Abstract
The study of heat transport has the potential to reveal new insights into the physics of mesoscopic systems. This is especially true of those that show the integer quantum Hall effect^{1}, in which the robust quantization of Hall currents limits the amount of information that can be obtained from charge transport alone^{2}. As a consequence, our understanding of gapless edge excitations in these systems is incomplete. Effective edgestate theory describes them as prototypical onedimensional chiral fermions^{3,4}—a simple picture that explains a large body of observations^{5} and suggests the use of quantum point contacts as electronic beam splitters to explore a variety of quantum mechanical phenomena^{6,7,8}. However, this picture is in apparent disagreement with the prevailing theoretical framework, which predicts in most situations^{9} extra gapless edge modes^{10}. Here, we present a spectroscopic technique that addresses the question of whether some of the injected energy is captured by the predicted extra states, by probing the distribution function and energy flow in an edge channel operated outofequilibrium. Our results show it is not the case and therefore that regarding energy transport, quantum point contacts do indeed behave as optical beam splitters. This demonstrates a useful new tool for heat transport and outofequilibrium experiments.
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The integer quantum Hall effect, discovered nearly thirty years ago^{1}, has recently experienced a strong revival driven by milestone experiments towards quantum information with edge states^{7,11,12}. Beyond Hall currents, new phenomena have emerged that were unexpected within the free onedimensional chiral fermions (1DCFs) model. The ongoing debate triggered by electronic Mach–Zehnder interferometer experiments^{7,13,14,15} vividly illustrates the gaps in our understanding. Coulomb interaction is seen as the key ingredient. In addition to its most striking repercussion, the fractional quantum Hall effect^{16}, the edge reconstruction turns out to have deep implications on edge excitations. This phenomenon results from the competition between Coulomb interaction that tends to spread the electronic fluid, and the confinement potential: as the latter gets smoother, the noninteracting edge becomes unstable^{17}. Theory predicts new branches of gapless electronic excitations in reconstructed edges^{10,18}, which breaks the mapping of an edge channel onto 1DCFs and, possibly, the promising quantum optics analogy. For most edges realized in semiconductor heterojunctions (except by cleaved edge overgrowth^{19}), edge reconstruction results in wide compressible edge channels separated by narrow incompressible strips^{9} and the new excited states are overall neutral internal charge oscillations across the edge channels’ width^{10}.
In practice, the predicted extra neutral modes are transparent to Hall currents. More surprisingly, a linear I–V characteristic is frequently observed for tunnel contacts (different behaviours were also reported, for example, ref. 20), whereas a nonlinear characteristic is predicted^{21,22,23}. This contradiction is resolved by assuming adhoc that only rigid displacements of compressible edge channels are excited by tunnel events, and not internal excitations^{10,21,22,24}. The rigid displacement model arguably relies on the overriding strength of Coulomb interaction that tends to orthogonalize bare tunnelling electrons and correlated electronic fluids^{24}. However, the above argument does not hold at arbitrary transmission probabilities, where several electron processes occur. Therefore, the role of predicted internal excitations has to be determined experimentally. The present work provides such a test. An edge channel is driven outofequilibrium with a quantum point contact (QPC) of arbitrary transmission, possibly exciting internal modes. A short distance away, the resulting energy distribution f(E) is measured with a tunnelcoupled quantum dot expected to probe only rigid displacement excitations, hereafter called quasiparticles. Consequently, the amount of energy injected into internal modes at the QPC would appear as an energy loss in f(E).
Measurements of the energy distribution in mesoscopic devices were first carried out in 1997 on metallic circuits using a superconducting tunnel probe^{25}. In twodimensional electrongas systems, nonFermi energy distributions could not be measured because transferring the techniques developed for metal circuits is technically challenging (although hot electrons have been detected, for example, ref. 26). Regarding the quantum Hall regime, the state of the art is the very recent qualitative probe of heating^{27}. Here, we demonstrate that f(E) can be fully extracted from the tunnel current across a quantum dot. In the sequential tunnelling regime, the discrete electronic levels in a quantum dot behave as energy filters^{28}, as previously demonstrated with double quantum dots^{29}. Assuming a single active quantumdot level of energy E_{lev}, and ignoring the energy dependence of tunnel rates and tunnelling density of states in the electrodes, the quantumdot current reads
where the subscript S (D) refers to the source (drain) electrode; f_{S,D} are the corresponding energy distributions and I_{QD}^{max} is the maximum quantumdot current. In practice, f_{S,D} are obtained separately by applying a large enough source–drain voltage (Fig. 1a,b) and the probed energy E_{lev}=E_{0}−e η_{G}V_{G} is swept using a capacitively coupled gate biased at V_{G}, with η_{G} being the gate voltagetoenergy lever arm and E_{0} an offset. Raw data ∂ I_{QD}/∂ V_{G} measured by lockin techniques are proportional to ∂ f_{D,S}(E)/∂ E.
A tunable nonFermi energy distribution is generated in an edge channel with a voltagebiased QPC. Similar setups were used previously to create imbalanced electron populations between copropagating edge channels^{30}, each characterized by a cold Fermi distribution. Only in a very recent experiment^{27} was an edge channel heated up. Beyond heating, f(E) is here controllably tuned outofequilibrium. Let us consider one edge channel and assume it can be mapped onto noninteracting 1DCFs. According to the scattering approach^{5}, the energy distribution at the output of a QPC of transmission τ is a tunable double step (Fig. 1c, left inset)
where f_{D1} (f_{D2}) is the equilibrium Fermi distribution function in the partially transmitted (reflected) incoming edge channel of electrochemical potential shifted by e V_{D1} (e V_{D2}). In the presence of edge reconstruction, the above energy distribution applies to the quasiparticles if internal modes are not excited at the QPC. On the other hand, if internal modes are excited, there are no theoretical predictions because a QPC is very difficult to treat nonperturbatively in their natural bosonic formalism.
The sample shown in Fig. 1c was tailored in a twodimensional electron gas realized in a GaAs/Ga(Al)As heterojunction, set to filling factor two and measured in a dilution refrigerator of base temperature 30 mK. The experiment detailed here focuses on the outer edge channel represented as a white line. The inner edge channel (not shown) is fully reflected by the QPC and the quantum dot. We checked that charge tunnelling between edge channels is negligible along the 0.8 μm propagation length from the QPC to the quantum dot.
We first carry out a standard nonlinear quantumdot characterization^{28} (Fig. 2, top left inset). The two large signal stripes are frontiers of consecutive Coulomb diamonds and are accounted for by a single active quantumdot level. Small contributions of three extra levels of relative energies {−95,30,130} μeV are also visible. The lever arm extracted from the stripes’ slopes is η_{G}≃0.052±9%.
Then, we test the spectroscopy with known Fermi functions by measuring ∂ I_{QD}/∂ V_{G}(V_{G}) at V_{D1}=V_{D2}=−88 μV for several temperatures (Fig. 2, bottom right inset). By fitting these data with equation (1) using Fermi functions, we extract a fit temperature scaled by the lever arm T_{fit}/η_{G}. The value η_{G}=0.057, compatible with the nonlinear quantumdot characterization, is found to reproduce best the mixingchamber temperature T with T_{fit}. The drain and source fit temperatures are shown in Fig. 2, together with T_{fit} obtained using the standard procedure^{28} from ∂ I_{QD}/∂ V_{D}(V_{G}) at V_{D}≃0. We find deviations mostly within ±10% (dashed lines in Fig. 2) except for a saturation at T_{fit}≈50 mK possibly owing to a higher electronic temperature. In the following, we use η_{G}=0.057 obtained here in the same experimental configuration as to measure unknown f(E)s.
Electrons are now driven outofequilibrium in the drain outer edge channel. In the following, the electrode D2 and the inner drain edge channel are voltage biased at V_{D2}=−88 μV and the source edge channels are emitted by a cold ground.
First, the bias voltage across the QPC is set to δ V_{D}≡V_{D1}− V_{D2}=36 μV and its conductance G_{QPC}=τ e^{2}/h is tuned by applying V_{QPC} to the bottom left gate in Fig. 1c (see Fig. 3a). Note that at 30 mK, we find the transmission τ is constant within 2% with the QPC voltage bias below 36 μV. Typical sweeps ∂ I_{QD}/∂ V_{G}(V_{G}) and the corresponding f_{D}(E) are shown in Fig. 3b and e, respectively. The quantumdot drain negative contribution transforms from a single dip at τ={0,1} into two dips separated by a fixed gate voltage and with relative weights that evolve monotonously with τ∈]0,1[. The solid lines are fits with equation (2) using for f_{D1,D2} two Fermi functions shifted by a fixed energy and weighted by the factors τ_{fit} and 1−τ_{fit}. The values of τ_{fit} are found to deviate by less than 0.03 from the measured transmission τ (Fig. 3c), in accurate agreement with the free 1DCF model. The plus symbols in Fig. 3 correspond to data obtained in a second cooldown.
In a second step, the QPC transmission is fixed to τ≈0.5 and the bias voltage δ V_{D} is changed. Typical raw data are shown in Fig. 4a. These were obtained in a third cooldown with a quantum dot renewed by the thermal cycle showing no signs of extra quantumdot levels in the probed energy range. The single dip in the quantumdot drain contribution (bright) at δ V_{D}=0 splits into two similar dips that are separated by a gatevoltage difference proportional to δ V_{D}. In contrast, the quantumdot source peak (dark) is mostly unchanged but slowly drifts parallel to one quantumdot drain dip owing to the capacitive coupling between the drain and the quantum dot. In the first cooldown, V_{D1} was kept within [−106,−34] μV to minimize complications related to extra quantumdot levels (lower bound) and to ensure wellseparated source and drain contributions (upper bound). The symbols in Fig. 4b and e are, respectively, data and extracted f_{D}(E) for the quantumdot drain contribution at δ V_{D}={−18,0,18,27,36,45,54} μV and τ=0.58. The solid lines in Fig. 4b are fits with equation (2) using the measured τ and for f_{D1,D2} two Fermi functions shifted in energy by the fit parameter −e η_{G}δ V_{G}. The resulting η_{G}δ V_{G} are plotted as symbols versus δ V_{D} in Fig. 4c. Those obtained in the third cooldown are shown as purple star symbols using the renewed lever arm η_{G}=0.062. We find η_{G}δ V_{G}≃δ V_{D} as expected from the noninteracting 1DCF model. Deviations are always smaller than 8 μV (5 μV) for the first (third) cooldown, a reasonable agreement regarding uncertainties in η_{G} of ±10% (±5%).
In the two experiments above, we found the measured quasiparticle f(E)s verify predictions of the scattering approach. To establish the QPC/beamsplitter analogy one also needs to demonstrate that internal edgechannel modes are not excited. A direct test consists of extracting the quasiparticle heat current J_{E}^{qp} from the data, and comparing it with the full edgeexcitations’ heat current J_{E} obtained from powerbalance considerations (see Supplementary Information for details):
The cancellation v ν=1/h of velocity (v) and density of states per unit length and energy (ν) that applies to the 1DCF quasiparticles permits us to obtain J_{E}^{qp} from the measured f(E) without any samplespecific parameters:
where μ is the electrochemical potential and θ(E) is the step function. Consequently, we measure quantitatively the quasiparticle heat current. The result of this procedure is shown as symbols in Figs 3d and 4d using the generalized nonequilibrium temperature together with the prediction if none of the injected power is carried on by internal modes (solid lines). We find a good agreement T_{qp}≃T_{1DCF} without fitting parameters and essentially in or close to error bars. Hence, within our experimental accuracy, the propagative internal modes do not contribute to heat transport and therefore are not excited. Note that the relatively small observed deviations are cooldown dependent, which suggests that the quantumdot detector is responsible for these deviations. Indeed, the data can be more accurately accounted for including a second active quantumdot level (see Supplementary Information). Last, preliminary data show a significant energy redistribution with the inner edge channel for propagations longer than 2 μm in the probed energy range. Therefore, the observed small discrepancies could also result from the finite 0.8 μm propagation length.
Overall, we demonstrate that QPCs in the quantum Hall regime are tunable electrical beam splitters for onedimensional fermions, that is, rigid edgechannel displacements, (1) by comparing the energy distribution at a QPC output with predictions of the scattering approach^{5}, and (2) by showing that internal edgechannel modes are not excited. This does not only rule out nonideal QPC behaviours to explain the surprising phenomena observed on electronic Mach–Zehnder interferometers^{7,13,14,15}, it also establishes a solid ground for future quantum information applications with edge states. Finally, an essential part of this work is the demonstration of a new technique to measure the fundamental energydistribution function. It makes f(E) accessible for most systems where quantum dots can be realized. We expect it will trigger many new experiments dealing with heat transport, outofequilibrium physics and quantum decoherence.
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Acknowledgements
The authors gratefully acknowledge discussions with M. Buttiker, P. Degiovanni, C. Glattli, P. Joyez, A. H. MacDonald, F. Portier, H. Pothier, P. Roche and G. Vignale. This work was supported by the ANR (ANR05NANO02803).
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Experimental work and data analysis: C.A., H.l.S. and F.P.; nanofabrication: C.A. and F.P. with input from D.M.; heterojunction growth: A.C. and U.G.; theoretical analysis and manuscript preparation: F.P. with input from coauthors; project planning and supervision: F.P.
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Altimiras, C., le Sueur, H., Gennser, U. et al. Nonequilibrium edgechannel spectroscopy in the integer quantum Hall regime. Nature Phys 6, 34–39 (2010). https://doi.org/10.1038/nphys1429
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DOI: https://doi.org/10.1038/nphys1429
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