Abstract
The presence of “upstream” modes, moving against the direction of charge current flow in the fractional quantum Hall (FQH) phases, is critical for the emergence of renormalized modes with exotic quantum statistics. Detection of excess noise at the edge is a smoking gun for the presence of upstream modes. Here, we report noise measurements at the edges of FQH states realized in dual graphitegated bilayer graphene devices. A noiseless dc current is injected at one of the edge contacts, and the noise generated at contacts at length, L = 4 μm and 10 μm away along the upstream direction is studied. For integer and particlelike FQH states, no detectable noise is measured. By contrast, for “holeconjugate” FQH states, we detect a strong noise proportional to the injected current, unambiguously proving the existence of upstream modes. The noise magnitude remains independent of length, which matches our theoretical analysis demonstrating the ballistic nature of upstream energy transport, quite distinct from the diffusive propagation reported earlier in GaAsbased systems.
Introduction
Transport in integer quantum Hall (QH) states occurs through onedimensional edge modes located at the edge of the sample with downstream chirality dictated by the magnetic field (Fig. 1a). This is also true for particlelike fractional quantum Hall (FQH) states^{1,2}. By contrast, socalled “holeconjugate” FQH states (i + 1/2 < ν < i + 1 with filling factor ν and i = 0, 1, 2,..) are expected to host counterpropagating chiral edge modes moving respectively along the downstream and upstream directions. A paradigmatic example is the ν = 2/3 bulk state. MacDonald and Johnson^{3,4} proposed that the edge supports two counterpropagating modes: a downstream mode, ν = 1, and an upstream ν = 1/3 mode (Fig. 1bmiddle panel). The existence of upstream modes is of fundamental importance^{5} and crucially affects transport properties including electrical and thermal transport^{6,7}, noise, and particle interferometry^{8,9}.
There has been an extensive effort over recent years to detect experimentally upstream modes and their properties in GaAs/AlGaAs quantum well based 2DEG. Two questions then come to mind. The first is whether the upstream modes can be detected by measuring the electrical conductance. Indeed, for the holeconjugate ν = 2/3 state, and for distances much smaller than the charge equilibration length, the twoterminal electric conductance G is expected^{10} to be \(4/3\frac{{e}^{2}}{h}\) (instead of \(2/3\frac{{e}^{2}}{h}\)), confirming the counterpropagating character of edge modes. This value was indeed measured, but only on engineered edges at interfaces between two FQH states^{11}. In experiments on a conventional edge (the boundary of a ν = 2/3 FQH state), \(G=2/3\frac{{e}^{2}}{h}\) is found to be robust and essentially equal to the filling factor ν = 2/3, implying that charge propagates only in the downstream direction. This topological value of \(2/3\frac{{e}^{2}}{h}\equiv \nu \frac{{e}^{2}}{h}\) is expected universally in the regime of strong charge equilibration^{10,12,13,14}; it does not tell anything about the presence or absence of upstream modes. For spinunpolarized ν = 2/3 state on a conventional edge in GaAs^{15} an increase of G from \(G=2/3\frac{{e}^{2}}{h}\) to \(\approx 0.73\frac{{e}^{2}}{h}\) was detected at shortest lengths, thus providing an indication of counterpropagating modes. However, the conductance remained far from \(4/3\frac{{e}^{2}}{h}\). For a more conventional spinpolarized ν = 2/3 state, also studied in Ref. ^{15}, G remained equal to \(2/3\frac{{e}^{2}}{h}\) down to the lowest distances. The second question is whether the upstream mode can be detected by measuring the thermal conductance. Thermal transport on an edge may be qualitatively different from charge transport^{10,13}. Even if charge propagates only downstream, energy may propagate upstream. Measurements of thermal conductance G_{Q} at ν = 2/3 and related fillings yielded results fully consistent with the theory expected from holeconjugated character of these states with upstream modes^{16,17,18,19}. Still, measurements of G_{Q} do not prove unambiguously the existence of upstream modes. Ideally, for ν = 2/3, holeconjugate state with counterpropagating 1 and 1/3 modes is favored by particlehole symmetry, which is however only approximate in any realistic situation. Note, however, that an edge with two copropagating ν = 1/3 modes is a fully legitimate candidate^{1,2} too. Such a model would be consistent not only with measured electrical conductance \(G=2/3\frac{{e}^{2}}{h}\) but also with the thermal conductance measurements reported in Ref. ^{18}. On a broader scope, several alternative approaches attempting to establish the presence of upstream current of heat at edges of a variety of FQH states in GaAs/AlGaAs structures were employed in Refs. ^{20,21,22,23,24}. Those studies used structures involving quantum point contacts or quantum dots. It is also worth noting that candidates to the nonAbelian ν = 5/2 state possess a different number of upstream modes, and intensive current efforts aim at understanding which of them is actually realized in experiment^{17,25,26,27,28,29,30}.
The emergence of the twodimensional graphene platform opened up a new era in the study of FQH physics^{31,32}, with different Landau level structure (as compared with traditional GaAs heterostructures), new fractions, and enriched family of quantum Hall states due to an interplay of spin, orbital, and valley degrees of freedom^{33,34,35,36,37,38}. Furthermore, graphene features an unprecedented sharp confining potential and is thus expected to exhibit bulkedge correspondence without additional complex edge reconstruction^{39,40}. Also, in view of the sharp edge, one expects strong interaction between the edge modes in graphene, which may give access to regimes that are difficult to reach in GaAs structures. Moreover, the presence of the layer degree of freedom offers richer tunability in bilayer graphene^{31,41}. Remarkably, for graphene or graphenebased hybrid structures, no direct evidence for the presence of upstream modes has so far been reported. Here, we report a smoking gun signature of upstream modes for holeconjugate FQH states in bilayer graphene and identify their nature, employing noise spectroscopy, which is a purely electrical tool. The essence of our approach is as follows^{42}. When a bias is applied to an FQH edge segment, the Joule heat is dissipated at the “hot spots” as shown in Fig. 1c. In the presence of upstream modes, heat is transported upstream to the socalled noise spot (Fig. 1c), where the heat partitions the charge current and thereby generates noise.
We have carried out electrical conductance together with noise measurements at integer QH states, electronlike state ν = 1/3 and holeconjugate states ν = 2/3 and ν = 3/5, realized in a dual graphite gated hexagonalboronnitride (hBN)encapsulated highmobility bilayer graphene (BLG) devices in a cryofree dilution fridge at a base temperature of ~20 mK. For ν = 1/3 state and integer QH states, we do not detect any excess noise along the upstream direction. This is expected because the corresponding edge states do not host upstream modes. By contrast, for ν = 2/3 and ν = 3/5 FQH states, a finite noise is detected which increases with increasing injected current. At the same time, the averaged current in the upstream direction is zero. Thus, noise detection unambiguously demonstrates that upstream modes exist for the holeconjugate FQH states in graphene and only carry heat energy. Moreover, the magnitude of the noise remains constant for two different lengths L = 10 μm and 4 μm between the current injecting contact and noise detection point in the same device. Moreover, our experimentally measured noise magnitude matches remarkably well with our theoretical analysis. This conclusively demonstrates the ballistic nature of upstream modes, implying the absence of thermal equilibration on the length scales employed in the experiment.
Results
Device and electrical response
For our measurements, we have used two dual graphite gated hexagonalboronnitride (hBN)encapsulated bilayer graphene devices, which are fabricated using the standard dry transfer pickup technique, see the Method section and Supplementary Information (SI), section S1. To observe, at modest magnetic fields, welldeveloped FQH states, which are highly susceptible to disorder (below ν < 1), screening layers are required. The presence of the top graphite gate serves that purpose and improves the quality of the device^{43}. The device schematic is shown in Fig. 1c. The zeromagneticfield gate response of the devices is shown in SI (S1). From this data, the mobility of the devices is extracted and found to be ~1.7 × 10^{5}cm^{2}V^{−1}s^{−1} and ~3 × 10^{5}cm^{2}V^{−1}s^{−1} for devices 1 and 2, respectively. First, we perform electrical conductance measurements at a fixed magnetic field of 10 T using standard lockin amplifier. The electron density is tuned by the back graphite gate, keeping the top graphite gate fixed at zero voltage. The measured transverse Hall resistance (R_{xy}) and longitudinal resistance (R_{xx}) of device1 are shown in Fig. 1d for ν ≤ 1. Clear plateaus in R_{xy} are developed for ν = 1, ν = 2/3, and ν = 1/3, accompanied with zeroes in R_{xx} for ν = 1 and ν = 2/3, and a very low (in comparison to h/e^{2}) resistance of 30 − 40Ω for ν = 1/3. Similarly, plateaus are seen in R_{xy} for ν = 1, ν = 2/3, ν = 3/5 and ν = 1/3 for device2, see SI (S10), where R_{xx} reaches zero for ν = 1, ν = 2/3 and ν = 1/3, and ~200 Ω for ν = 3/5.
Next, we perform noise measurements at ν = 2/3, ν = 1/3, and integer QH states for two different lengths L = 10 μm and 4 μm of device1. The Measurement scheme for L = 10 μm is shown in Fig. 2a, where the device is set into ν = 2/3 QH state with d and u representing counterpropagating downstream and upstream eigenmodes. The measurement scheme for shorter length L = 4 μm is obtained in the same device by changing the chirality (see SI, S5). Before the noise measurements, we perform two crucial checks: biasdependent response of the ν = 2/3 FQH state and chirality of charge transport. For this, a 100 pA AC signal is injected on top of the DC bias current at contact C, keeping contact D at the ground, and the AC voltage at contacts C and B is measured. The voltage at contact C, shown as V_{C} in Fig. 2c, remains flat as a function of the DC bias current, which means the conductance of the ν = 2/3 state does not change with the bias current. This kind of response, implying that there is no transport through the bulk (via states above the bulk gap), is a prerequisite check for the noise measurements. This is in full consistency with the value of the gap of ν = 2/3 at 10 T, which is around 5 K as determined from the activation plot of R_{xx} (see SI, S9), i.e., well above our largest bias voltage. Further, no detectable voltage is measured at contact B (shown as V_{B} in Fig. 2c), which demonstrates that the charge propagates downstream only. Similar results are obtained for the shorter length (4 μm) (see SI, S5). Thus, if our ν = 2/3 edge hosts counterpropagating modes (which is demonstrated below), the charge equilibration length is much smaller than the size of our device.
Noise and analysis
To unambiguously demonstrate the existence of an upstream mode, we injected a noiseless dc current at contact C and measured the voltage noise, S_{V} at contact B along the upstream direction (Fig. 2a). For the noise measurement, an LCR resonant circuit with the resonance frequency of ~763 kHz is utilized together with an amplifier chain and a spectrum analyzer (see methods and SI(S4))^{44,45,46,47}. At zero bias, the equilibrium voltage noise measured at the amplifier contact is given by^{18,44}
where k_{B} is the Boltzmann constant, T is the temperature, R is the resistance of the QH state, G is the gain of the amplifier chain, and BW is the bandwidth. The first term, 4k_{B}TR, corresponds to the thermal noise, and \({V}_{n}^{2}\) and \({i}_{n}^{2}\) are the intrinsic voltage and current noise of the amplifier. At finite bias, the upstream mode carries heat from the hot spot to the noise spot near the amplifier contact. This leads to the enhancement of stochastic intermode tunneling processes in the noisespot region, which creates an excess noise, see below for detail. At the same time, the intrinsic noise of the amplifier remains unchanged. Frequency independence of the thermal noise and the excess noise allows us to operate at a higher frequency (~763 kHz) so that we can eliminate the contribution from flicker noise (1/f) which usually becomes negligible for frequency above a few tens of Hz. The excess noise (δS_{V}) due to bias current is obtained by subtracting the noise value at zero bias from the noise at finite bias, i.e δS_{V} = S_{V}(I) − S_{V}(I = 0). The excess voltage noise δS_{V} is converted to excess current noise S_{I} according to \({S}_{I}=\frac{\delta {S}_{V}}{{R}^{2}}\), where \(R=\frac{h}{\nu {e}^{2}}\) is the resistance of the considered QH edge.
In the absence of an upstream mode, the energy cannot flow from the hot spot near C to the noise spot near B, see Fig. 2a, so that no noise is expected. This is precisely what is observed for ν = 1 and ν = 1/3 states, see Fig. 2d. At the same time, it is shown in Fig. 2d that for the ν = 2/3 state there is a strong noise which increases almost linearly with current. This is quite striking as at contact B the timeaveraged current is zero (Fig. 2c). This clearly demonstrates that the ν = 2/3 edge hosts an upstream mode that leads to an upstream propagation of energy, even though the charge propagates downstream only. The mechanism of the noise generation is as follows^{42}. The heat propagating upstream from the hot spot near C reaches the noise spot near B, inducing their creation of particlehole pairs propagating in opposite directions. If the particle (or hole) is absorbed at contact B, while the hole (or, respectively, particle) flows downstream, there will be a voltage fluctuation at B detected by our noise measurement scheme. Similarly, the noise along the upstream direction is detected for ν = 2/3 and ν = 3/5 of device2 as shown in SI(S10). Note that the 1/3 state gives rise to detectable noise at a larger bias current (Fig. 2d), which is, however, very weak: the noise magnitude remains ~15 times smaller than the noise measured for 2/3 state. This tiny increase in noise for 1/3 as compared to ν = 1 in Fig. 2d may be attributed to a minute amount of bulk contribution for 1/3 state. This is consistent with the bias response of the 1/3 state shown in SI(S12) where the conductance barely changes with the applied bias, thus confirming that the bulk contribution to transport in the 1/3 state is indeed tiny.
To verify that the heat propagates from the hot spot to the noise spot entirely via the edge, we have measured the noise in an alternative configuration. In this setting, a noiseless dc current is injected at contact A, while the contact C is electrically floating as shown in Fig. 2b. In this situation, in order to induce the noise, the heat would have to propagate upstream from the hot spot near D to the noise spot near B. However, this path is “cut” by the metallic contact C held at the base temperature. Thus, the only way for the heat to propagate is via the bulk. As shown in Fig. 2e, no detectable noise is measured at contact B, which rules out any sizeable bulk conduction of heat in our device.
To inspect the length dependence of the noise, we have also studied it for L = 4 μm of the same device1 (see SI, S5). The data are shown in Fig. 2d. It can be seen that the noise amplitude is nearly identical for L = 10 μm and 4 μm. This is striking since it shows that the heat propagates upstream ballistically and without losses along the ν = 2/3 edge, from the hot spot to the noise spot. There are two distinct mechanisms that could suppress the heat propagation and thus the noise: (i) thermal equilibration between the counterpropagating modes^{10,13,42,48} and (ii) dissipation of energy from the edge to other degrees of freedom, including phonons, photons, and Coulombcoupled localized states^{49,50,51}. Our results show that none of these mechanisms is operative at bath temperature (T_{bath}) ~ 20 mK on the length scale of L = 10 μm. The absence of thermal equilibration on the edge is in a striking contrast with the very efficient electric equilibration emphasized above.
Up to now, all the data were at T_{bath} ~ 20 mK. Now, we explore the effect of temperature. Figure 3a and b show the evolution of the ν = 2/3 noise at L = 4 μm and L = 10 μm, respectively, with increasing T_{bath}. At each temperature, the noise data in Fig. 3a and b are fitted linearly for both positive and negative bias current, and from the averaged slopes we quote the noise value at I_{SD} = ± 1nA in Fig. 3c for both L = 4 μm and L = 10 μm. The linear fittings are shown in SI (S7). It is seen that the noise remains constant and equal for both lengths up to T_{bath} ~ 50 mK. We have verified [see SI (S6)] that the cooling of the electron system remains efficient down to our lowest temperature, i.e., the electron temperature T_{e} is very close to T_{bath} in the whole temperature range of our measurements. In particular, for T_{bath} of 40 mK and 50 mK, the electron temperature T_{e} is equal to T_{bath}. For the lowest temperature, T_{bath} = 20 mK, we find, for the ν = 2/3 state, T_{e} = 22 mK, i.e., only slightly higher than T_{bath}.
For higher temperature, the noise decays with T_{bath} (or, equivalently, with T_{e}), and the decay is substantially faster for the larger L. This decay can be attributed to one of two mechanisms mentioned above: thermal equilibration within the edge (which would imply a crossover from ballistic to diffusive regime of heat flow) or loss of heat to the bulk. Both mechanisms are expected to be enhanced at higher temperature^{18,49,50,51}, but further work is needed to understand which of them is dominant.
Our experiments thus clearly indicate that at low temperatures, T_{bath} ≤ 50 mK, the upstream heat transport is ballistic and lossless. To support this conclusion, we have calculated theoretically the expected noise S_{I} on the ν = 2/3 edge in this regime. The theory extends that of Refs. ^{42,48} to the ballistic (rather than diffusive) regime of heat transport corresponding to vanishing thermal equilibration. In this regime, the backscattering of heat takes place only at interfaces with the contact regions^{10,19}. We assumed the bias voltage V = (3h/2e^{2})I_{SD} to be much larger than T, which is well fulfilled for our typical current I_{SD} ~ 1nA. The result (see method section and SI (S13) for detail)
is shown in Fig. 2d and is an excellent agreement with the experimental data, thus giving a further strong support to our interpretation of the experiment.
Discussion
Our measurements of noise present an unambiguous demonstration of the presence of an upstream mode in ν = 2/3 and ν = 3/5 FQH edges in graphene. This mode is responsible for the upstream heat transport that is at the heart of the noise generation mechanism. Remarkably, the noise is temperatureindependent for T_{bath} ≤ 50 mK and remains the same for L = 4 μm and L = 10 μm, demonstrating the ballistic and lossless character of heat transport. The ballistic heat transport implies the absence of thermal equilibration on the edge, in contrast to full charge equilibration revealed by electric conductance measurements. This is entirely consistent with the data of Ref. ^{18} on thermal conductance in graphene. There, a dramatic difference between the charge and heat equilibration lengths was explained by the vicinity of a system to a stronginteraction fixed point, where the bare modes of the ν = 2/3 edge are renormalized into a charge and a neutral mode. Very recently, the absence of thermal equilibration (notwithstanding very efficient electric equilibration) was also reported for GaAs samples^{19}.
Methods
Device fabrication and measurement scheme
For making encapsulated devices, we used the standard dry transfer pickup technique^{52,53}. Fabrication of these heterostructures involved mechanical exfoliation of hBN and graphite crystals on an oxidized silicon wafer using the widely used scotchtape technique. BLG and graphite flakes were exfoliated from natural graphite crystals. Suitable flakes were identified under the optical microscope. The thickness of the top and bottom graphites were ~5 nm and ~20 nm, respectively, and the thickness of of the top and bottom hBN flakes were of the order of ~20 nm. The smaller distance between graphite gate and BLG layer (~20 nm thick bottom hBN) was a similar range to the magnetic length scale of our experiment at 10 T, implying a sharp confining potential at the physical edge of BLG. Details of the fabrication procedure are in SI (S1). The BLG channel area of the stack was microscopically ironed using an AFM (atomic force microscopy) tip in contact mode^{54}, to remove any atomic level strain or ripples or small bubbles from the channel area, which can arise due to the stacking process. After this, for making contacts we used electron beam lithography (EBL). After EBL, reactive ion etching (mixture of CHF_{3} and O_{2} gas) was used to define the edge contact. Then, thermal deposition of Cr/Pd/Au (4/12/60 nm) was done in an evaporator chamber having a base pressure of ~1 − 2 × 10^{−7} mbar. The optical image of the two measuring devices are shown in SI (S1). The schematic of the device and measurement setup are shown in Fig. 1c. All the measurements are done in a cryofree dilution refrigerator having a base temperature of ~20mK. The electrical conductance was measured using the standard Lockin technique. For R_{xy}, an ac current is injected at contact C (Fig. 1c), contact A is kept at the ground, and we measure the potential difference across the contacts D and B. For R_{xx}, we inject current at C, contact D is kept at ground, and we measure the potential difference across the contacts A and B. The resistance value of R_{xy} for ν = 2/3 is ≈39 kΩ = (3/2)h/e^{2}, implying full charge equilibration among the counterpropagating eigenmodes. Under no charge equilibration, one would get a very different value of the resistance according to LandauerButtiker formalism, see SI (S11) for detail. The noise is measured employing noise thermometry based on an LCR resonant circuit at the resonance frequency of ~763 kHz and amplified by homemade preamplifier at 4K followed by room temperature amplifier, and finally measured by a spectrum analyzer. The details of the noise measurement technique are mentioned in the SI (S4).
Theoretical calculation of noise
We assume the regime of strong charge equilibration along the edge segment, i.e., \(L\gg {l}_{{{{{{{{\rm{eq}}}}}}}}}^{C}\), where \({l}_{{{{{{{{\rm{eq}}}}}}}}}^{C}\) is the charge equilibration length. Then, the dc noise S generated due to intermode tunneling along this segment is given by
Here, h is the Planck constant, and ν_{+} and ν_{−} are the total filling factors associated with the downstream and upstream edge modes, respectively, with the bulk filling factor ν = ν_{+} − ν_{−}. The exponential factor in the integral is a result of chiral charge transport, ν_{+} ≠ ν_{−}, and implies that the dominant noise contribution comes from the noise spot—a region of size \(\sim {l}_{{{{{{{{\rm{eq}}}}}}}}}^{C}\) near the contact that is located on the upstream side of the segment (the voltage probe where the noise is measured).
The noise kernel Λ(x) is calculated by assuming a thermally nonequilibrated regime. In this case, Λ becomes xindependent within the edge segment, so that the xintegral is straightforwardly calculated, yielding for the ν = 2/3 edge the noise
where the dependence of Λ on the bias voltage V and the interaction between edge modes (parameter Δ) is noted. We take Δ ≈ 1, which corresponds to the stronginteraction fixed point^{5}, at which the heat equilibration length is much larger than the charge equilibration length^{18}, as observed experimentally. To determine the dependence of Λ on the voltage, we first calculate the effective temperatures of the downstream and upstream modes on the edge. They are found from the system of energy balance equations that include the Joule heating at the hot spot as well as a partial reflection of heat at interfaces between the interacting segment of the edge and the contact region. The contacts are modeled in terms of noninteracting one and 1/3 modes (see SI, S13). For Δ = 1, the corresponding reflection coefficient is \({{{{{{{\mathcal{R}}}}}}}}=1/3\). The resulting temperatures of the modes are
The noise kernel Λ is now calculated by using the Green’s function formalism for the chiral Luttinger liquid and the Keldysh technique. Expressing the result in terms of the bias current I_{SD} = (2e^{2}/3h)V, we come to the final result given by Eq. (2) of the manuscript. An analogous calculation for ν = 3/5 edge yields S_{I} = 0.138 eI_{SD}, which is in a very good agreement with the experiment too (see SI, S13).
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Beenakker, C. Edge channels for the fractional quantum Hall effect. Phys. Rev. Lett. 64, 216 (1990).
Wen, X.G. Chiral luttinger liquid and the edge excitations in the fractional quantum Hall states. Phys. Rev. B 41, 12838 (1990).
MacDonald, A. H. Edge states in the fractionalquantumHalleffect regime. Phys. Rev. Lett. 64, 220–223 (1990).
Johnson, M. & MacDonald, A. Composite edges in the ν = 2/3 fractional quantum Hall effect. Phys. Rev. Lett. 67, 2060 (1991).
Kane, C., Fisher, M. P. & Polchinski, J. Randomness at the edge: theory of quantum Hall transport at filling ν = 2/3. Phys. Rev. Lett. 72, 4129 (1994).
Altimiras, C. et al. Chargeless heat transport in the fractional quantum Hall regime. Phys. Rev. Lett. 109, 026803 (2012).
Jezouin, S. et al. Quantum limit of heat flow across a single electronic channel. Science 342, 601–604 (2013).
Ji, Y. et al. An electronic mach–zehnder interferometer. Nature 422, 415–418 (2003).
Nakamura, J. et al. Aharonov–Bohm interference of fractional quantum Hall edge modes. Nat. Phys. 15, 563–569 (2019).
Protopopov, I., Gefen, Y. & Mirlin, A. Transport in a disordered ν = 2/ 3 fractional quantum Hall junction. Ann. Phys. 385, 287–327 (2017).
Cohen, Y. et al. Synthesizing a ν = 2/3 fractional quantum Hall effect edge state from counterpropagating ν = 1 and ν = 1/3 states. Nat. Commun. 10, 1920 (2019).
Kane, C. & Fisher, M. P. Impurity scattering and transport of fractional quantum Hall edge states. Phys. Rev. B 51, 13449 (1995).
Nosiglia, C., Park, J., Rosenow, B. & Gefen, Y. Incoherent transport on the ν = 2/3 quantum Hall edge. Phys. Rev. B 98, 115408 (2018).
Spånslätt, C., Gefen, Y., Gornyi, I. & Polyakov, D. Contacts, equilibration, and interactions in fractional quantum Hall edge transport. Phys. Rev. B 104, 115416 (2021).
Lafont, F., Rosenblatt, A., Heiblum, M. & Umansky, V. Counterpropagating charge transport in the quantum Hall effect regime. Science 363, 54–57 (2019).
Banerjee, M. et al. Observed quantization of anyonic heat flow. Nature 545, 75–79 (2017).
Banerjee, M. et al. Observation of halfinteger thermal Hall conductance. Nature 559, 205–210 (2018).
Srivastav, S. K. et al. Vanishing thermal equilibration for holeconjugate fractional quantum Hall states in graphene. Phys. Rev. Lett. 126, 216803 (2021).
Melcer, R. A. et al. Absent thermal equilibration on fractional quantum Hall edges over macroscopic scale. Preprint at https://arxiv.org/abs/2106.12486 (2021).
Bid, A. et al. Observation of neutral modes in the fractional quantum Hall regime. Nature 466, 585–590 (2010).
Dolev, M. et al. Characterizing neutral modes of fractional states in the second Landau level. Phys. Rev. Lett. 107, 036805 (2011).
Gross, Y., Dolev, M., Heiblum, M., Umansky, V. & Mahalu, D. Upstream neutral modes in the fractional quantum Hall effect regime: heat waves or coherent dipoles. Phys. Rev. Lett. 108, 226801 (2012).
Venkatachalam, V., Hart, S., Pfeiffer, L., West, K. & Yacoby, A. Local thermometry of neutral modes on the quantum Hall edge. Nat. Phys. 8, 676–681 (2012).
Gurman, I., Sabo, R., Heiblum, M., Umansky, V. & Mahalu, D. Extracting net current from an upstream neutral mode in the fractional quantum Hall regime. Nat. Commun. 3, 1289 (2012).
Ma, K. K. & Feldman, D. Partial equilibration of integer and fractional edge channels in the thermal quantum Hall effect. Phys. Rev. B 99, 085309 (2019).
Asasi, H. & Mulligan, M. Partial equilibration of antipfaffian edge modes at ν = 5/2. Phys. Rev. B 102, 205104 (2020).
Simon, S. H. & Rosenow, B. Partial equilibration of the antipfaffian edge due to Majorana disorder. Phys. Rev. Lett. 124, 126801 (2020).
Park, J., Spånslätt, C., Gefen, Y. & Mirlin, A. D. Noise on the nonabelian ν = 5/2 fractional quantum Hall edge. Phys. Rev. Lett. 125, 157702 (2020).
Dutta, B. et al. Novel method distinguishing between competing topological orders. Preprint at https://arxiv.org/abs/2101.01419 (2021).
Yutushui, M., Stern, A. & Mross, D. F. Identifying the ν = 5/2 topological order through charge transport measurements. Preprint at https://arxiv.org/abs/2106.07667 (2021).
Li, J. et al. Evendenominator fractional quantum Hall states in bilayer graphene. Science 358, 648–652 (2017).
Kim, Y. et al. Even denominator fractional quantum Hall states in higher Landau levels of graphene. Nat. Phys. 15, 154–158 (2019).
Yang, K., Sarma, S. D. & MacDonald, A. Collective modes and skyrmion excitations in graphene SU (4) quantum Hall ferromagnets. Phys. Rev. B 74, 075423 (2006).
Sodemann, I. & MacDonald, A. H. Broken SU (4) symmetry and the fractional quantum Hall effect in graphene. Phys. Rev. Lett. 112, 126804 (2014).
Kharitonov, M. Phase diagram for the ν = 0 quantum Hall state in monolayer graphene. Phys. Rev. B 85, 155439 (2012).
Weitz, R. T., Allen, M., Feldman, B., Martin, J. & Yacoby, A. Brokensymmetry states in doubly gated suspended bilayer graphene. Science 330, 812–816 (2010).
Young, A. F. et al. Spin and valley quantum Hall ferromagnetism in graphene. Nat. Phys. 8, 550–556 (2012).
Kumar, C., Srivastav, S. K. & Das, A. Equilibration of quantum Hall edges in symmetrybroken bilayer graphene. Phys. Rev. B 98, 155421 (2018).
Li, G., LuicanMayer, A., Abanin, D., Levitov, L. & Andrei, E. Y. Evolution of landau levels into edge states in graphene. Nat. Commun. 4, 1–7 (2013).
Hu, Z.X., Bhatt, R. N., Wan, X. & Yang, K. Realizing universal edge properties in graphene fractional quantum Hall liquids. Phys. Rev. Lett. 107, 236806 (2011).
Maher, P. et al. Tunable fractional quantum Hall phases in bilayer graphene. Science 345, 61–64 (2014).
Spånslätt, C., Park, J., Gefen, Y. & Mirlin, A. D. Topological classification of shot noise on fractional quantum Hall edges. Phys. Rev. Lett. 123, 137701 (2019).
RibeiroPalau, R. et al. Highquality electrostatically defined Hall bars in monolayer graphene. Nano Lett. 19, 2583–2587 (2019).
Srivastav, S. K. et al. Universal quantized thermal conductance in graphene. Sci. Adv. 5, eaaw5798 (2019).
Sahu, M. R. et al. Enhanced shot noise at bilayer graphene–superconductor junction. Phys. Rev. B 100, 235414 (2019).
Paul, A. K. et al. Interplay of filling fraction and coherence in symmetry broken graphene pn junction. Commun. Phys. 3, 1–7 (2020).
Sahu, M. R. et al. Quantized conductance with nonzero shot noise as a signature of Andreev edge state. Phys. Rev. B. 104, L081404 (2021).
Park, J., Mirlin, A. D., Rosenow, B. & Gefen, Y. Noise on complex quantum Hall edges: chiral anomaly and heat diffusion. Phys. Rev. B 99, 161302 (2019).
Schmidt, D., Schoelkopf, R. & Cleland, A. Photonmediated thermal relaxation of electrons in nanostructures. Phys. Rev. Lett. 93, 045901 (2004).
Gutman, D. B. et al. Energy transport in the Anderson insulator. Phys. Rev. B 93, 245427 (2016).
Rosenblatt, A. et al. Energy relaxation in edge modes in the quantum Hall effect. Phys. Rev. Lett. 125, 256803 (2020).
Purdie, D. et al. Cleaning interfaces in layered materials heterostructures. Nat. Commun. 9, 1–12 (2018).
Pizzocchero, F. et al. The hot pickup technique for batch assembly of Van der Waals heterostructures. Nat. Commun. 7, 1–10 (2016).
Kim, Y., Herlinger, P., Taniguchi, T., Watanabe, K. & Smet, J. H. Reliable postprocessing improvement of Van der Waals heterostructures. ACS nano 13, 14182–14190 (2019).
Acknowledgements
C.S., Y.G., and A.D.M. acknowledge support by DFG Grants MI 658/101 and MI 658/102, and by the GermanIsraeli Foundation Grant No. I1505303.10/2019. Y.G. acknowledges support by the Helmholtz International Fellow Award, and by DFG RO 2247/111, and CRC 183 (project C01), and the Minerva Foundation. Y.G. further acknowledges the support of the Infosys Foundation. C.S. further acknowledges funding from the Excellence Initiative Nano at the Chalmers University of Technology and the 2D TECH VINNOVA competence Center (Ref. 201900068). K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), JST. A.D. thanks the Department of Science and Technology (DST), India for financial support (DSTO2051) and acknowledges the Swarnajayanti Fellowship of the DST/SJF/PSA03/201819. A.D. further thanks for the Institute funding under the Institute of Eminence (IoE). R.K. and S.K.S acknowledge Inspire fellowship, DST and Prime Minister’s Research Fellowship (PMRF), Ministry of Education (MOE), for financial support, respectively.
Author information
Authors and Affiliations
Contributions
R.K. and S.K.S. contributed to device fabrication, data acquisition, and analysis. A.D. contributed in conceiving the idea and designing the experiment, data interpretation, and analysis. K.W. and T.T. synthesized the hBN single crystals. C.S., A.D.M., and Y.G. contributed in the development of theory, data interpretation, and all the authors contributed in writing the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review information
Nature Communications thanks Xu Du and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kumar, R., Srivastav, S.K., Spånslätt, C. et al. Observation of ballistic upstream modes at fractional quantum Hall edges of graphene. Nat Commun 13, 213 (2022). https://doi.org/10.1038/s41467021278054
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467021278054
This article is cited by

Direct determination of the topological thermal conductance via local power measurement
Nature Physics (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.