Strongly coupled edge states in a graphene quantum Hall interferometer

Abstract

Electronic interferometers using the chiral, one-dimensional (1D) edge channels of the quantum Hall effect (QHE) can demonstrate a wealth of fundamental phenomena. The recent observation of phase jumps in a Fabry-Pérot (FP) interferometer revealed anyonic quasiparticle exchange statistics in the fractional QHE. When multiple integer edge channels are involved, FP interferometers have exhibited anomalous Aharonov-Bohm (AB) interference frequency doubling, suggesting putative pairing of electrons into $2e$ quasiparticles. Here, we use a highly tunable graphene-based QHE FP interferometer to observe the connection between interference phase jumps and AB frequency doubling, unveiling how strong repulsive interaction between edge channels leads to the apparent pairing phenomena. By tuning electron density in-situ from filling factor $\nu <2$ to $\nu >7$ , we tune the interaction strength and observe periodic interference phase jumps leading to AB frequency doubling. Our observations demonstrate that the combination of repulsive interaction between the spin-split $\nu =2$ edge channels and charge quantization is sufficient to explain the frequency doubling, through a near-perfect charge screening between the localized and extended edge channels. Our results show that interferometers are sensitive probes of microscopic interactions and enable future experiments studying correlated electrons in 1D channels using density-tunable graphene.

Fig. 1. Highly tunable Fabry-Pérot interferometer in graphene.

a False-color scanning electron microscopy image of a Fabry-Pérot (FP) device identical to the device measured here. The graphite top-gate layer is selectively etched to form 8 separated top-gates (purple). Metal bridges (blue) connect to each graphite top-gate region and two additional bridges (yellow) suspend over the quantum point contacts (QPCs). The lithographic area of the interferometer cavity (area $A=1.16\mu {\text{m}}^2$) is defined by the central hexagonal top-gate. Scale bar: $1\mathrm{\mu }\mathrm{m}$. b Simplified schematic of the FP tuned so that filling factors ${\nu }_{\mathrm{LG}}={\nu }_{\mathrm{MG}}{=\nu }_{\mathrm{RG}}=2$ and ${\nu }_{\mathrm{SG}1}={\nu }_{\mathrm{PG}}{=\nu }_{\mathrm{SG}2}=0$ illustrating interference of the partitioned outer edge channel (EC) (red) while the inner EC (blue) forms a closed annulus inside the FP. Voltage $V_{\mathrm{i}}$ applied to the top-gate labeled ‘i’ tunes the local filling factor ${\nu }_{\mathrm{i}}$. Voltages $V_{\mathrm{QPC}1}$ and $V_{\mathrm{QPC}2}$ applied to the suspended metal bridges selectively gate the QPC constrictions through the etched graphite gaps, tuning the QPC transmissions. We measure the diagonal conductance $G_{\mathrm{D}}=I_{\mathrm{d}}/\left(V_{\mathrm{D}}^{+}-V_{\mathrm{D}}^{-}\right)$, where $V_{\mathrm{D}}^{\pm }$ and $I_{\mathrm{d}}$ are measured voltages in $\left(\pm \right)$ probes and drained current, respectively. See Supplementary Fig. 1 for the full device details. In addition to magnetic field, we tune the interference phase using voltage $V_{\mathrm{MG}}$ on the ‘middle gate’ or $V_{\mathrm{PG}}$ on the ‘plunger gate’. c Conductance as a function of $V_{\mathrm{QPC}1}$ with $V_{\mathrm{QPC}2}=7\mathrm{V}$ (i.e. $T_{\mathrm{QPC}2}=2$) demonstrating QPC1 tunings to interfere outer EC (red dot) and inner EC (blue dot) in $\nu =2$. See Supplementary Fig. 3 for QPC tuning details and voltages set on the other gates to form the necessary QPC saddle-points to acquire this data. d Same type of plot as (c), but demonstrating QPC2 operation instead of QPC1. e, f Characteristic FP oscillations as a function of $V_{\mathrm{PG}}$ for the inner EC and outer EC, respectively, at the QPC tunings indicated in (c) and (d). Vertical dashed lines indicate edges of plateaus of filling factor ${\nu }_{\mathrm{PG}}$. All data is at fixed magnetic field $B=6\mathrm{T}$.

Fig. 2. Density-tuned Aharonov-Bohm frequency doubling transition of outer EC.

a Hall conductance $G_{\mathrm{xy}}$ in the region gated by $V_{\mathrm{LG}}$, demonstrating that $V_{\mathrm{LG}}$ (equivalently, any of the top gates) tunes the filling $\nu$ underneath it at a fixed magnetic field $B=6\mathrm{T}$. Colored dots indicate the filling (set by equivalent $V_{\mathrm{MG}}$ voltages) at which interference data are shown in (b–e); vertical dashed lines show the range of $V_{\mathrm{MG}}$ swept for (f). Top inset: schematic of compressible regions expected in the FP cavity when $V_{\mathrm{MG}}$ is swept. b–e Conductance $G_{\mathrm{D}}$ oscillations on the outer EC with $V_{\mathrm{PG}}$ and $B$, for each of the indicated $V_{\mathrm{MG}}$ values. f $G_{\mathrm{D}}$ oscillations on the outer EC with $V_{\mathrm{PG}}$ and $V_{\mathrm{MG}}$, for $V_{\mathrm{MG}}$ swept continuously over the transition from apparent $h/e$ to $h/2e$ oscillations periodicity, at $B=6\mathrm{T}$. $G_{\mathrm{D}}$ is plotted as a percentage of $\frac{e^2}{h}$ deviation from the average value calculated for each fixed $V_{\mathrm{MG}}$ linecut and subtracted off. Further phase jumps or periodicity changes are not observed past $V_{\mathrm{MG}}\approx 1.7\mathrm{V}$ (checked up to $\nu =7$). QPCs are retuned to maintain transmissions $T_{\mathrm{QPC}1}=T_{\mathrm{QPC}2}=0.5$ over the dataset while ${\nu }_{\mathrm{LG}}{=\nu }_{\mathrm{RG}}=2$ and ${\nu }_{\mathrm{SG}1}{=\nu }_{\mathrm{SG}2}=0$ are fixed.

Fig. 3. Phase jump extraction in the transition regime.

a Phase $\theta$ of the 1D FFT extracted along linecuts parallel to the phase jumps in (b). The phase is evaluated at the dominant frequency in the FFT amplitude spectrum for the linecuts in between phase jumps. A linear increase in phase extracted from regions without phase jumps is subtracted off to make the phase jump magnitude evident as the vertical shift between plateaus in panel (a). From this data we extract an average $\mathrm{\Delta }\theta /2\pi \approx -0.47,$ reflecting approximately half of an electron repelled from the outer EC for each charge added to the inner EC within this range of $V_{\mathrm{MG}}$. Inset: illustration of the coupling $K_{12}$ between the outer and inner ECs contributing to the phase jumps. b Conductance $G_{\mathrm{D}}$ oscillations on the outer EC with $V_{\mathrm{PG}}$ and $V_{\mathrm{MG}}$ near the center of the transition regime showing periodic phase jumps along the dashed black lines. Note that increasing $V_{\mathrm{MG}}$ adds electrons to the system or equivalently increases phase, so the phase jumps correspond to negative shifts in phase i.e., repulsion of electrons from the FP cavity. Similar interference patterns are observed in both the strong and weak QPC backscattering regimes (Supplementary Fig. 4) as well as at elevated temperatures (Supplementary Fig. 5).

Fig. 4. Comparison of inner and outer EC interference and couplings across transition.

a Conductance $G_{\mathrm{D}}$ oscillations on the inner EC ($T_{\mathrm{QPC}1}=T_{\mathrm{QPC}2}=1.5$) with $V_{\mathrm{PG}}$ and $B$, for $V_{\mathrm{MG}}=1.2\mathrm{V}$. Dotted black lines highlight conductance maxima. Left inset: illustration of inner EC interference configuration. b 2D FFT of the $G_{\mathrm{D}}$ oscillations in (a) showing peak ${\mathit{f}}_{\mathrm{i}}$ (vector corresponding to blue arrows) and its harmonics, where ${\mathrm{\Phi }}_0\equiv h/e$. c $G_{\mathrm{D}}$ oscillations on the outer EC ($T_{\mathrm{QPC}1}=T_{\mathrm{QPC}2}=0.5$) at the same density set by $V_{\mathrm{MG}}$. Dotted black lines with identical slope to (a) highlight phase jumps. Left inset: illustration of outer EC interference configuration. d 2D FFT of oscillations in (c) showing the peaks ${\mathit{f}}_{\mathrm{o}}$ (red arrows), ${\mathit{f}}_{\mathrm{o}+\mathrm{i}}$, and ${\mathit{f}}_{\mathrm{o}-\mathrm{i}}$ and their harmonics. e Magnitude of the phase jump on the outer EC as a function of $V_{\mathrm{MG}}$. Each data point is averaged over a $~0.25\mathrm{V}$ range in $V_{\mathrm{MG}}$; error bars indicate $\pm 1$ standard deviation in this range. Unfilled data points represent zero observable phase jumps over the range, hence we infer a magnitude of $0$ or $-1$. $G_{\mathrm{xy}}$ of the device taken in an identical measurement to Fig. 2a, reflecting the expected filling ${\nu }_{\mathrm{MG}}$, is plotted for reference. Top inset: cartoon of the outer and inner EC evolution with increasing $V_{\mathrm{MG}}$. f Magnitudes $I_{\mathrm{o}}$, $I_{\mathrm{o}+\mathrm{i}}$, and $I_{\mathrm{o}-\mathrm{i}}$ of the respective peaks ${\mathit{f}}_{\mathrm{o}}$, ${\mathit{f}}_{\mathrm{o}+\mathrm{i}}$, and ${\mathit{f}}_{\mathrm{o}-\mathrm{i}}$ as a function of $V_{\mathrm{MG}}$. $I_{\mathrm{o}}$, $I_{\mathrm{o}+\mathrm{i}}$, and $I_{\mathrm{o}-\mathrm{i}}$ are normalized by the sum $I_{\mathrm{o}}+I_{\mathrm{o}+\mathrm{i}}+I_{\mathrm{o}-\mathrm{i}}$ to show their relative contributions. Each data point is extracted from a 2D FFT (Supplementary Fig. 7). g Magnetic field frequency multiplied by ${\mathrm{\Phi }}_0$ for peaks ${\mathit{f}}_{\mathrm{o}}$, ${\mathit{f}}_{\mathrm{i}}$, ${\mathit{f}}_{\mathrm{o}+\mathrm{i}}$, and ${\mathit{f}}_{\mathrm{o}-\mathrm{i}}$ tracked through the transition. Note that ${\mathit{f}}_{\mathrm{i}}$ is measured from a separate measurement of interference on the inner EC (Supplementary Fig. 8). h Same as (g) but for plunger gate frequency. Horizontal dashed lines in (g, h) indicate the corresponding ${\mathit{f}}_{\mathrm{o}}$ and ${2\mathit{f}}_{\mathrm{o}}$ values before the transition. Black (red) dots show calculated ${\mathit{f}}_{\mathrm{o}}\pm {\mathit{f}}_{\mathrm{i}}$ from outer and inner EC data, which match the peaks identified as ${\mathit{f}}_{\mathrm{o}+\mathrm{i}}$ and ${\mathit{f}}_{\mathrm{o}-\mathrm{i}}$, respectively.