Observation of ballistic upstream modes at fractional quantum Hall edges of graphene

Department of Physics, Indian Institute of Science, Bangalore, 560012, India.

Department of Microtechnology and Nanoscience (MC2),Chalmers University of Technology, S-412 96 Göteborg, Sweden.

Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany.

Institut für Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany.

National Institute of Material Science, 1-1 Namiki, Tsukuba 305-0044, Japan.

Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel.

Petersburg Nuclear Physics Institute, 188300 St. Petersburg, Russia.

L. D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia.

Introduction: Transport in integer quantum Hall (QH) states occurs through one-dimensional edge modes located at the edge of the sample with downstream chirality dictated by the magnetic field (Fig. 1a). This is also true for particle-like fractional quantum Hall (FQH) states [1, 2]. By contrast, so called “hole-conjugate” FQH states ($i+1/2<\nu<i+1$ with filling factor $\nu$ and $i$=0,1,2,..) are expected to host counter-propagating chiral edge modes moving respectively along the downstream and upstream directions. A paradigmatic example is the $\nu=2/3$ bulk state. MacDonald and Johnson [3, 4] proposed that the edge supports two counter-propagating modes: a downstream mode, $\nu=1$, and an upstream $\nu=1/3$ mode (Fig. 1b). The existence of upstream modes is of fundamental importance and crucially affects transport properties including electrical and thermal transport [5, 6], noise, and particle interferometry [7, 8].

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 hole-conjugate $\nu=2/3$ state, and for distances much smaller than the charge equilibration length, the two-terminal electric conductance $G$ is expected [9] to be $4/3\frac{e^{2}}{h}$ (instead of $2/3\frac{e^{2}}{h}$), confirming the counter propagating character of edge modes. This value was indeed measured, but only on engineered edges at interfaces between two FQH states [10]. In experiments on a conventional edge (the boundary of a $\nu=2/3$ FQH state), $G=2/3\frac{e^{2}}{h}$ is found to be robust and essentially equal to the filling factor $\nu=2/3$, implying that charge propagates only downstream direction; one cannot tell anything about the presence or absence of upstream modes [9, 11, 12]. 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 [9, 11]. Even if charge propagates only downstream, energy may propagate upstream. Measurements of thermal conductance $G_{Q}$ at $\nu=2/3$ and related fillings yielded results fully consistent with the theory expected from hole-conjugated character of these states with upstream modes [13, 14, 15, 16]. Still, measurements of $G_{Q}$ do not prove unambiguously the existence of upstream modes. Ideally, for $\nu=2/3$, hole-conjugate state with counter-propagating 1 and 1/3 modes is favored by particle-hole symmetry, which is however only approximate in any realistic situation. Note, however, that an edge with two co-propagating $\nu=1/3$ modes is a fully legitimate candidate [2, 1] 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. [15]. 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. [17, 18, 19, 20, 21]. Those studies used structures involving quantum point contacts or quantum dots. It is also worth noting that candidates to the non-Abelian $\nu=5/2$ state possess different number of upstream modes, and intensive current efforts aim at understanding which of them is actually realized in experiment [14, 22, 23, 24, 25, 26, 27].

The emergence of the two-dimensional graphene platform opened up a new era in the study of FQH physics [28, 29], 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 [30, 31, 32, 33, 34, 35]. Furthermore, graphene features an unprecedented sharp confining potential and is thus expected to exhibit bulk-edge correspondence without additional complex edge reconstruction [36, 37]. 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. Remarkably, for graphene or graphene-based hybrid structures, no direct evidence for the presence of upstream modes has so far been reported. Here, we, for the first time, report a smoking gun signature of upstream modes for hole-conjugate FQH states in graphene and identify their nature, employing noise spectroscopy, which is a purely electrical tool. The essence of our approach is as follows [38]. When a bias is applied to a 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 so-called 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, electron-like state $\nu=1/3$ and hole-conjugate states $\nu=2/3$ and $\nu=3/5$, realized in a dual graphite-gated hexagonal-boron-nitride (hBN)-encapsulated high-mobility bilayer graphene (BLG) devices in a cryo-free dilution fridge at base temperature of $\sim 20$ mK (electron temperature $\sim 28$ mK). For $\nu=1/3$ 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 $\nu=2/3$ and $\nu=3/5$ FQH states, a finite noise is detected which increases with increasing injected current. At the same time, the averaged current in upstream direction is zero. Thus, noise detection unambiguously demonstrates that upstream modes exist for the hole-conjugate FQH states in graphene and only carry heat energy. Moreover, the magnitude of the noise remains constant for two different lengths $L=10\,\mu$m and $4\,\mu$m 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.

Experiment: We turn now to a more detailed exposition of our results from two devices. The device schematic and noise measurement set-up is shown in Fig. 1c. The device fabrication details are mentioned in the method section and Supplementary Material (SM, S1). First, we perform electrical conductance measurements at a fixed magnetic field of $10$ T using standard lock-in 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 device-1 are shown in Fig. 1d for $\nu\leq 1$. Clear plateaus in $R_{xy}$ are developed for $\nu=1$, $\nu=2/3$, and $\nu=1/3$, accompanied with zeroes in $R_{xx}$ at the same gate voltage. Similarly, for $\nu=1$, $\nu=2/3$, $\nu=3/5$ and $\nu=1/3$ for device-2, see SM (S7). Next, we perform noise measurements at $\nu=2/3$, $\nu=1/3$, and integer QH states for two different lengths $L=10\,\mu$m and $4\,\mu$m of device-1. The Measurement scheme for $L=10\,\mu$m is shown in Fig. 2a, where the device is set into $\nu=2/3$ QH state with $d$ and $u$ representing counter-propagating downstream and upstream eigenmodes. The measurement scheme for shorter length $L=4\,\mu$m is obtained in the same device by changing the chirality (see SM, S4). Before the noise measurements, we perform two crucial checks: bias-dependent response of the $\nu=2/3$ 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 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 $\nu=2/3$ 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 $\nu=2/3$ at 10 T, which is around 5 K as determined from the activation plot of $R_{xx}$ (see SM, S6), 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 ($4\,\mu$m) (see SM, S4). Thus, if our $\nu=2/3$ edge hosts counter-propagating modes (which is demonstrated below), the charge equilibration length is much smaller than the size of our device.

To unambiguously demonstrate the existence of upstream mode, a noiseless dc current is injected at contact C, and noise $S_{I}$ is measured at contact B along the upstream direction. For the noise measurement, a LCR resonant circuit with resonance frequency of $\sim$770 kHz is utilized together with an amplifier chain and a spectrum analyzer (see methods and SM(S1)) [39, 40, 41, 42]. 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 $\nu=1$ and $\nu=1/3$ states, see Fig. 2d. At the same time, it is shown in Fig. 2d that for the $\nu=2/3$ state there is a strong noise which increases almost linearly with current. This is quite striking as at contact B the time-averaged current is zero. This clearly demonstrates that the $\nu=2/3$ 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 [38]. The heat propagating upstream from the hot spot near C reaches the noise spot near B, inducing there creation of particle-hole 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 $\nu=2/3$ and $\nu=3/5$ of device-2 as shown in SM(S7).

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 $L=4\,\mu$m of the same device-1 (see SM, S4). The data are shown in Fig. 2d. It can be seen that the noise amplitude is nearly identical for $L=10\,\mu$m and $4\,\mu$m. This is striking since it shows that the heat propagates upstream ballistically and without losses along the $\nu=2/3$ 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 counter-propagating modes [9, 11, 43, 38] and (ii) dissipation of energy from the edge to other degrees of freedom, including phonons, photons, and Coulomb-coupled localized states [44, 45, 46]. Our results show that none of these mechanisms is operative at $T\sim 20$ mK on the length scale of $L=10\,\mu$m. 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 $T\sim 20$ mK. Now, we explore the effect of temperature. Figure 3a shows the evolution of the $\nu=2/3$ noise at $L=4\,\mu$m with increasing $T$. In Fig. 3b we display the temperature dependence of the noise for $L=4\,\mu$m and $L=10\,\mu$m at $1nA$ current. It is seen that the noise remains constant and equal for both lengths up to $T=50$ mK. For higher temperature, the noise decays with $T$, 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. Further work is needed to understand which of them is dominant.

Our experiments thus clearly indicate that at low temperatures, $T\leq 50$ mK, the upstream heat transport is ballistic and lossless. To support this conclusion, we have calculated theoretically the expected noise $S_{I}$ on the $\nu=2/3$ edge in this regime. The theory extends that of Refs. [43, 38] 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 [9, 16]. 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 $I_{SD}\sim 1nA$. The result (see method section and SM (S9) 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 $\nu=2/3$ and $\nu=3/5$ 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 temperature-independent for $T\leq 50$ mK and remains the same for $L=4\,\mu$m and $L=10\,\mu$m, 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. [15] 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 strong-interaction fixed point, where the bare modes of the $\nu=2/3$ edge are renormalized into a charge and a neutral mode. Very recently, absence of thermal equilibration (notwithstanding very efficient electric equilibration) was also reported for GaAs samples [16].