Nonequilibrated Counterpropagating Edge Modes in the Fractional Quantum Hall Regime

It is well established that density reconstruction at the edge of a two-dimensional electron gas takes place for hole-conjugate states in the fractional quantum Hall effect (such as $v=2/3$, $3/5$, etc.). Such reconstruction leads, after equilibration between counterpropagating edge channels, to a downstream chiral current edge mode accompanied by upstream chiral neutral modes (carrying energy without net charge). Short equilibration length prevented thus far observation of the counterpropagating current channels—the hallmark of density reconstruction. Here, we provide evidence for such nonequilibrated counterpropagating current channels, in short regions ($l=4\mu \mathrm{m}$ and $l=0.4\mu \mathrm{m}$) of fractional filling $v=2/3$ and, unexpectedly, $v=1/3$, sandwiched between two regions of integer filling $v=1$. Rather than a two-terminal fractional conductance, the conductance exhibited a significant ascension towards unity quantum conductance ($G_Q=e^2/h$) at or near the fractional plateaus. We attribute this conductance rise to the presence of a nonequilibrated channel in the fractional short regions.

Figure 1

Samples are fabricated in heterostructures embedding 2DEG with density $1.5\times {10}^{11}{\mathrm{cm}}^{-2}$ with mobility $6\times {10}^6{\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$ buried 85 nm below the surface. Each sample embeds two fractional regions, a long region (LR) of $45\mu \mathrm{m}$ and a short region (SR) of either $4\mu \mathrm{m}$ or $0.4\mu \mathrm{m}$. Current is injected from the source contact S1 (S2), towards a short (long) top-gated fractional region. The filling of the fractional region is varied by applying negative bias to the gate (colored in dark green). The current flows along a defining gate (dark blue) negatively biased and depleted from electrons. The transmitted current is deduced from the measured voltage between $P1$ ($P2$) and the ground.

Figure 2

Conductance vs top-gate voltage of three different fractional region sizes: long region (LR), $45\mu \mathrm{m}$ (black solid line); short region (SR), $4\mu \mathrm{m}$ (blue dotted line); and short region (SR), $0.4\mu \mathrm{m}$ (dark cyan dashed line). (a) Measurements performed at 8.5 T and bulk filling factor ${\nu }_{\mathrm{B}}=1$. The LR conductance shows a few fractional plateaus. For $\mathrm{SR}=4\mu \mathrm{m}$, the conductance near ${\nu }_{\mathrm{f}}=2/3$ rises up to $0.71\times e^2/h$, and of $\mathrm{SR}=0.4\mu \mathrm{m}$ peaks at $0.93\times e^2/h$. Similarly, the conductance near ${\nu }_{\mathrm{f}}=1/3$ peaks at $0.51\times e^2/h$ for $\mathrm{SR}=4\mu \mathrm{m}$ and $0.73\times e^2/h$ for $\mathrm{SR}=0.4\mu \mathrm{m}$. Conductance plateaus at ${\nu }_{\mathrm{f}}=2/5$, $3/5$ and a signature of ${\nu }_{\mathrm{f}}=2/7$ also seen at LR. For ${\nu }_{\mathrm{f}}=2/5$, a reminiscent signature of reconstruction is also seen in the SR, while at ${\nu }_{\mathrm{f}}=3/5$, any signature is absorbed by the conductance peak of ${\nu }_{\mathrm{f}}=2/3$. The graphs are shifted by $0.05\times e^2/h$ for convenience. (b) Similar measurements at 4.3 T at ${\nu }_{\mathrm{B}}=2$. Conductance of LR (black solid line) shows plateaus at values of $(2,5/3,4/3,1,2/3,1/3)\times e^2/h$. The $\mathrm{SR}=4\mu \mathrm{m}$ shows weaker conductance peaks near ($5/3$, $2/3$, $1/3$) while around the integer value of $1\times e^2/h$ the conductance is flat or monotonic, without peaks. For the $\mathrm{SR}=0.4\mu \mathrm{m}$, the conductance increases near all the fractional fillings but remains constant $1\times e^2/h$ near ${\nu }_{\mathrm{f}}=1$.

Figure 3

Dependence of conductance on magnetic field and electron density. (a) Conductance of $\mathrm{LR}=45\mu \mathrm{m}$ and of $\mathrm{SR}=0.4\mu \mathrm{m}$ as a function of the top-gate voltage at different magnetic fields $B=(8.5,7.7,7.0)\mathrm{T}$, drawn in blue (solid line), green (dashed line), and red (dash-dotted line), respectively. As the magnetic field decreases, the conductance near ${\nu }_{\mathrm{f}}=1/3$, $2/3$ rises slightly. Similar behavior was observed for $\mathrm{SR}=4\mu \mathrm{m}$ (see the Supplemental Material [34]). (b) Conductance of $\mathrm{LR}=45\mu \mathrm{m}$ (inset) and of $\mathrm{SR}=0.4\mu \mathrm{m}$ as function of the top-gate voltage at $B=7\mathrm{T}$ and at different bulk densities of $n_{\mathrm{B}}=(1.7,1.8,1.9,2)\times {10}^{11}{\mathrm{cm}}^{-2}$, corresponding to different bias of the back gate, (0.4, 0.6, 0.8, 1.0) V, drawn in blue (solid line), green (dashed line), purple (dash-dotted line), and red (dotted line), respectively. The conductance peaks for ${\nu }_{\mathrm{f}}=1/3$, $2/3$ remain nearly unchanged.

Figure 4

Temperature dependence of $\mathrm{SR}=0.4\mu \mathrm{m}$, 30 mK–4.2 K at ${\nu }_{\mathrm{B}}=1$ and ${\nu }_{\mathrm{B}}=2$. The bottom $x$ axis denotes the voltage on the top gate, the top $x$ axis is the corresponding filling factor ${\nu }_{\mathrm{f}}$ at SR. (a) Conductance of $\mathrm{SR}=0.4\mu \mathrm{m}$ as a function of top-gate voltage at different temperatures (32 mK, 200 mK, 500 mK, 1.2 K, 4.2 K). The apparent conductance peaks near ${\nu }_{\mathrm{f}}=1/3$, $2/3$ decay as temperature rises, with that of the ${\nu }_{\mathrm{f}}=1/3$ being much more prominent. (b) Similar conductance measurements at ${\nu }_{\mathrm{B}}=2$ with temperature range of 40 mK, 420 mK, 600 mK, 1.2 K, 4.2 K. As temperature increases throughout the sample, the transmitted current in ${\nu }_{\mathrm{f}}=1$ rises due to increase of electron tunneling through the gated regions, as expected. Inset: comparison of the classical simulation and the high temperature measurement at $\mathrm{SR}=4\mu \mathrm{m}$—being qualitatively similar.