Abstract
Electron pairing is a rare phenomenon appearing only in a few unique physical systems; for example, superconductors and Kondocorrelated quantum dots. Here, we report on an unexpected electron pairing in the integer quantum Hall effect regime. The pairing takes place within an interfering edge channel in an electronic Fabry–Perot interferometer at a wide range of bulk filling factors, between 2 and 5. We report on three main observations: highvisibility Aharonov–Bohm conductance oscillations with magnetic flux periodicity equal to half the magnetic flux quantum; an interfering quasiparticle charge equal to twice the elementary electron charge as revealed by quantum shot noise measurements, and full dephasing of the pairs' interference by induced dephasing of the adjacent inner edge channel—a manifestation of interchannel entanglement. Although this pairing phenomenon clearly results from interchannel interaction, the exact mechanism that leads to electron–electron attraction within a single edge channel is not clear. We believe that substantial efforts are needed in order to clarify these intriguing and unexpected findings.
Introduction
The Fabry–Perot interferometer (FPI) is one of several interferometers implemented in optical^{1,2} as well as in electronic systems^{3,4,5,6,7}. Its electronic version^{6,7} is an excellent platform to demonstrate the waveparticle duality of electrons: interference^{4}—a manifestation of wavelike behaviour—alongside with currentfluctuation^{8} due to particles discreteness. Moreover, the electronic version of the FPI harbours an exceedingly more complex behaviour due to the interacting nature of electrons. Two distinct regimes of the FPI operating in the QHE regimes have been observed^{6,7} and theoretically studied^{9,10,11,12}: the coherent Aharonov–Bohm (AB) regime, where interactions were thought to play only a minor role and the Coulombdominated (CD) regime, where interactions are dominant. In the rather difficult to achieve AB regime, the conductance of the FPI obeys strictly AB interference of independent electrons^{13,14}, with conductance oscillation phase following ϕ_{AB}=2πAB/φ_{0}, where A denotes the area of the interferometer, B the applied magnetic field and φ_{0}=h/e=41 μm^{2}G the magnetic fluxquantum with h the Planck constant and e the electron charge. Alternatively, in the CD regime, conductance oscillation is due to periodic charging (and discharging) of the FPI with single electrons. Noting that the AB regime is highly desirable as it had been proposed to be a sensitive marker to fractional statistics (abelian or nonabelian) of fractionally charged quasiparticles^{15,16,17,18}.
We address the FPI in the AB regime, which is formed by two quantum point contacts (QPCs) that serve as electronic beam splitters (Fig. 1a). A chiral edge channel impinging from the left is transmitted with amplitude , where r_{1} (r_{r}) and t_{1} (t_{r}) are the reflection and transmission amplitudes of the left (right) QPC, respectively. Each term in the above sum stands for a different number n of windings that a coherent electron undergoes. Hence, the transmission through the interferometer is an oscillating function of the acquired AB phase in a single winding in the FPI:
To first order in the transmission coefficient of the QPCs (namely, ), we have . Evidently, one expects an AB oscillation period in area ΔA=φ_{0}/B and in magnetic field ΔB=φ_{0}/A.
As described in some detail below, the predicted behaviour of equation 1 was precisely observed in a range of bulk filling factors ν_{B}=1–2; however, at fillings around ν_{B}=3–4, the AB periodicity halved, namely, , with , without an observable sign of φ_{0} periodicity. Since similar visibilities (contrast, defined as the oscillation amplitude divided by its average), reaching ∼60%, were observed in the two filling regimes, interference of only even windings as the cause of the halved periodicity can be excluded. Testing the possibility of electron pairing in the interfering channel, the interfering charge e* was determined by measuring the quantum shot noise, indeed leading to e*∼2e. These observations were found to be robust and reproducible in FPIs of very different sizes (2–12 μm^{2}), fabricated in different heterostructures (twodimensional electron gas, 2DEG, density ranged 1.3–2.4 × 10^{11} cm^{−2}), and at a temperature range 30–130 mK. Moreover, two very different screening methods were employed in order to suppress the CD regime and operate the FPI in the AB regime, both leading to similar results.
Results
Measurement setup
Our FPIs were realized in a ubiquitous highmobility 2DEG embedded in AlGaAsGaAs heterostructure(s), employing optical and Ebeam lithography. The FPIs were fabricated as a semiclosed stadium (Fig. 1), terminated with two QPCs; one at the source side and the other at the drain side. A charged modulation gate (MG) allowed varying the FPI’s area. Only the most outer edge channel was partitioned by the QPCs, whereas the impinging inner ones were fully reflected from the QPCs and, likely, the trapped ones circulated inside the FPI. Coulomb interactions were suppressed via two different configurations (illustrated in Fig. 1b,c), both leading to similar results: the first configuration employs a metallic (TiAu) topgate covering nearly the whole area of the FPI; the second configuration employs a small, grounded, ohmiccontact (alloyed NiGeAu) in the centre of the incompressible bulk of the FPI. Although the screening process achieved by the grounded ohmic contact is less transparent, it was found to be more effective in suppressing interaction, allowing the fabrication of smaller FPIs operating in the AB regime. This said, the phenomenon reported here is a direct result of interedge interactions, while the grounded ohmic contact, as well as the top gate, lower the charging energy of the FPI as a whole (increasing the total selfcapacitance) without much effect on the interchannel capacitance. Differential sourcedrain conductance was measured by applying an AC voltage of 1 μVrms at 800 kHz; whereas excess quantum shot noise at the drain was measured by driving a variable DC source current, and measuring the noise at a centre frequency of 800 kHz and bandwidth of 10 kHz. For further details see Methods.
Conductance measurements
The conductance of the FPI was measured as a function of the magnetic flux ϕ, either by varying the magnetic field or the enclosed area (via MG). In the following, we present data taken with two FPIs, each with a centreohmic contact, and areas ∼2 μm^{2} and ∼12 μm^{2}. Figure 2 presents the most important data of this work. Starting with the smallest FPI, we plot in Fig. 2a the fluxdependent conductance at ν_{B}≅2 (being representative of similar results in ν_{B}≅1–2.5). Constant phase lines correspond to constant flux; namely, increasing magnetic field necessitates reducing the FPI area. An AB area of 2.1 μm^{2} is deduced from ΔB period. However, at ν_{B}≅3 (being a representative of ν_{B}≅2.5–4.5), an unexpected halving of the oscillation periods in B and in V_{MG} is observed (Fig. 2b). This is clearly evident in the B dependence oscillations (at constant V_{MG}) shown in Fig. 2c and their Fourier transform in Fig. 2d. Higher harmonics, corresponding to multiple windings, are also visible in the two regimes. Note an absence of any sign of the fundamental periodicity h/e in the h/2e regime.
The universality of the results is presented in Fig. 3, this time with the larger FPI (A=12.5 μm^{2}), where the dependence of the AB frequencies (φ_{0}/ΔB in Fig. 3a; 1/ΔV_{MG} in Fig. 3b) is plotted as a function of bulk filling ν_{B}. In filling ranges ν_{B}<2.5 and ν_{B}>4.5, the normalized oscillation frequency in magnetic field is φ_{0}/ΔB^{e}=12 μm^{2}—agreeing with FPI area, and is independent of the filling factor. The frequency in MG 1/ΔV_{MG} depends linearly on B, , as expected. This dependence can be easily understood by recalling that the incremental change in charge δQ is related to δV_{MG} via the capacitance . Moreover, its relation to the incremental change in area is δQ=en_{e}δA, with n_{e} the carrier density. Hence, where e* is the interfering quasiparticle charge in the most outer channel, being in this filling range e. However, in the filling range ν_{B}=3–4.5, the Bdependent normalized frequency is doubled, φ_{0}/ΔB^{2e}=25 μm^{2}, and again independent of filling factor. This doubling suggests halved flux quantum, namely, . Similarly, the AB frequency in 1/ΔV_{MG} has now twice the slope, , again leading to The small deviation from an exact in the region ν_{B}∼2.5–3 (Fig. 3a,b) is observed in all the tested samples. The behaviour in the transition region near ν_{B}∼2.5 and ν_{B}∼4.5 is not universal and will not be discussed here.
Analysis refuting a possibility of evenwinding processes
Although the doubling of the AB frequencies can be attributed to , with , one can also envision a peculiar preference of only even number of interfering windings, namely, A^{*}=2A. To distinguish between these two scenarios, we address the visibility of the h/2e oscillations compared with that of the h/e oscillations. With the visibility of the h/e periodicity in the ∼12 μm^{2} FPI being ν^{(e,1)}_{max}≈60%, we estimated a dephasing length L_{γ}≈35 μm (see Supplementary Note 1). This value sets an upperbound for the highest visibility of the second harmonic (two windings in the h/e regime) ν^{(e,2)}_{max}≈10%, which had been confirmed experimentally. However, the highest visibility in the h/2e regime was found to be ν^{(2e,1)}_{max}≈50%, refuting a peculiar favouring of even number windings.
Charge determination via quantum shot noise measurements
Shot noise measurements were found to provide an excellent method for a determination of the partitioned charge^{19,20,21,22,23,24,25}. For independent and stochastic partitioning events by any scatterer, the expected zero frequency (hω<<eV_{SD}) excess noise (added noise due to driven current) is given by equation 2, which had been modified to allow e*≠e (refs 23, 24):
where t=τ^{2} is the transmission coefficient, I the impinging current, T the electrontemperature, V_{SD} sourcedrain DCbias and .
Noise measurements were first conducted with a single QPC partitioning the outer edge channel at v_{B}=2 and v_{B}=3. A partitioned charge e*=e was extracted in a wide range of transmission coefficients using equation 2 (Fig. 4a,b). The FPI was then formed by pinching the second QPC with noise measured in the h/e regime, leading to e*=e (Fig. 4c, Supplementary Note 2 and Supplementary Fig. 1). In the h/2e regime, on the other hand, the quasiparticle charge was found consistently to be e*∼2e (Fig. 4d). Note that very large Fano factors, defined here as , were found when the visibility was very high, which is attributed to phase noise due to charge fluctuations^{26,27}, as, when visibility is high, a small variation in the phase might lead to large variation of the transmission.
Interedge interaction
To test the effect of interchannel interaction, we modified further the FPI by adding an additional centreQPC between the edge of the FPI and the centreohmic contact (see scanning electron microscopy image in Fig. 5a). This configuration allows selective reflection of inbound edge channels to the centrecontact; thus dephasing them and fixing their Fermi energy at ground. Starting at filling ν_{B}=2 with oscillation periodicity h/e, we plotted the transmission of the centreQPC (to be distinguished from the reflection to the centreohmic) and the oscillation visibility of the most outer edge channel as function of the centreQPC gate voltage (Fig. 5b). As expected, the visibility quenches when the interfering edge channel is fully reflected to the centreohmic contact.
Now, in ν_{B}∼3, with oscillation periodicity h/2e, the behaviour is entirely different. The visibility of the h/2e periodicity (with all its harmonics) quenches rapidly as soon as the adjacent second edge channel is reflected to the centrecontact—with no sign of interference at all (Fig. 5c). We wish to stress that when the h/2e periodicity quenches, the outer edge channel is uninterruptedly fully transmitted through the centreQPC. As no tunnelling was observed between these two adjacent edge channels (with an upper limit of 2pA; see Supplementary Note 3), this result demonstrates that the coherence of the second edge channel is indispensable for the observation of h/2e periodicity of the outer edge channel.
Employing charge measurement throughout the abovedescribed dephasing process, we plot again in an expanded scale the visibility and the FPI conductance (proportional to the transmission of the outer edge channel) as function of V_{centreQPC} in Fig. 6a, alongside with the partitioned charge in Fig. 6b. Since the conductance maintains its slope, as the second channel is gradually reflected to ground, it is clear that no current from the outer channel is being lost to the second channel. However, as the coherence of the h/2e periodicity gradually diminishes, the quasiparticle charge drops gradually from e*∼2e to e*=e. Noting, that as the transmission of the FPI is highly sensitive to those of the two QPCs and the AB phase, each charge value was determined by averaging the charge measured in a wide span of AB phases at a fixed setting of QPCs. It is hence clear that an interfering charge e*∼2e is always associated not only with the doubling of the AB frequency, but also with a significant visibility (coherence).
Discussion
Observation of h/2e periodicity (and higher harmonics) in other electronic systems, such as a CD FPI^{6,7} and a CD quantum antidot^{28,29,30}, have been reported in the past. In these examples, the nonlinear conductance exhibited the ubiquitous Coulomb diamonds^{28,31}, and the oscillation periodicity with respect to magnetic field scaled inversely with the number of fully transmitted channels through the device^{6,30,32,33,34}. These two features clearly differ from our observation in the AB–FPI.
It may be useful to recapitulate the main results in this work for the interference of the most outer edge channel of the FPI: Although the bare FPI is heavily dominated by Coulomb interactions, thus fully masking the AB interference, our FPI had been modified to suppress the interactions (mostly the intrachannels) and thus showed clear AB behaviour. Two methods were implemented for screening the FPI: a grounded small centreohmic contact added to the bulk, and a topgate covering the device (for more information see Supplementary Notes 4 and 5 and Supplementary Figs 2 and 3); different area interferometers, ranging from ∼2 to ∼12 μm^{2} (for more information see Supplementary Note 5 and Supplementary Fig. 4), fabricated on different heterostructures, showed qualitatively similar results; the FPIs were tested in a wide range of QPCs transmissions (see Supplementary Note 6 and Supplementary Fig. 5); at bulk fillings ν_{B}=1–2.5, the AB periodicity in magnetic field and in MG voltage (affecting the AB area) corresponded to flux periodicity of φ_{0}=h/e; namely, single electron interference. At this range, the interfering charge, determined by shot noise measurement, was e*=e; at bulk fillings ν_{B}=2.5–4.5, the periodicity in magnetic field and MG voltage halved; namely, with a flux periodicity φ_{0}=h/2e. The interfering charge, measured by shot noise, was e*=2e; dephasing the second edge channel (second spinsplit Landau level) in filling range ν_{B}=1–2.5 (by shorting the second channel to the centreohmic) did not affect the h/e oscillation and the interfering charge; dephasing the second edge channel in filling range ν_{B}=2.5–4.5, fully dephased the h/2e oscillations and simultaneously lowered the partitioned charge to e*=e; tunnelling current between adjacent edge channels was not observed (see also Supplementary Note 3 and Supplementary Fig. 6); the temperature dependence of the visibility in the h/e and the h/2e regimes was very similar (see Supplementary Fig. 7); decaying exponentially , with T_{0}∼25–40 mK; the dependence of the visibility on the transmission of the QPCs was very similar in both regimes (see Supplementary Fig. 8); although the transition regions near ν_{B}=2.5 and ν_{B}=4.5 are complex and sample dependent, near ν_{B}=4.5 a coexistence of the two periods, h/e and the h/2e—being in phase—was observed (see Supplementary Note 7 and Supplementary Fig. 9). Moreover, dephasing the h/2e periodicity did not affect the h/e periodicity (see Supplementary Note 7 and Supplementary Fig. 10); allowing the second edge channel to interfere (while the outer channel is fully transmitted), we find only the ubiquitous h/e oscillation in all bulk filling factors (see Supplementary Fig. 8).
Our results reveal an emergent, and robust, electron pairing in a coherent chiral edge channel, hence, intimately tied to an interfering process. Clear evidence of interchannel entanglement between the interfering channel and the adjacent one takes place under the pairing conditions. Although Cooper pairing is phonon mediated, here the exact mechanism that leads to intrachannel twoelectron attraction is not understood. An important question remains whether the observed phenomenon is general and can be reproduced in other correlated quantum systems.
Methods
Screening techniques
As mentioned in the text, all measurements are performed in the AB dominated regime, as opposed to the CD regime. The CD regime is avoided by reducing the charging energy of the FPI. This was done by two different techniques, which we detail here further to that discussed in the text.
The first, and previously reported^{6,7}, technique consists of placing a top gate, namely thin layers of 50 Å Ti 150 Å Au on top of the heterostructure, covering the whole area of the FPI, being 73 nm away from the 2DEG. The top gate accommodates imagecharge of the charge variations in the 2DEG, and thus induces effective screening and suppression of Coulomb interaction. In other words, the top gate increases the capacitance of the device, and reduces the charging energy of the FPI. Nonetheless, although this technique works well for the bigger FPIs utilized (areas larger than 12 μm^{2}), it does not work for arbitrarily small device. For example, devices of area smaller than 4 μm^{2} diameter have been found to be CD, even if covered by top gate^{6,7}. This was attributed to the different scaling of the capacitances with diameter^{10}.
The second technique, unreported previously to our knowledge, consists of 106 nm Au/53 nm Ge/40 nm Ni ohmic contacts (OC) alloyed to the heterostructure. Although contacts act normally as sources and drains, here we employ them not for their resistive coupling but for the capacitive, namely, for screening. Clearly, as we see coherent oscillations, the LLL, the interfering one, is not resistively connected to the OC. This crucial point is verified by measuring the current flowing into the OC, as detailed in Supplementary Note 7.
Measurement techniques
Our electronic setup for measuring conductance and noise is detailed briefly in the Measurement Setup. We note that two amplifiers were utilized in the course of our measurements: the first, a homemade voltage preamplifier cooled to T=1 K having a gain factor 11 and a commercial amplifier (NF SA220F5) at room temperature having a voltage gain 200.
Additional information
How to cite this article: Choi, H.K. et al. Robust electron pairing in the integer quantum hall effect regime. Nat. Commun. 6:7435 doi: 10.1038/ncomms8435 (2015).
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Acknowledgements
We thank Y. Gefen, W.R. Lee and Y. Cohen for useful discussions and ideas. We also thank a numerous number of colleagues we consulted with in the effort to understand the mechanism that governs the observed effects. We acknowledge the partial support of the Israeli Science Foundation (ISF), the Minerva foundation, the USIsrael BiNational Science Foundation (BSF) and the European Research Council under the European Community’s Seventh Framework Program (FP7/20072013)/ERC Grant agreement No. 227716.
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H.K.C., I.S. and M.H. contributed to paper writing. H.K.C., I.S. and A.R. contributed to sample design, device fabrication and data acquisition. D.M. contributed to electron beam lithography. V.U. grew the heterostructures.
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Supplementary Figures 110, Supplementary Notes 17 and Supplementary References. (PDF 830 kb)
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Choi, H., Sivan, I., Rosenblatt, A. et al. Robust electron pairing in the integer quantum hall effect regime. Nat Commun 6, 7435 (2015). https://doi.org/10.1038/ncomms8435
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