Abstract
Electron correlation in a quantum manybody state appears as peculiar scattering behaviour at its boundary, symbolic of which is Andreev reflection at a metalsuperconductor interface. Despite being fundamental in nature, dictated by the charge conservation law, however, the process has had no analogues outside the realm of superconductivity so far. Here, we report the observation of an Andreevlike process originating from a topological quantum manybody effect instead of superconductivity. A narrow junction between fractional and integer quantum Hall states shows a twoterminal conductance exceeding that of the constituent fractional state. This remarkable behaviour, while theoretically predicted more than two decades ago but not detected to date, can be interpreted as Andreev reflection of fractionally charged quasiparticles. The observed fractional quantum Hall Andreev reflection provides a fundamental picture that captures microscopic charge dynamics at the boundaries of topological quantum manybody states.
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Introduction
When a twodimensional electron system (2DES) is subjected to a perpendicular magnetic field at low temperatures, electrons condense into the strongly correlated phase of the fractional quantum Hall (FQH) state^{1}. Quasiparticles in FQH systems have fascinating properties, such as fractional charge^{2} and anyonic statistics^{3}. Furthermore, for particular states such as that at Landaulevel filling factor v = 5/2, the theory predicts that quasiparticles obey nonAbelian braiding statistics that provide the basis of faulttolerant quantum computation^{4,5}. The fractional charge^{6,7,8,9} and anyonic nature^{10,11,12} of the quasiparticles have been revealed experimentally by shotnoise measurements and Fabry–Pérot interferometry. These studies have elucidated the behaviour of quasiparticles within the FQH state—either bulk or edges—that gives their defining properties. On the other hand, one may expect the quasiparticles to exhibit unique behaviour at an interface between the FQH state and another topologically distinct system, in a similar way as the Cooperpair correlation in a superconductor manifests itself as Andreev reflection, where an electron incident from a normal metal to a superconductor is reflected as a hole^{13,14}. This, in turn, poses a fundamental question as to whether electron correlation in a topological quantum manybody state shows up as a unique interface phenomenon. FQH Andreev reflection, which we demonstrate in this paper, is an elementary process that answers this question.
The FQH Andreev process has been predicted by theories examining charge transport across a narrow junction between quantum Hall (QH) states with different filling factors. The most intensively studied system is one comprised of the v = 1/3 Laughlin state and the v = 1 integer QH (IQH) state^{15,16}. The charge transport can be modelled as the tunnelling between the v = 1/3 and 1 edge channels, which can be treated as a chiral Luttinger liquid and a Fermi liquid, respectively^{17}. When the channels are coupled through a single scatterer, the problem can be solved analytically by transforming it into that of tunnelling between edge states with Luttinger parameter g = 1/2^{18,19}. The exact solution predicts that in the strongcoupling regime the twoterminal conductance G exceeds the conductance e^{2}/3h (e: electron charge, h: Planck’s constant) of the v = 1/3 state, reaching e^{2}/2h in the strongcoupling limit^{15,16,18,19,20,21}. The enhancement of G can be interpreted as the result of the Andreev process, where two incoming chargee/3 quasiparticles are scattered into a transmitted electron with charge e and a reflected quasihole with charge −e/3^{15}. This theoretical prediction, however, has not yet been confirmed experimentally, despite recent progress in experiments on related systems^{22,23,24,25,26}.
In this paper, we present evidence of the FQH Andreev process, namely G exceeding e^{2}/3h in a narrow junction between v = 1/3 and 1 states. As the junction width is varied using the splitgate voltage applied to form the junction, G oscillates around e^{2}/3h, exhibiting several peaks where G overshoots the bulk conductance e^{2}/3h, reaching G ≅ 1.2 × e^{2}/3h. The conductance oscillations indicate several Andreev processes at multiple scatterers present between the v = 1/3 and 1 edges. The evidence is also reinforced by demonstrating that the junction operates as a dcvoltage transformer generating a negative voltage output for positive input.
Results
FQHIQH junction
Our QH device, formed in a Hall bar containing a 2DES in a GaAs quantum well, has several top gates and pairs of split gates in between (Fig. 1a). A perpendicular magnetic field of B = 9 T sets the bulk of the 2DES at v = 1. We then use the leftmost top gate (V_{L} = −0.42 V) to form a v = 1/3 region underneath (see the inset in Fig. 2a). A narrow 1/31 junction is formed by applying a negative gate bias V_{S} to both electrodes of the split gate located immediately to the right of v = 1/3 region and depleting the 2DES underneath (Fig. 1b). In this situation, the setup for transport measurements can be expressed schematically as in Fig. 1c [see Supplementary Note 1]. We measured the twoterminal differential conductance dI/dV_{in} by applying a sourcedrain voltage V_{in} = V_{in}^{dc} + V_{in}^{ac} on the v = 1/3 side of the junction and measuring the transmitted current I on the v = 1 side using a standard lockin technique.
Enhanced twoterminal conductance
Figure 2a presents the central result of this paper, where we plot the zerobias conductance G, i.e., dI/dV_{in} at V_{in}^{dc} = 0 V, as a function of V_{S}. A narrow junction forms at V_{S} < −0.55 V. As V_{S} is decreased below −0.55 V, the junction width decreases and G starts to oscillate around e^{2}/3h with the amplitude growing with decreasing V_{S}. The most striking observation is that G overshoots e^{2}/3h at several oscillation peaks before the junction is pinched off at V_{S} ≅ −1.4 V. The maximum G reaches 1.2 × e^{2}/3h at V_{S} ≅ −1.1 V. Such a twoterminal conductance, enhanced by narrowing the junction and exceeding the conductance of the constituent element, is nontrivial and counterintuitive. We note that these features appear only in 1/31 junctions and not in 1/31/3 or 13 junctions (see Supplementary Note 7).
The peculiarity of the chargetransfer process is revealed alternatively by probing the potentials, or voltages V_{i} (i = 1–4) of the incoming and outgoing edge channels. Figure 2b, c displays V_{i} measured at V_{in}^{dc} = 0 V normalised by V_{in}, plotted as a function of V_{S}. The voltages V_{1} and V_{3} of the incoming channels are, respectively, equal to potentials V_{in} and 0 V of the electrodes on their upstream, independent of V_{S}. In contrast, the voltages V_{2} and V_{4} of the outgoing channels vary with V_{S}. The most remarkable feature is the negative voltage that appears in V_{2}. Phenomenologically, this demonstrates that the junction operates as a dcvoltage transformer generating negative voltage output (V_{2} < 0) for a positive input (V_{in} > 0).
From the Landauer–Büttiker formalism, V_{2} and V_{4} are related to G as
These formulas show that both V_{2} < 0 and V_{4} > V_{in}/3 correspond to G > e^{2}/3h. Within the picture of Andreev reflection, the negative voltage (V_{2} < 0) of the backreflected channel is a direct manifestation of the quasihole reflection.
Bias and temperature dependence
While it is evident that the Andreev reflection is responsible for the observed G > e^{2}/3h, to understand microscopic processes therein, we need to explain the origin of the conductance oscillations, which is not predicted from the original models based on the tunnelling through a single scatterer^{18,19}. Resonant tunnelling through unintentional discrete levels in the junction, which are responsible for the oscillations in the lowconductance regime near V_{S} = −1.3 V (see Supplementary Note 4), cannot account for the conductance oscillations with G > e^{2}/3h. In the following, we present the dependence of the conductance on V_{in} and temperature T and discuss the oscillation mechanism.
Figure 3a displays a colour plot of differential conductance dI/dV_{in} as a function of V_{S} and V_{in}^{dc}. The oscillations with dI/dV_{in} > e^{2}/3h are seen only at V_{in}^{dc} < 40 μV. For illustration, we plot in Fig. 3b the pinchoff trace at V_{in}^{dc} = 100 μV, where dI/dV_{in} < e^{2}/3h over the entire range of V_{S}. Figure 3c shows the V_{in}^{dc} dependence of dI/dV_{in} at V_{S} = −1.113 and −0.985 V. At V_{S} = −1.113 (−0.985) V, which corresponds to the peak (valley) of the oscillations in Fig. 3b, we observe a pronounced zerobias enhancement (suppression) of the conductance. In contrast, at V_{S} = 0 V, where the v = 1/3 and 1 regions form a long junction spanning across the 80μmwide Hall bar, dI/dV_{in} remains constant at e^{2}/3h. These results clearly show that the Andreev process is observed only in narrow junctions at low bias. The data also reveal that not only the conductance enhancement but also its suppression are lowbias anomalies.
Figure 3d shows the T dependence of the conductance oscillations. The oscillation amplitude decreases with increasing T, and the signature of the Andreev process, G > e^{2}/3h, disappears above 200 mK. We focus on two single periods of the oscillations near V_{S} = −1.113 and −1.095 V and extract the amplitude A as the peaktovalley value of G in each period. The two sets of A vs. T data are well fitted by an exponential function A_{0}exp(−T/T_{0}), as shown in Fig. 3e, where A_{0} is the amplitude at T = 0 and T_{0} is the characteristic temperature. The exponential temperature dependence bears analogy with that seen in various electronic interferometers^{27}. The T_{0} values (170 and 190 mK) are close to each other, indicating that these oscillations share the same origin in nature. The data in Fig. 3 also demonstrate that the conductance oscillations with G > e^{2}/3h are highly reproducible (similar conductance oscillations were reproduced for different cooldowns and in different samples, see Supplementary Notes 3, 5, and 6).
Multiplescatterer model
We argue that several Andreev processes in the junction are responsible for the conductance oscillations, as predicted in theories involving multiple scatterers or a line junction of finite width^{18,19,28,29,30,31,32}. We consider N scatterers along the counterpropagating v = 1/3 and 1 channels and incoherent transport between them. N is proportional to the junction width (i.e., length of the counterpropagating channels) and hence varies with V_{S}. Here, “incoherent transport” means that the N scatterers give independent scattering events, where the outgoing channels are characterised by a chemical potential that defines the input for the next scatterer. With this assumption, the voltage of the v = 1/3 (1) channel incoming to the nth scatterer is given by V_{n−1} = i_{n−1} × 3h/e^{2} (W_{N−n} = j_{N−n} × h/e^{2}), where i_{n−1} (j_{N−n}) is the current in the incoming channel (see Supplementary Figure 7). With this setup, one can use the exact solution for the singlescatterer model^{19} to define the conductance g_{n} for each scatterer as a function of the applied bias V_{n−1}–W_{N−n} and evaluate the current I_{n} flowing through it. Notably, as shown for the singlescatterer case, the charge conservation law and the requirement that the outgoing power be equal to or less than the incoming one lead to 0 ≤ g_{n} ≤ 1/2^{21}. Here, the charge transport through the nth scatterer becomes dissipationless only when g_{n} = 0 or 1/2. The latter (former) corresponds to the strongcoupling limit (complete decoupling). Namely, tunnelling for any intermediate g_{n} values is accompanied by energy dissipation.
The conductance G of the whole junction is obtained by solving a nonlinear system of equations numerically. The results are shown in Fig. 4a for three representative cases: strong (T_{k} = 0, black open circles), intermediate (T_{k} = 1.5 mK, red filled circles), and weak couplings (T_{k} = 36 mK, blue diamonds) under the experimental condition with an applied voltage of 20 µV and a temperature of 9 mK. Here, T_{k} is the crossover energy scale between strong and weakcoupling regimes^{19} (for details, see Supplementary Note 8). In the strongcoupling limit, where we have g_{n} = 1/2 for all n, each scatterer only switches the sign of the voltage between the channels without causing energy dissipation. Consequently, G oscillates as a function of N between 0 (N even) and e^{2}/2 h (N odd) (black circles). This oscillation can be regarded as the result of successive dissipationless Andreev processes, where the tunnel current switches direction at each scatterer without changing magnitude. When the coupling weakens to give g_{n} < 1/2, each scatterer equilibrates the channels, which results in the reduced output voltage at each scatterer and hence damping of the conductance oscillations (red circles). The damping is significant particularly for large N, where the channels experience equilibration many times. When the coupling weakens further to give g_{n} < 1/3 for all n, the tunnel current flows only in one direction. In this case, G monotonically increases with N, asymptotically approaching G = e^{2}/3h^{30} (blue diamonds). The simulation for the intermediate coupling (red circles) captures essential features of the experimental data in Fig. 2a. If we take into account more realistic experimental situations, including the confining potential of the split gate and randomness in the positions of the scatterers, the simulation can even better reproduce the experimental features (Fig. 4b). In the simulation, the confining potential controls the effective width of the junction by multiplying a positiondependent window function to the coupling strength of the scatterers, resulting in weaker coupling near the junction ends and hence reduced oscillation amplitude (for details, see Supplementary Note 8).
While the above multiplescatterer model well explains the V_{S} dependence of G, it still fails to account for the observed V_{in} and T dependence. Since the model inherits the V_{in} and T dependence of the conductance from the singlescatterer model^{18,19}, which gives dI/dV_{in} as an increasing function of V_{in} and T, it remains incapable of reproducing oscillations decaying with V_{in} or T. This, in turn, suggests that coherent processes neglected in the above model, such as interference between successive scattering events^{33}, play an important role. We speculate that constructive (destructive) interference of tunnelling amplitudes can enhance (suppress) the coupling strengths of several scatterers in some range of V_{S}. Indeed, a theory considering coherent interference predicts that a 1/31 junction with Coulomb interaction shows conductance oscillations up to e^{2}/2h as a function of the junction width^{28}. In this view, conductance enhancement or suppression at the extrema of the oscillations is partly due to the interference enhancement of the coupling. This picture explains why the oscillations appear only at low V_{in} and T. Furthermore, it helps to understand why the simulation for the weakcoupling regime of the incoherent model can mimic the V_{S} dependence of G at high V_{in} (Fig. 3b) or high T (Fig. 3d).
Discussion
Finally, we discuss a related interesting issue, namely the mixing of the v = 1/3 and 1 edge modes expected for a wide junction. Counterpropagating v = 1/3 and 1 channels studied here is a basic setup in the model for the edge modes of the holeconjugate v = 2/3 FQH state^{34}. There, interchannel Coulomb interaction and disorderassisted tunnelling govern the mixing of the channels and thus determine their fate in the lowtemperature and longchannel limit. The conductance oscillations with G exceeding e^{2}/3h observed in our experiment can be interpreted as a precursory phenomenon of the mixing process, namely the “mesoscopic fluctuation” predicted in ref. ^{32}, suggesting the presence of the neutralmode physics of counterpropagating channels at the 1/31 junction^{31,32}. Our findings indicate that the Andreev process is a vital ingredient therein.
We have demonstrated FQH Andreev reflection, which is one of the essential concepts for understanding edge transport at the boundaries of topological quantum manybody systems. We expect to observe similar Andreev processes in various FQH junctions with different electronic systems, including nonQH systems such as normal metals and superconductors^{35,36}.
Methods
Sample fabrication
We fabricated the sample in a 2DES in a GaAs quantum well of 30 nm width. The centre of the well is located 190 nm below the surface. The sample was patterned using ebeam lithography for fine gate structures and photolithography for chemical etching, coarse metalized structures, and ohmic contacts formed by alloying Au–Ge–Ni on the surface.
Measurement setup
We set electron density in the 2DES at 2.2 × 10^{11} cm^{−2} by applying a backgate voltage of 1.29 V at a refrigerator temperature of 9 mK, except for the data in Fig. 3d, e. A perpendicular magnetic field B = 9 T was applied from back to the front of the sample. The lockin measurements were performed with the ac modulation of V_{in}^{ac} = 20 μV RMS at 31 Hz. The experimental results demonstrated in the main text were obtained for the splitgate device with an opening of 300 nm. The data from the devices with wider apertures are available in Supplementary Note 6.
Data availability
All data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors thank T. Ito, N. Shibata, and T. Fujisawa for fruitful discussions and H. Murofushi for technical support. This work was supported by GrantsinAid for Scientific Research (Grant nos. JP16H06009, JP15H05854, JP26247051, and JP19H05603) and JST PRESTO Grant no. JP17940407.
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M.H. conceived the experiment. M.H., T.A., and S.S. fabricated the sample. M.H. performed the measurement and analysed the data. T.J. and K.M. performed the simulations. M.H., T.J., and K.M. interpreted the results with help from J.R. and T.M. M.H. wrote the paper with help from T.J. and K.M. All authors discussed the results and commented on the paper.
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Hashisaka, M., Jonckheere, T., Akiho, T. et al. Andreev reflection of fractional quantum Hall quasiparticles. Nat Commun 12, 2794 (2021). https://doi.org/10.1038/s41467021231606
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DOI: https://doi.org/10.1038/s41467021231606
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