Research ArticleCONDENSED MATTER PHYSICS

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
  • 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.

  • 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.

  • 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.

  • 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.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/3/eaao2848/DC1

    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)

  • Supplementary Materials

    This PDF file includes:

    • 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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