We begin the Fabry–Pérot interferometry study of even-denominator states at the filling factor \(
u =-\,\frac{1}{2}\) owing to its simple edge structure, which consists only of fractional modes. Figure 2a shows the longitudinal resistance R xx and Hall resistance R xy measured at 11 T on the right side of the FPI. The data clearly reveal fully developed integer and fractional quantum Hall states at ν = −1, \(-\frac{2}{3}\), \(-\frac{1}{2}\) and \(-\frac{1}{3}\). Figure 2b presents an R xx fan diagram, which we use to extract the constant-filling-factor trajectories. We define \({\alpha }_{{\rm{c}}}=\frac{{\varPhi }_{0}}{
u e}C\), with \(C=\frac{1}{A}\frac{{\rm{d}}Q}{{\rm{d}}{V}_{{\rm{RG}}}}\) the capacitance per unit area between the right gate and the bilayer graphene underneath, extracted from the Streda formula for each fractional state as the centre of the incompressible region, whose boundaries are indicated by dashed red lines (see Supplementary Information Section 5).
Fig. 2: Even-denominator Aharonov–Bohm interference. a, Longitudinal resistance R xx and Hall resistance R xy measured at 11 T on the right side of the FPI, clearly showing fully developed even-denominator and odd-denominator quantum Hall states at \(
u =-\,\frac{2}{3}\), \(-\frac{1}{2}\) and \(-\frac{1}{3}\). b, R xx fan diagram performed on the right side of the FPI between 10.5 and 11.5 T. Dashed red lines indicate the boundaries for each quantum Hall state. c, ΔR D at \(
u =-\,\frac{1}{2}\) shown as a \(B{| }_{{\alpha }_{{\rm{c}}}}-{V}_{{\rm{PG}}}\) pajama plot, showing clear Aharonov–Bohm oscillations. Inset, 2D-FFT analysis used to extract the magnetic-field periodicity \(\frac{{\varPhi }_{0}}{\Delta B}\) shown on the lower-right side of the pajama. d,e, Same as a,b for the electron-doped filling factors \(
u =\frac{4}{3}\), \(\frac{3}{2}\) and \(\frac{5}{3}\). f, Same as c for \(
u =\frac{3}{2}\) with partitioning of the fractional inner mode. a.u., arbitrary units. Full size image
Focusing on the \(-\frac{1}{2}\) state, Fig. 2c shows the interference pattern as a function of V PG and \(B{| }_{{\alpha }_{{\rm{c}}}}\), in which the α c constraint indicates that V CG is adjusted to maintain constant filling. Specifically, we present the data as ΔR D = R D − ⟨R D ⟩, subtracting the average value at each magnetic field. The positive slope of the pajama indicates Aharonov–Bohm-dominated interference, because increasing V PG decreases the interference area for hole-doped states. The measured visibility, defined as Visibility = (G max − G min )/(G max + G min − 2G outer ), in which G max and G min are the maximum and minimum diagonal conductance values, respectively, and G outer represents the conductance of any fully transmitted outer edge modes, is around 1.9%, comparable with that at integers and odd-denominator states (see Supplementary Information Section 6), and the corresponding edge mode velocity of v edge = 7.95 × 103 m s−1, approximately an order of magnitude smaller than that observed at integer filling, is extracted from source-drain bias V SD -dependent R D (see Supplementary Information Section 7). To extract the flux periodicity, we perform a 2D fast Fourier transform (2D-FFT), shown in the inset of Fig. 2c as a function of \(\frac{{\varPhi }_{0}}{\Delta B}\) and \(\frac{1}{\Delta {V}_{{\rm{PG}}}}\). From the magnetic-field periodicity, we extract \(A\frac{{\varPhi }_{0}}{\Delta \varPhi }\approx -\,0.53\,{{\rm{\mu }}{\rm{m}}}^{2}\). The lithographic area A ≈ 1 μm2 agrees with that extracted from interference at ν = −1 to within 2% (see Supplementary Information Section 8). Using the same area at \(
u =-\,\frac{1}{2}\) yields the unexpected flux periodicity ΔΦ = (1.89 ± 0.26)Φ 0 ≈ 2Φ 0 . We found that this unexpected 2Φ 0 flux periodicity was robust against changes in the compressibility of the bulk, magnetic field and in plunger-gate spectroscopy13 (see Supplementary Information Sections 10, 11 and 12). Furthermore, our transmission study (Supplementary Information Section 9) shows qualitatively similar interference patterns over the experimentally available t R,L range. At very low t, at which we expect electron-dominated tunnelling, the visibility is lost.
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