi wavenumber in a 1D (2D) system is related to its equilibrium 1D (2D) electron density n_{1D} (n_{2D}) as follows:
Hence, the equilibrium electron densities in the upper and lower wells, for the wire (W) and lead (L) regions of the device, can be estimated from the values of and in Fig. 3. These are shown in Fig. 4. The wire gates efficiently tune electron density in the wire (W) regions of the upper well (Fig. 4a), but have little influence on densities in the lower well and in the lead (L) regions of the upper well (Fig. 4b). Note that the LW density in the W regions is only slightly smaller than that in the L regions (Fig. 4b), and notably higher than the UW density in the W regions (Fig. 4a).
Regime of a single 1D wire subband filled
The rate of electron tunnelling between the upper and lower wells depends on the overlap between the respective spectral functions, which can be varied by tuning the magnetic field and the dcbias. The magnetic field B and the dcbias V_{dc} shift the UW and LW spectral functions relative to each other, in the momentum and energy directions, by ħΔk=eBd and ħΔω=eV_{dc}, respectively. Hence, by mapping out the tunnelling conductance G(B, V_{dc}), the spectral characteristics of the UW wires, UW leads and lower well are probed. Figure 5 shows a map corresponding to the singlesubband regime (V_{wg}=−0.69 V). It displays quasiparabolic resonances, corresponding to tunnelling from ground states of the source (spectrometer) to excited states of the drain (probe) occurring with conservation of energy and momentum, that is, the tunnelling condition
is satisfied, with either or , where is the Fermi wavenumber of the lower well (upper well) in the wire or lead regions of the device. Here, ɛ_{UW} (k) and ɛ_{LW} (k) are the dispersions of the elementary excitations in the upper and lower wells, respectively, in the wire or lead regions. The map reflects tunnelling events occurring between ground states of the upper well and excited states of the lower well, and between ground states of the lower well and excited states of the upper well. The wire (W) and lead (L) regions of the device contribute distinct signals. The solid (dashed) green lines in Fig. 5 mark resonances corresponding to tunnelling in W (L) regions, from UW ground states to LW states; they reveal the LW dispersion in W (L) regions of the device. The dashdotted green lines mark resonances corresponding to tunnelling in L regions from LW ground states to UW states; they reveal the UWlead dispersion. Most importantly here, the resonances marked by the black and white arrows correspond to tunnelling in W regions, from LW ground states to UWwire states. They reveal 1D UWwire dispersions: a holon band in the particle sector (h+), and a spinon band in the hole sector (s−), as elucidated below. Remarkably, an extra wire band (s+), symmetric of s−, is also observed in the wires’ particle sector. Note that the particle and hole sectors for the upper well are inverted relative to those for the lower well. The width of 1D bands in Fig. 5 is to a large extent associated to intrinsic physical effects. The width parallel to the magneticfield axis correlates to momentum uncertainty, which (according to the Heisenberg’s principle) is inversely proportional to the wire length. Wire shortness is thus a cause of resonance broadening. 1D bands are also broad parallelly to the dcbias axis, because of the powerlaw (slowdecaying) shape of 1D singularities. Tunnelling maps are symmetric with respect to the sign of the magnetic field. Note, however, that the map in Fig. 5 is not symmetric with respect to the sign of the dcbias. The dispersions of the s− and s+ bands are symmetric, but this is just a local symmetry. The map does not exhibit general symmetry with respect to V_{dc}. This indicates that potentials are different in the upper and lower layers. Note that resonances in the map of Fig. 5 do not cross the V_{dc}=0 axis at B=0. That is, . This indicates that equilibrium electron densities, and Fermi wavenumbers, are different in the upper and lower layers.
Regime of three 1D wire subbands filled
Maps corresponding to the regime of three 1D subbands filled (V_{wg}=−0.55 V) are shown in Fig. 6. To better discriminate the location of resonances, the derivatives dG/dV_{dc} and dG/dB are plotted. Mapping of dG/dB (dG/dV_{dc}) emphasizes conductance modulations as a function of momentum (excitation energy). In Fig. 6, tunnelling resonances correspond to peaks of enhanced conductance, across which colour contrast changes quite abruptly. These peaks define quasiparabolic lines on the maps (see the guidetotheeye lines drawn). Note that valleys, with colour contrast sequence opposite to that of peaks, do not correspond to resonances, but to regions of low conductance. The resonances marked by the green lines in Fig. 6a,b have the same meanings as in Fig. 5. Most importantly here, the black lines mark 1D UWwire bands. The dashedblack line drawn in Fig. 6a,b for V_{dc}>0 (lower half of the plots) marks the (first) spinon hole subband (s−), best discriminated in Fig. 6a,c as a line with red/blue contrast. Remarkably, a band with symmetric (inverted) dispersion is seen in the particle sector (dashedblack line for V_{dc}<0, upper half of the plots), as in the singlesubband regime (s+). The inverted band is best discriminated in Fig. 6b,d as a line with red/white contrast. The (first) holon subband (h+, marked by the solidblack line) appears well discriminated in the particle sector (V_{dc}<0) at high energies, but hardly discernible in the hole sector (V_{dc}>0). In addition, Fig. 6b,d reveals quasihorizontal features. Similar states are seen in the regime of fully depleted wires. The quasihorizontal features seen in Fig. 6b,d possibly are superpositions of contributions from (i) the bottoms of the second and third UW wire subbands, and from (ii) localized states^{29,30,31} likely formed at intersection and/or bending points of the UW leads (Lb, Lc and/or Ld regions in Fig. 1c). Momentumconserving resonances in Figs 5 and 6 (green and black lines) are seen not to be perfectly symmetric in their right and left sides, as opposed to the symmetric dispersions to which they correspond. The small asymmetry is known to be caused by capacitance effects^{32}, for which we account on the basis of a simple model previously used^{15} (see Methods section for details).
Conductance oscillations
When tunnelling occurs through a barrier of finite (short) length l in the inplane direction perpendicular to the magnetic field, momentum is not strictly conserved in tunnelling and, as a consequence, the tunnelling conductance exhibits oscillations^{32,33,34,35}. The maps in Figs 5 and 6 exhibit oscillations of this kind (diagonal shortperiod conductance oscillations), associated to tunnelling in (central or peripheral) La regions of the device (see Fig. 1c). In these regions, the UW states are laterally confined under the action of the wire gates, in the inplane direction perpendicular to the magnetic field. For square confinement, the magneticfield period ΔB of tunnelling conductance oscillations has been shown^{34,35} to be ΔB≈φ_{0}/(ld), where φ_{0}=2πħ/e is the quantum of flux. For soft confinement, the period of the oscillations has been shown^{34,35} to be ΔB≈φ_{0}/(Δx d), where Δx is the distance between the classical turning points. In our device, La regions are nominally 0.6 μmwide, corresponding to the gap between the wire gates. In the maps of Figs 5 and 6, the oscillation period is ΔB≈0.26 T, which corresponds to a confinement width Δx≈0.5 μm. This is just a bit smaller than the gap between the wire gates, consistent with the small lateral depleting effect that the gates produce.
Band assignment
Within the linear theory of Luttinger liquids, the spectral function of repulsive spinful 1D fermions has powerlaw singularities at linear spinon and holon mass shells and at the inverted holon mass shell near the right (+) and left (−) Fermi points^{22,23}. Spinons and holons travel at different velocities , respectively (). Within the nonlinear theory^{22,23,24}, the excitation with lowest possible energy for a given momentum (edge of support) is predicted to coincide, in the main region of the hole sector (ω<0), with the spinon massshell ɛ_{s}(k). In the main region of the particle sector (ω>0), the edge is predicted to coincide with the inverted spinon mass shell−ɛ_{s}(k). For arbitrary k in the regions defined by integer j, the edge is predicted to coincide with periodic shifts of the functions ±ɛ_{s}(k), see Fig. 1a in ref. 22. Holon singularities may exist at the holon mass shell ɛ_{c}(k), as well as at shifted holon lines. The leading nonlinear correction to the holon dispersion near the Fermi points is quadratic^{23,24}:
whereas that to the spinon dispersion is cubic:
Here, is the holon effective mass, and ξ is a positive parameter. Note that the nonlinear corrections have opposite sign for holons and for spinons in the particle sector , and same sign in the hole sector . The precise shapes of the spinon and holon dispersions, away from the Fermi points, are generally unknown theoretically.
According to the nonlinear theory^{22,23}, states at the edge of support are spinon states (not holon states); spinon excitations are protected from decay, but holon excitations are subjected to decay. Hence, the resonance labelled s− in Fig. 5 (as well as that marked by the dashedblack line in the lower half of Fig. 6a,b) should correspond to the main spinon band. Consistent with the nonlinear theory, the symmetric inverted replica seen in the particle sector (s+ in Fig. 5, and the upperhalf dashedblackline resonance in Fig. 6a,b) appears to be its shadow (spinon) band. On the other hand, given its positive curvature (upwards bending), the resonance labelled h+ in Fig. 5 (as well as that marked by the solidblack line in the upper half of Fig. 6a,b) should correspond to a holon band. It cannot correspond to a spinon band, because the dispersion of spinons has opposite curvature in this sector (see Fig. 1a in ref. 22).
Spinon and holon effective masses
To extract quantitative information, we simulate the experimental holon dispersion (h+, solidblack lines) with a quadratic functional form (equation (6)). It has been shown^{26} that the spinon dispersion in the nonlinear regime is approximately parabolic. Therefore, we simulate the spinon dispersion (s−, dashedblack lines) in the hole sector (lower half of the plots) of the main region with a quadratic functional form, analogous to that of holons, but with a different effective mass :
We express the holon and spinon effective masses as , where are phenomenological parameters, accounting for renormalization of the effective mass due to 1D confinement^{26}, and is the effective mass of the noninteracting electronlike quasiparticles in the parent UW Fermi liquid. Note that a change in the nature of the elementary excitations occurs upon imposing lateral 1D confinement, but not upon 2D confinement. thus account for mass renormalization in going from a 2D Fermi liquid to a 1D Luttinger liquid. The spinon dispersion is found to be characterized by an effective mass similar to that of noninteracting quasiparticles in the UWlead and LW Fermi liquids, that is, , corresponding to . (See Methods section for details of simulations of the LW and UWlead dispersions.) Note that the spinon momentum is bounded^{23}, and so the spinon quasiparabolas (lowerhalf dashedblack lines in Fig. 6) do not continue beyond , but theoretically replicate through shifts and inversions (see Fig. 1a in ref. 22). In contrast, the holon momentum is not bounded^{23}, and so the holon quasiparabolas (solidblack lines in Fig. 6) theoretically extend from the particle sector ω>0 (V_{dc}<0) into the hole sector ω<0 (V_{dc}>0) without changing the sign of their curvature (see Fig. 1a in ref. 22). Fitting of the holon dispersion is carried out in the particle sector (upper half of the plots), where it is well discriminated. The holon dispersion is found to be characterized by an effective mass notably smaller than that of noninteracting quasiparticles in the UWlead and LW Fermi liquids. The mass renormalization parameter is found to be () in the regime of one (three) subband(s) filled. This is consistent with the expected effect of repulsive interactions, for which the theoretical Luttingerliquid parameter K_{c} is in the range 0<K_{c}<1. The higher value of in the regime of three subbands filled is understood considering that the second and third subbands somewhat screen interactions in the first subband. Nevertheless, observation of the inverted spinon shadow band (s+), when more than one subband is filled (Fig. 6), indicates that the first 1D subband remains highly correlated even in the presence of screening by higher subbands. We emphasize that the wire bands seen in the hole sector (s− in Fig. 5, lowerhalf dashedblack lines in Fig. 6a,b) and in the particle sector (h+ in Fig. 5, upperhalf solidblack lines in Fig. 6a,b) cannot be adequately described with a single parabola. A continuation of the lowerhalf dashedblack lines—dotted yellow lines in Fig. 6a,b—is clearly unsatisfactory in fitting the resonance observed in the particle sector at high energies (upper half of the plots, V_{dc} << 0). The spinon (s−, lowerhalf dashedblackline) and holon (h+, upperhalf solidblackline) bands are characterized by notably different effective masses.
Discussion
We finally comment on the physical meaning of the observed 1D bands in the context of the new nonlinear theory^{22,23,24}. The injection of a spinup (spindown) electron into a wire leads to the creation (annihilation) of a spinon and the creation of a holon. If the energy of the incoming electron is ħω=ɛ_{s}(k), the final state will contain a spinon carrying the wholeenergy ħω, and a holon carrying no energy. Similarly, if the energy of the incoming electron is ħω=ɛ_{c}(k), the final state will contain a holon with energy ħω, and a spinon with no energy. Spinon excitations are protected from decay by energy and momentum conservation laws. Therefore, the (spinon) edge singularity is predicted to be robust. Holon excitations are subjected to decay to some (not accurately known) extent, and so it is theoretically unclear whether holon singularities should be observable away from the Fermi points^{23,24}. Here, the holon band appears experimentally well discriminated in the particle sector, but hardly discernible in the hole sector. That is, holon particles appear longlived, but holon holes shortlived. Phenomenological theoretical expressions yield the j and kdependent exponents of the powerlaw singularities at the spinon and holon mass shells^{22,23,24}. The spinon exponents in the particle sector are similar to those in the hole sector, except for different selection rules m_{±}. Hence, the (main region) spinon singularity is theoretically predicted to be a divergent one in the hole sector (thus easy to observe), and a convergent one in the particle sector (thus difficult to observe). This would explain why observation of the spinon shadow band in the particle sector of the main region, reported here, has been previously elusive. Our results indicate that this shadow band does exist, supporting the predictions of the new nonlinear theories^{22,23,24}. They show that the shadow band has appreciable weight for short wires (∼1 μm long), wire length appearing as a relevant parameter modulating the spectral weight of shadow bands for electronic (spinful) systems. Theoretical studies^{27,28} indicate that the spectral weight of shadow bands for spinless systems increases strongly with decreasing wire length. Here, the experimental observation of the inverted spinon shadow band (s+) appears to be facilitated by (i) the wire shortness, (ii) the array design of our device that boosts the wire signal and (iii) the homogeneity of the lithography.
Methods
Experimental device
The vertical structure of the device, grown by molecularbeam epitaxy, comprises two 18 nmwide GaAs quantum wells, a 14 nmwide tunnel barrier in between, and spacer and Sidoped Al_{0.33}Ga_{0.67}As layers at both sides. The structure terminates with a 10 nmwide GaAs cap layer. The tunnel barrier is a 10 × [(0.833 nm)Al_{0.33}Ga_{0.67}As/(0.556 nm)GaAs] superlattice. The lower (upper) spacer is 40 nm (20 nm) wide and the Sidoped layers (10^{18} donors cm^{−3}) are 40 nm wide. The electrical (surface) structure of the device was fabricated on a 200 μmwide Hall bar (Fig. 1b). Electronbeam lithography was used to pattern surface gates, that is, split (SG), midline (MG), bar (BG) and cutoff (CG) gates (Fig. 1a)—used to define the experimental area and to setup tunnelling conditions—and a 30 × 200 array of airbridged wire gates (Fig. 1d and Fig. 2)—used to define the quantum wires in the upper well. The wire gates are 1 μmlong and 0.3 μmwide. They are separated by 0.19 μmwide gaps (W wire regions) in the transverse direction and by 0.6 μmwide gaps (L lead regions) in the longitudinal direction (Fig. 1c). The device dimensions were chosen carefully to achieve minimal modulation of the LW carrier density by the negative wire–gate voltage.
Tunnelling setup
The two quantum wells are separately contacted with AuGeNi Ohmic contacts by using a surfacegate depletion technique. At one side of the device (Fig. 1a), gate SG is negatively biased to pinch off both the upper and lower layers, while gate MG is positively biased to open a narrow conducting channel in the upper layer only. At the other side of the device, gates BG and CG are negatively biased to pinch off just the upper layer. Hence, the current injected through one of the Ohmic contacts is forced to tunnel between the upper and lower layers, before flowing out through the other Ohmic contact.
Modelling of the LW and UWlead dispersions
The electron density in the lower well (UW leads) is n_{LW}=1.5 × 10^{11} cm^{−2} (n_{UW,L}=2.7 × 10^{11} cm^{−2}), obtained from the crossing magneticfield values . For these densities, 2D electron systems are known to be Fermi liquids with effective mass renormalized by interactions^{36,37}. We thus simulate the LW and UWlead dispersions with the parabolic functional form
characteristic of Fermiliquid quasiparticles, where is the effective mass in the lower well or in the UW leads. Replacing the corresponding dispersions in equation (5), one obtains functions of the type
that we use to simulate the tunnelling resonances corresponding to the LW (+) and UWlead (−) dispersions. By setting d=32 nm (nominal value), the LW and UWlead dispersions are found to be well described by an effective mass , somewhat smaller than the band mass of GaAs (), where m_{0} is the electron rest mass. This is consistent with previous measurements and theoretical predictions for quantum wells with similar densities^{38,39,40,41,42}. If the bands were substantially bent across the quantum wells, the wavefunctions would be somewhat displaced relative to flatband conditions, and the distance d (between the centres of the UW and LW wavefunctions) would somewhat differ from the distance between the centres of the quantum wells (32 nm). Equally good fits are obtained if we impose the effective mass to be equal to the band mass () and allow for variation of d. The best fit is then achieved for d=37 nm. We cannot determine the individual contributions of and d to the curvature of tunnelling resonances. Nevertheless, we note that the mass is in very good agreement with the careful measurements carried out by Hayne et al.^{38,39}.
Modelling of capacitance effects
Tunnel transport between the upper and lower layers of the device, across its dielectric barrier, is affected by capacitance effects, which cause a small deformation of the observed dispersions. The finite capacitance of the device (C/A per unit area, C/L per unit length) leads to a small increase/reduction of the electron density ±δn_{2D/1D} at each side of the barrier (eδn_{2D}=V_{dc}C/A for 2D systems, and eδn_{1D}=V_{dc}C/L for 1D systems), and consequently to changes of the Fermi wavenumbers of the source and drain systems ( for 2D systems, and for 1D systems). To account for capacitance effects^{15} in simulations of the measured resonances, we replace the Fermi wavenumbers, corresponding to zero capacitance, by modified ones
for 2D systems, and
for 1D systems, in the expressions describing tunnelling conditions (equation (5)). Hence, dispersions V_{dc}(B) initially symmetric become somewhat asymmetric. Satisfactory fits are achieved with capacitance values C/A≈0.004 F m^{−2}, C/L≈0.1 nF m^{−1}.
Data availability
Data associated with this work are available at the University of Cambridge data repository (http://dx.doi.org/10.17863/CAM.797; see ref. 43).
Additional information
How to cite this article: Moreno, M. et al. Nonlinear spectra of spinons and holons in short GaAs quantum wires. Nat. Commun. 7:12784 doi: 10.1038/ncomms12784 (2016).
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Acknowledgements
This work was supported by the UK EPSRC [Grant Nos. EP/J01690X/1 and EP/J016888/1]. M.M. thanks J. Waldie for her assistance with device fabrication and transport measurements, and P. See for his advice on device fabrication.
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Project planning: C.J.B.F. and A.J.S.; MBE growth: I.F. and D.A.R.; optical and electronbeam lithography: M.M. and J.P.G., with assistance from Y.J. and G.A.C.J.; transport measurements: M.M., C.J.B.F. and Y.J.; analysis of results and theoretical interpretation: M.M., C.J.B.F., O.T. and A.J.S.; manuscript preparation: M.M. with contributions from C.J.B.F.
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Correspondence to M Moreno.
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Moreno, M., Ford, C., Jin, Y. et al. Nonlinear spectra of spinons and holons in short GaAs quantum wires. Nat Commun 7, 12784 (2016). https://doi.org/10.1038/ncomms12784
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