Abstract
Photoemission and tunnelling spectroscopies measure the energies at which single electrons can be added to or removed from an electronic system^{1}. Features observed in such spectra have revealed electrons coupling to vibrational modes of ions both in solids^{2} and in individual molecules^{3}. Here we report the discovery of a sharp resonance in the tunnelling spectrum of a twodimensional electron system. Its behaviour suggests that it originates from vibrational modes, not involving ionic motion, but instead arising from vibrations of spatial ordering of the electrons themselves. In a twodimensional electronic system at very low temperatures and high magnetic fields, electrons can either condense into a variety of quantum Hall phases or arrange themselves into a highly ordered ‘Wigner’ crystal lattice^{4,5,6}. Such spatially ordered phases of electrons are often electrically insulating and delicate, and have proven very challenging to probe with conventional methods. Using a pulsed tunnelling method capable of probing electron tunnelling into insulating phases, we observe a sharp peak with dependencies on energy and other parameters that fit to models for vibrations of a Wigner crystal^{7,8}. The remarkable sharpness of the structure presents strong evidence of the existence of a Wigner crystal with long correlation length.
Main
Theory suggests that a Wigner crystal (WC) may exist near integer quantum Hall states as an insulating phase with expected transition temperature in the range of a few hundred millikelvin or below^{5,6,9,10,11,12}. Measurements in these regimes have demonstrated insulating phases^{13,14,15}, microwave pinning resonances^{12,16}, Knight shifts^{17}, and compressibility features^{18} suggestive of a Wigner solid. However, it has proven difficult to demonstrate a longrange ordered crystal; the observed features could arise from localization of electrons or a shortrange ordered glassy phase. One method of detecting crystalline order is by means of observation of the crystal’s vibrations. In prior measurements^{1}, strong coupling of tunnelling electrons to vibrational modes of ions have given rise to ‘bosonic peaks’ in the tunnelling spectra. Observation of such structure was essential to verifying the Bardeen–Cooper–Schrieffer theory of superconductivity^{1}. In this letter, we report the observation of a sharp peak in tunnelling spectra that displays the characteristics of phonons of an electron crystal.
As with a lattice formed by ions in normal materials, the WC has gapless phonon modes from transverse and longitudinal phonons. However, application of a magnetic field forces hybridizes the two modes into magnetoplasmon and magnetophonon modes^{7,19}, with the magnetophonon mode remaining as a gapless Goldstone mode that signals the emergence of crystalline order (spontaneous translational symmetry breaking)^{7,8}. A measurement revealing the sharp magnetophonon mode would constitute direct and conclusive evidence of the existence of the WC. In this study, we detect the presence of the magnetophonon spectrum through its modification of the tunnelling density of states (TDOS), and we control the energy scale of the magnetophonon spectrum by tuning the electronic carrier density and applied magnetic field. Our measurements rely on a pulsed tunnelling method, capable of precisely measuring the TDOS in both conductors and insulators^{20} to measure the TDOS of a twodimensional (2D) hole system under high perpendicular magnetic fields (see the caption of Fig. 1 and Methods for further details).
The TDOS of a holedoped quantum well (QW) measured at B = 8 T and T = 25 mK are shown in Fig. 1b. In addition to the general spectral structure reported previously^{20,21}, these lowertemperature measurements reveal very sharp resonances around the filling factor ν = 1, whose full width is about 0.2 meV. The resonances display an antisymmetry around the Fermi level and filling factor ν = 1 (see Fig. 1c, f); that is, for ν < 1 (ν > 1) the tunnelling conductance peaks (dips) for energies greater than the Fermi level and dips (peaks) for energies below the Fermi level.
We examine scenarios to explain this signal. First of all, the observation of the lowenergy resonance signals in tunnelling near ν = 1 may suggest an origin in spinrelated excitations due to skyrmions^{22,23}. However, while the strongest resonances in our spectrum appear near ν = 1, we also see resonances with nearly identical energies and dispersions near ν ∼ 0 and ν ∼ 2 (see Fig. 1e and Supplementary Fig. 12 of Supplementary Section 8). To our knowledge, there has been no proposal so far for spinrelated excitations near filling factors 0 and 2. Secondly, we rule out disorderinduced localization and oscillations of locally trapped particles due to the observed sharpness of the resonance and because these modes should soften due to screening as the quasiparticle density increases, in contrast to our observation.
It thus seems reasonable to focus on a possible electron–phonon interaction in the system. Tunnelling studies show that, in metals and semiconductors, phonons can strongly couple to electrons and display resonances in electron tunnelling spectrum^{1}. While it is well known that tunnelling electrons that inelastically scatter phonons can generate an increase in tunnelling conductance symmetrically in bias voltage around the Fermi level, an antisymmetric feature in the TDOS can arise from elastic coupling of electrons to the phonon degrees of freedom^{1,2,24}. Formally speaking, the electron–phonon interaction generates a nonzero selfenergy Σ of tunnelling electrons, whose antisymmetric real part is strongly peaked at the energy of the van Hove singularity ε_{vH}. The selfenergy can be interpreted as an additional complexvalued potential existing only at an energy near ±ε_{vH} that modulates the tunnelling matrix element and the TDOS (see Fig. 1d and Supplementary Section 1).
The emergence of these features arising from an electron–phonon interaction in the TDOS appears surprising because no strong ionic phonon DOS exists in the energy scale of a few meV—the charged carriers in this system are effectively decoupled from ionic lattice dynamics. More importantly, the resonance energy ε_{r} in the spectrum is a strong function of filling factor ν, pointing to the electron–electron interaction as the origin of the feature. These observations lead us to consider that the phonons generated from ordered states of electrons themselves affect the tunnelling spectrum. In Landau levels, the existence of robust energy gaps around integer filling factors permits a description in terms of quantum Hall quasiparticles; a system with an integer filling factor ν = l (with l = 0, 1, 2, …) contains l ‘inert’ filled (spinresolved) Landau levels, and as the filling factor ν is tuned away from integer values, the added or subtracted particles can be described as quasiparticles of effective filling factor ν_{qp} = l − ν. When ν_{qp} is small, these quasiparticles resemble bare electrons and holes with Coulombic interactions that may lead to the formation of a WC^{6,9}.
To examine this idea quantitatively, we explore the hypothesis that the spectrum of the resonance feature near ν = 1, at low effective quasiparticle filling factor ν_{pq} = 1 − ν, follows the energy scaling law of the magnetophonon of a WC formed by quantum Hall quasiparticles^{7,8} (see Fig. 2). In this picture, the strong resonance at ε_{r} in the TDOS results from the peaked DOS of the magnetophonons originating from a van Hove singularity (at ε_{vH}) near the Brillouin zone boundary of the hexagonal WC. The energy of the resonance appears to follow ε_{r} = Δ + ε_{vH}, where Δ accounts for the magneticfieldinduced Coulomb gap Δ_{c} (and a spin gap in the case of injecting or ejecting minority spins—see Supplementary Section 5) near the Fermi level^{20,25}. Generally, in a Coulombic system we expect that the energy scale of the interaction would increase along with the effective filling factor of quasiparticles ν_{qp} = 1 − ν as the interparticle distance l_{s} between the quasiparticles shrinks. In the case of a crystal, as l_{s} decreases, the energy scales of the entire phonon dispersion and of the van Hove singularity ε_{vH} increase as a result. To compare our data with theoretically expected dispersions, in Fig. 2a we have overlaid our data with semiclassical and fully quantum mechanical curves calculated for phonon energy versus filling factor. The semiclassical harmonic approximation is given by , where l_{B} is the magnetic length, ε_{s} is the dielectric constant, is a dimensionless dynamic matrix whose determinant is an order of unity, and k is the phonon wavevector^{7,8}. The van Hove singularity ε_{vH} is located at k = k_{M} (Mpoint in the hexagonal Brillouin zone; see Fig. 2b) with (ref. 8). This semiclassical approximation applies only when the quasiparticle density is low (ν_{qp} = 1 − ν ≪ 1). As shown in Fig. 2a, the semiclassical predictions (blue curves), after multiplication by an overall scaling factor of 1.5, fit well in the range of the filling factor 0.88 < ν < 1; however, as l_{s} becomes smaller and ν_{qp} becomes larger, the quasiparticle wavefunctions have more overlap, and higherorder, nonharmonic, quantum mechanical effects become more important^{8}.
To account for this additional stiffening of the resonance, one would require the full quantum mechanical treatment. In the lowest Landau level, Chang et al. have shown that crystals described with a composite fermion (CF) wavefunction correctly account for quantum correlations and become the lowest energy state even at very low ν_{qp} (ref. 10). For comparison with data, we have plotted curves with additional quantum stiffening expected from the shear modulus of a crystal of composite fermions with four flux quanta attached^{11}, which is shown to be the ground state below ν_{qp} ∼ 0.18 down to ν_{qp} ∼ 0.12 (see red curves in Fig. 2; see Supplementary Section 4 for more detail). The theoretical curves capture the qualitative aspect of the stiffening of the resonance that evolves as ν_{qp} increases; the nontrivial sudden increase of the stiffness in our data can be explained by the characteristic of the CF correlation near the phase transition. We also note that the short distance to the tunnelling electrode (∼35 nm from the well centre) means that the effect of screening diminishes at higher quasiparticle densities, thus further stiffening the crystal.
To investigate the nature of the observed resonance as further evidence of the magnetophonons and the quasiparticle crystal, we studied the temperature dependence (see Fig. 3). As temperature is increased from the base electron temperature (25 mK), the resonance diminishes in strength, and disappears completely above ∼200 mK. This observation agrees with previous reports of the temperatureinduced disappearance of the insulating phases and pinning resonances, and is consistent with the thermal melting of a WC of quasiparticles and the resulting disappearance of its magnetophonons^{14,16}. At 25 mK, on increasing the density of quasiparticles by changing the filling factor away from ν = 1, the resonance features terminate. This suggests that the WC undergoes another melting transition—one that is a purely densitydriven quantum mechanical phase transition.
How the behaviour of the phonon energy ε_{vH} changes as the crystal melts (either via tuning ν or T) can give hints on the nature of the phase transition. For a secondorder transition, theories predict that the shear modulus (and phonon frequency) would diminish to zero at the transition^{26}; however, we observe that the magnetophonon energy ε_{vH} does not decrease towards zero in either the thermal or quantum melting transition. This strongly suggests that the phase transition is of first order, with hysteresis involved at the transition^{26}. Although we were not able to directly measure hysteresis involved with the firstorder transition, the vertical features extending from the resonances in the dI/dV_{t} versus (E_{t}, ν) plot (see yellow arrows in Fig. 1f) are consistent with a picture of increased measurement noise due to the presence of hysteresis at the transition (see Supplementary Section 7). According to a calculation of Maki et al. ^{9,26} that did not account for fractional quantum Hall (FQH) order, the shear modulus of the crystal has a peak value at ν_{qp} ≡ ν − 1 ∼ 0.27, then decreases to zero at ν_{qp} ∼ 0.45, where the crystal would undergo a secondorder phase transition to a liquid phase, in an apparent disagreement with our observation. However, as the authors noted, the crystalline phase competes with FQH liquid states. If the ground state energy of the ν = 4/5 (ν_{qp} = 1/5) FQH state is lower than that of the WC, the crystalline phase would abruptly terminate before ν_{qp} reaches 0.2, with a potentially firstorder phase transition. Very interestingly, Archer et al. predict that the shear modulus would have discontinuous jumps at phase transitions^{11} due to correlations of CFs within the crystal. We observe that the filling factor needed for the disappearance of the phonon feature is ν_{qp} > ∼0.17, which is in rough quantitative agreement with the theory of Archer et al. and with the termination of insulating phases previously reported in transport studies and microwave pinning measurements^{11,14,16}. We are also able to identify a weaker but clear resonance signal near ν ∼ 0 in the spectrum (see the white square in Fig. 1e and Supplementary Fig. 12a of Supplementary Section 8) that corroborates the fact that the WC is indeed the ground state as ν → 0. It is crucial to note that we measure the internal spectrum of the crystal, distinct from transport or microwave measurements of the insulating phase that indicate localization and pinning of electrons but do not demonstrate a crystalline order.
Even though we observed the resonance signals at ν = 0 and 1, there still remains a question of whether the quasiparticles near ν = 1 are comprised of skyrmions or normal QH quasiparticles because, in principle, phonon modes can exist in either case^{27}. Skyrmions were proposed as energetically stable quasiparticles near ν = 1 in systems where the Coulomb exchange energy scale E_{C} dominates the Zeeman energy E_{Z} (that is, quantum Hall ferromagnet)^{28}. Thus, some hints can be provided by measuring the electron system, where E_{Z}/E_{C} is tunable by external magnetic fields, that changes the nature of the particle near ν = 1. As an investigation into the possibility of the skyrme lattice^{27}, we performed similar measurements on an electrondoped sample with and without inplane magnetic field in an attempt to tune E_{Z}/E_{C}. An inplane field is expected to shrink the size of skyrmions and, at sufficiently high inplane fields, skyrmions will no longer be the ground state quasiparticles^{29}. In Fig. 4a, b, we plot spectra measured, respectively, at B_{⊥} = 6 T, B_{∥} = 0 T (where E_{Z}/E_{C} is 0.036) and B_{⊥} = 5.8 T, B_{∥} = 8.2 T (where E_{Z}/E_{C} is 0.062). Thus, we expect to have smallsize skyrmions for the former case and usual quasiparticles with singlespin flip for the latter case^{29}. We observe that, when measured without an inplane field, the magnetophonon resonance was absent around ν = 1. The application of inplane magnetic fields creates visible resonances around ν = 1, suggesting that elimination of skyrmions in favour of ordinary quasiparticles allows observation of magnetophonon resonances in the TDOS.
Remarkably, we find that the locations of ε_{vH} of the electron system are very close to those of the hole system (within 20%), although the effective mass of the electron systems is about five times smaller. This provides strong evidence of the characteristic mass independence of the magnetophonon spectrum in the extreme quantum limit of high magnetic fields^{19} (see Supplementary Section 6).
The observed sharpness of the resonance features suggests welldefined phonon energies, and hence a long lattice correlation length. For the magnetophonon mode, we expect that the energy of the resonance follows ε_{vH} ∝ 1/l_{s}^{3}. Attributing the entire width of the observed resonance to the broadening due to lattice inhomogeneity, it leads to the relationship Δl_{s}/l_{s} ∼ Δε_{vH}/3ε_{vh}. Very roughly, for the hole sample, the resonance shown in Fig. 3b gives Δε_{vH}/ε_{vH} ∼ 0.3, and thus Δl_{s}/l_{s} ∼ 0.1, or a lattice correlation length of about 10 lattice spacings. In Supplementary Section 3, we obtain a better lower bound estimate of lattice correlation length by numerically simulating the electron selfenergy and fitting to the data. At ν = 0.88, this calculation gives a lower bound of the correlation length L_{c} ∼ 15l_{s} ∼ 970 nm. A similar calculation for the electron sample gives L_{c} ∼ 3l_{s} ∼ 250 nm. We conclude by noting that this observation suggests that tunnelling spectra can probe other ordered electronic states, such as stripe and bubble phases in the quantum Hall regime^{30}.
Methods
The measurement employs a sequence of highfrequency square pulses using a Tektronix DTG5274 pulse generator and a homemade pulse shaper which minimizes distortions of the pulses (see Supplementary Fig. 1a). The initial pulse induces a tunnelling voltage across the tunnelling barrier. Subsequently applied opposite polarity pulses are used to retrieve the tunnelled electrons out of the QW.
Using a biastee, we are able to apply a d.c. voltage and pulses to the samples at the same time. The d.c. voltage capacitively tunes the carrier density in the 2D system (see Supplementary Fig. 1a). As there is no leakage through the sample (the thick AlGaAs barrier prevents leakage), this d.c. voltage changes only the carrier density in the quantum well and induces no current. The filling factor (proportional to density) is calibrated precisely by independently measuring and integrating the capacitance of the samples, as shown in Supplementary Fig. 1d, e.
The pulse amplitude defines the tunnelling energy of electrons; the tunnelling energy scale is given by the pulse amplitudes multiplied by the geometric lever arm of a heterostructure (the distance between QW to top electrode divided by the distance between top and bottom electrode), which is ∼0.21 for the hole wafer and ∼0.25 for the electron wafer, respectively. The pulse parameters are adjusted to ensure that the system returns to the initial equilibrium state between pulses with a predefined density of the 2D system (that is, the pulse sequence returns the system back to the same density of electrons or holes in the 2D system, with the 2D system in equilibrium with the 3D tunnelling electrode—there is also a delay of approximately 300 μs between pulses, and this allows the 2D system back down to its base temperature). The rise time of the pulses is ∼2 ns, during which only a small number of particles (due to the slow tunnelling rate in our samples—the effective RC time for equilibration is roughly on the order of 1 ms, which is about five orders of magnitude longer than the rise time) tunnel before the actual determination of tunnelling conductance starts. This amounts to a change of 10^{4} cm^{−2} in electron density. The tunnelling current is determined by measuring the linear increase of the displacement charge in response to the applied initial pulse in ∼2 ms (during which the electron density change of ∼10^{7} cm^{−2} can occur, less than a part per thousand of the equilibrium density) before the 2D system is disturbed. This means that the measurement of the tunnelling current accurately measures the tunnelling density of states, a true equilibrium property of the sample.
The amount of charge that tunnels in and out of the QW is measured by HEMT amplifiers at the mixing chamber, and further amplified by cryogenic Si–Ge transistors (Weinreb amplifiers) and roomtemperature broadband amplifiers. All amplifiers are 50 Ω impedancematched. The pulse sequence is repeated and averaged over 100,000 ∼ 1,000,000 times for a good signaltonoise ratio, generating a number that represents a single pixel of data in a 2D spectrum. As positive and negative energy sides are measured separately, there exists some noise near E = 0. This noise does not affect our study, and can always be effectively suppressed by measuring with finer voltage steps. To maintain low electronic temperature in the system, we thermally anchored attenuators at 1 K and 8 mK stages and used NbTiNbTi coaxial cables between the stages. In addition, the amplifiers are thermally decoupled from the samples by means of a resistive manganin wire (∼20 Ω) wound around an oxygenfree highconductivity copper heat sink post (∼8 mK). Furthermore, the power dissipation of the amplifier is maintained in the range of less than a few hundred nanowatts. By measuring the strongly temperaturedependent suppression of the zerobias tunnelling in a perpendicular magnetic field, we have confirmed that an electronic temperature of ∼25 mK (±5 mK) was achieved.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The work at MIT was funded by the BES Program of the Office of Science of the US DOE, contract no. FG0208ER46514, and the Gordon and Betty Moore Foundation, through grant GBMF2931. The work at Princeton University was funded by the Gordon and Betty Moore Foundation through the EPiQS initiative Grant GBMF4420, and by the National Science Foundation MRSEC Grant DMR1420541. We thank P. A. Lee and I. Sodemann for helpful conversations. We thank N. Staley for a careful proofreading of the manuscript and A. Demir for assistance in amplifier design.
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J.J. and B.M.H. performed measurements. J.J. and R.C.A. analysed data. K.W.W. and L.N.P. grew GaAs/AlGaAs heterostructures. All authors discussed the results. J.J. and R.C.A. wrote the paper, and R.C.A. supervised the overall project.
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Jang, J., Hunt, B., Pfeiffer, L. et al. Sharp tunnelling resonance from the vibrations of an electronic Wigner crystal. Nature Phys 13, 340–344 (2017). https://doi.org/10.1038/nphys3979
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