Wigner solid pinning modes tuned by fractional quantum Hall states of a nearby layer

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Science Advances  15 Mar 2019:
Vol. 5, no. 3, eaao2848
DOI: 10.1126/sciadv.aao2848


We studied a bilayer system hosting two-dimensional electron systems (2DESs) in close proximity but isolated from one another by a thin barrier. One 2DES has low electron density and forms a Wigner solid (WS) at high magnetic fields. The other has much higher density and, in the same field, exhibits fractional quantum Hall states (FQHSs). The WS spectrum has resonances which are understood as pinning modes, oscillations of the WS within the residual disorder. We found the pinning mode frequencies of the WS are strongly affected by the FQHSs in the nearby layer. Analysis of the spectra indicates that the majority layer screens like a dielectric medium even when its Landau filling is ~1/2, at which the layer is essentially a composite fermion (CF) metal. Although the majority layer is only ~ one WS lattice constant away, a WS site only induces an image charge of ~0.1e in the CF metal.


Wigner solids (WSs) occur when an electron-electron interaction dominates the zero-point or thermal motion of the carriers. They can be accessed in extremely dilute systems in the absence of a magnetic field or in a high magnetic field (B) at sufficiently low Landau level filling, ν, near the termination of the fractional quantum Hall state (FQHS) series, where WSs have long been expected (13). The magnetic field–induced WS in a two-dimensional electron system (2DES) is of great interest and has been studied experimentally by a variety of different techniques, including pinning mode spectroscopy (410), photoluminescence (11), transport (1215), nuclear magnetic resonance (16), and time-dependent tunneling (17). As a state stabilized by electron-electron interactions, it can be expected that a WS is strongly affected by nearby screening layers or its dielectric environment. There are theoretical works (18, 19) concerning the phase diagram of a 2DES in the presence of a nearby metal gate, for which the gate carries image charges that render electron-electron interactions dipolar at distances exceeding the gate separation. For a WS near a higher–dielectric constant substrate, the screening is less strong, and the magnitude of an image charge is less than |e|, as was studied (18, 20) for electrons separated from these substrates by thin He films.

Through pinning mode measurements (410), we study here a quantum 2D WS screened by a 2DES with a much larger density in a neighboring quantum well (QW). Previous dc-transport studies (15) of such density-asymmetric double wells have demonstrated the existence of a triangular-lattice WS in close proximity to a majority layer comprising a composite fermion (CF) (21) metal, by means of geometric resonance oscillations of the CFs acted on by the WS. Our work considers the reverse and examines the effect of the CF metal and majority-layer FQHSs on the statics and pinning-mode dynamics of the WS.

We find pinning modes signifying the presence of a WS both when the majority layer is a CF metal and when it is in a gapped FQHS. The difference between the pinning modes in the presence of these majority-layer states is remarkably slight. Even for a majority-layer CF metal, screening is closest to that expected from a dielectric substrate rather than that of a nearby metal gate, and we show that such screening can be modeled by image charges of only around 10% of a WS site charge, as illustrated in Fig. 1. This result is unexpected because the CF metal and solid are so close together, only about one lattice constant of the solid away. If a normal metal were at that distance, then the WS would be drastically different than one that is in the presence of a nearly inert, gapped FQHS at low temperature; instead, we find that the 2/3 FQHS and the CF metal have pinning mode frequencies different by at most ~ 10%. The finding is even more surprising in light of the geometric resonance results (15), which show that trajectories of CFs are substantially modified by the presence of a WS.

Fig. 1 WS close to CF Fermi sea.

The bilayer system has a high-density (majority) top layer that hosts a CF Fermi sea when its Landau filling νH is around 1/2 and exhibits FQHSs at odd-denominator fillings. The low-density (minority) bottom layer has a much smaller density compared to the majority layer and forms a WS when the majority layer is in the regime of FQHSs. (A to C) Schematic sketches of local charge densities of the minority layer and majority layer, ρL(x, y) and ρH(x, y), respectively. (A) Charge densities without screening by the majority layer. ρL shows the characteristic triangular Wigner lattice, but ρH remains uniform, as in an incompressible liquid state. (B) Same as in (A), but now, the majority-layer density screens the WS and develops dimples, regions of locally reduced charge density, which act as opposite-signed “image” charges. (C) Three panels show cuts of (A) and (B), through a line of WS electrons of charge e. The left panel is the incompressible–majority layer situation as in (A). The middle panel shows the static dielectric response of a compressible majority layer to the WS of the minority layer. The dimples, each with charge enIstat/nL, develop in the majority layer, where nL is the areal density of minority-layer electrons, and enIstat is the image charge density. The right panel illustrates our model of the screened WS: The charge of the image is modeled as summing with the charge at its WS lattice site, creating an effective WS charge per site of (nLnIstat)e/nL. (D) Schematic of the sample used for microwave spectroscopy of WS pinning modes. The dark area is a metal-film coplanar waveguide (CPW) transmission line through which microwaves are propagated. The CPW has a driven center conductor and grounded side planes and is capacitively coupled to the electrons in QWs. A backgate on the bottom of the sample allows the minority-layer density to be varied.

Experimental setup

Our samples contain two 30-nm-wide GaAs QWs separated by a 10-nm-thick, undoped barrier layer of Al0.24Ga0.76As, giving a center-to-center separation of 40nm. The QWs are modulation-doped with Si δ layers asymmetrically: The bottom and top spacer layer thicknesses are 300 and 80nm, respectively. This asymmetry leads to the different 2D electron densities in the QWs. As cooled, the densities of the top, high-density layer and the bottom, low-density layer are nH ~ 15 and nL ~ 5.0, in units of 1010cm− 2, which will be used for brevity in the rest of the paper. A bottom gate is used to control nL. As detailed in the Supplementary Materials, we obtained nH and nL following the procedure of Deng et al. (15, 22), adapted for microwave conductivity measurements using the setup in Fig. 1D. B-dependent charge transfer between layers for samples like ours is possible and occurs mainly for νH > 1. To account for this, the total density (ntot), which does not change with B, is obtained from low-B Shubnikov–de Haas oscillations, nH comes from high-B majority-layer FQHS positions, and nL in the B range of interest is found by taking the difference between ntot and nH.


The main result of this paper is illustrated in Fig. 2, which shows pinning modes exhibited by the WS in the minority layer, as B and hence the majority-layer filling, νH, are varied. The notable feature is that, although the WS resides in the minority layer, the pinning modes are clearly responding to the FQHSs of the majority layer, whose filling νH is marked at the right side in the figure. The pinning mode is clearly affected by the majority-layer state but, even in the presence of the CF metal at νH12, remains strong. The effect of the majority-layer state on the pinning modes makes it clear that screening of the WS is present. Throughout the measurement range, the minority-layer filling νL ≤ 0.113, well within the filling-factor range of WS for high-quality, single-layer 2DESs (7, 8, 12, 14).

Fig. 2 Pinning mode spectra are affected by the majority layer but are present for all majority-layer states, including νH12.

Re(σxx) versus frequency, f, spectra at many magnetic fields for majority- and minority-layer densities nH = 15 and nL = 2.20, respectively. Data were recorded in the low-power limit and at the bath temperature of 50 mK. Traces are vertically offset for clarity and were taken in equal steps of νH in the range 0.35 ≤ νH ≤ 0.75 (0.053 ≤ νL ≤ 0.113). The majority-layer filling νH is labeled on the right axis.

Figure 3A illustrates the effect of varying nL on the pinning mode of the minority layer. Re(σxx) versus f spectra are shown at different nL values, produced by changing backgate voltage bias. A typical characteristic of pinning modes in a single-layer WS at low ν (6, 7, 10) is that, when nL decreases, the peak frequency fpk increases, and the resonance becomes broader and weaker. We will refer to this behavior as the density effect. Its explanation in the weak-pinning theory (2325) is that, as the WS softens at lower density, the correlation length of crystalline order in the WS decreases, and the carrier positions become more closely associated with disorder and so, on average, experience a larger restoring force due to a small displacement. The inset of Fig. 3A shows the extracted fpk versus nL. The lines are fits to fpkn−1/2; this dependence has been observed previously (6, 7, 10) for single-layer samples at low densities in the low-ν WS range.

Fig. 3 Pinning mode as minority-layer electron density, nL, is varied.

(A) Re(σxx) versus f spectra at fixed νH=12 (nH = 15) at different nL values. Traces are offset vertically for clarity. The density effect, typical for pinning modes, is evident as decreasing nL increases fpk, while the resonance amplitude decreases and the resonance broadens. Inset: Extracted fpk versus nL, on a log-log scale, at νH=12 (circles) and 23 (triangles). Dashed lines are fits to fpknL−1/2. (B) fpk versus νH at nH = 15 and different nL; traces not vertically offset. Vertical dashed lines mark rational fractional fillings (νH) of FQHSs. The vertical dotted line marks νH=12. The upward overall increase of fpk for each step down in nL is accompanied by the oscillations of fpk versus νH, with minima at majority-layer FQHSs.

To highlight the clear response of the pinning mode to the majority-layer state, including the reduction of fpk when a FQHS develops in the majority layer at its odd-denominator fillings νH=25,37,47,35 and 23, in Fig. 3B, we show fpk as a function of νH for various nL. As nL decreases, the overall fpk curves shift upward over the entire νH range. The oscillation amplitudes of fpk seen in Fig. 3B at FQHSs of the majority layer become more pronounced when nL decreases. This is occurring as the spacing of the minority-layer WS electrons exceeds the 40-nm interlayer separation of the double-QW structure. For example, at nL = 3.25 and 1.02, the triangular WS lattice constant is a = 60 and 106 nm, respectively.

Figure 3B shows that, for νH on majority-layer FQHSs, there are minima in fpk.When the majority layer is in an FQHS, its ability to screen the interaction stabilizing the minority-layer WS is weakest, because the energy gap of the FQHS leaves few majority-layer charge-carrying excitations available to screen the WS. Moving νH away from an FQHS increases the screening due to the majority layer and reduces the correlation length of crystalline order, just as reducing the density does in the density effect, and results in a larger fpk. Theories of weak pinning (2325) all show that fpk increases with larger-crystal elastic moduli.

The FQHS minima in Fig. 3B appear on top of a weak decreasing background: For each trace, the fpk oscillations, and also its featureless region between νH = 0.46 and 0.54, are superimposed on a gradual decrease with νH. The decrease is similar for each trace, hence insensitive to nL. In light of this insensitivity, we ascribe the decreasing background to effects intrinsic to the minority layer. For example, these effects could be a change in the WS stiffness (3) or a change in the disorder coupling (2325) due to a change in the magnetic length (size of the carrier). Single-layer WSs are known to show weak dependence of fpk on B over wide ranges of Landau filling (7).


Our interpretation of the data relies on the illustration in Fig. 1, in which, above the pinned WS lattice sites in the minority layer, the majority-layer local charge density develops “image” charge minima. The amount of charge in each image depends on the static dielectric response of the majority layer, not on its conductivity. The ability of the image charge to follow the WS site charge dynamically as the pinning mode is driven, on the other hand, depends on the local conductivity of the majority layer as well. At each WS lattice site, there is then a combination of an image charge with the corresponding charge in the WS. This combined object has a dipole moment, but because of the finite majority-layer local compressibility, it can also have a nonzero charge. We will characterize our pinning mode data in terms of charge densities. nstat denotes the static charge density of the combined charges, and ndyn denotes the (dynamic) areal charge density that moves as the pinning mode is driven. Like nL, nstat and ndyn are given in units of 1010 cm− 2.

By means of the pinning mode sum rule (26), ndyn = (2Be)(S/ fpk ), where S is the integrated Re(σxx) versus frequency, f, for the resonance. Figure 4 (A to C) shows, for nL = 2.20, how ndyn is determined: fpk versus νH in Fig. 4A and S in Fig. 4B produce ndyn in Fig. 4C by use of the sum rule. S tends to increase as fpk decreases and vice versa. S is increased near the majority-layer FQHSs, reflecting a lack of available cancelling image charge at these low-compressibility states. In Fig. 4C, near the peaks at the most developed FQHSs (νH=23 and 25), ndyn approaches nL, which is shown as a horizontal line. The difference of nL and ndyn is the image charge density in the majority layer that is moving along with the electrons of the WS, reducing the total current driven by the resonance. We call ( nLndyn ) the dynamic image charge density, nIdyn. It is graphed versus νH for nL = 1.02, 2.20 and 2.84 in Fig. 3D. nIdyn shows minima at the majority-layer FQHSs, reflecting their small compressibility and small conductivity.

Fig. 4 Effective WS density and image charge densities.

(A to C) Plots of several quantities versus νH for nL = 2.20, to illustrate the determination of the dynamic exffective WS density ndyn : (A) shows fpk, and (B) shows S and the integrated Re[σxx] versus f, and (C) shows ndyn deduced from the pinning-mode sum rule ndyn = (2Be)(S/ fpk ). The overall downward or upward drifts respectively in fpk and S versus νH are removed in (C), and a comparatively flat ndyn versus νH is observed, in which the FQHSs appear as peaks. (D) Density, nIdyn=(nLndyn), of the image charge that moves as the pinning resonance is excited, for three nL values, plotted versus νH.The data are offset for clarity, and the respective zeroes of the traces are shown as lines with error bars. (E) Variation of the static image charge density, nIstat, with νH. Traces are offset for clarity.

The static image charge density nIstat, obtained as ( nLnstat ), is of particular interest because of its sensitivity to the dielectric response of the majority layer without the influence of the conductivity. It is plotted in Fig. 4E. While there is no direct method to measure nstat or nIstat, we can estimate their variations as νH sweeps through the FQHSs of the majority layer. We obtain nstat independently of ndyn, from the fpk data of Fig. 3B alone. This is possible because in weak-pinning theories (2325), fpk is solely determined by the stiffness of the WS and the disorder acting on it. Increasing the density of a WS raises its stiffness. As described in the Supplementary Materials, the density-effect law, fpknL− 1/2, is inverted to find nIstat to within an additive constant. By obtaining nstat from the density-effect law, we are treating nstat as if the image charge were on the same layer as the WS; because there is an interlayer separation on the order of the WS lattice constant, this may slightly overestimate the effect of nIstat so that the nIstat that we obtain is a lower-limit estimate of the true image charge density. Neglecting the finite compressibility of the majority layer at ν=23 and taking nIstat(νH=23)=0, we find that this low-estimate nIstat(νH=12) is about 10% of nL for the three nL values of Fig. 4: 1.02, 2.20, and 2.84. Overall, we find the variations of nIstat and nIdyn to be of similar size for most νH values. This implies that the image charge in the majority layer moves with the WS as the resonance is driven.

This estimated nIstat(νH=12) is in rough agreement with estimates of nIdyn(νH=12), which are on the order of their error, about 10% of nL as well. It is unexpected that FQHSs at νH=23 and 25 at a temperature well below their energy gaps have image charge densities so close to those at νH=12 at which the majority-layer state is well described by a CF metal. If the CF metal were able to respond to a WS site with image charges equal to an electronic charge, as one might expect, the pinning mode for νH12 would be much more strongly affected than that observed here and might not even be present. Thus, the CF metal is remarkably inefficient at screening. Figure 4 (B and C) shows that the image charge density, hence the screening, is far larger in the transition regions between the FQHSs, where it is expected that there is a network of edge states that can respond to the WS charge (27).

In summary, we study a WS separated from FQHSs by a distance comparable to its lattice constant. We observe a pinning mode from the minority-layer WS, indicating its existence even in the presence of the nearby, screening majority layer. The pinning mode is strongly affected by the majority-layer FQHSs, exhibiting a reduction in fpk with an increase in S around FQHSs. We find that these phenomena can be modeled by considering image charges in the majority layer and regarding them as reducing the WS charge. The image charge is assessed to be only about 10% of the WS charge even near νH=12. It is substantially larger at the transitions between FQHSs. Comparing static and dynamic estimates of image charge indicates that, in large part, the image charge oscillates as the pinning mode is driven.


We performed microwave spectroscopy (610) using a coplanar waveguide (CPW) patterned in a Cr:Au film on the top surface of the sample. A top view schematic of the measurement is shown in Fig. 1D. We calculate the diagonal conductivity as σxx(f) = (s/lZ0)ln(t/t0), where s = 30 μm is the distance between the center conductor and ground plane, l = 28 mm is the length of the CPW, Z0 = 50 ohm is the characteristic impedance without the 2DES, t is the transmitted signal amplitude, and t0 is the normalizing amplitude. The microwave measurements were carried out in the low-power limit, such that the results were not sensitive to the excitation power at our bath temperature of T = 50 mK.


Supplementary material for this article is available at

Obtaining nH and nL

The possibility of interlayer charge transfer changing the minority-layer density due to the majority-layer FQHSs

Determination of nstat and nIstat from fpk versus νH

Fig. S1. Data used to determine electron densities of the double QW.

Fig. S2. Fits used to obtain the static image charge density, nIstat.

References (28, 29)

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Acknowledgments: We thank J.-H. Park and G. Jones for their expert technical assistance and J. P. Eisenstein for discussions. Funding: The microwave spectroscopy work at the National High Magnetic Field Laboratory (NHMFL) was supported through Department of Energy Basic Energy Sciences (DOE-BES) grant DE-FG02-05-ER46212 at NHMFL/FSU. The NHMFL is supported by National Science Foundation (NSF) Cooperative Agreement nos. DMR-1157490 and DMR-1644779, by the State of Florida, and by the DOE. The work at Princeton University was funded by the Gordon and Betty Moore Foundation through the EPiQS initiative grant GBMF4420 and by the DOE-BES grant DE-FG02-00-ER45841, the NSF grant DMR-1709076, and the MRSEC grant DMR-1420541. Author contributions: A.T.H. conceived and designed the experiment, performed the microwave measurements, analyzed the data, and co-wrote the manuscript. L.W.E. conceived and designed the experiment, discussed data analysis, and co-wrote the manuscript. H.D., Y.L., and M.S. conceived the experiment, discussed data analysis, and co-wrote the manuscript. L.N.P., K.W.W., and K.W.B. were responsible for the growth of the samples. Competing interests: The authors declare that they have no competing interests. Data and materials availability: Data displayed in this manuscript will be available by email request to engel{at} All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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