Abstract
Quantum physics emerge and develop as temperature is reduced. Although mesoscopic electrical circuits constitute an outstanding platform to explore quantum behaviour, the challenge in cooling the electrons impedes their potential. The strong coupling of such micrometrescale devices with the measurement lines, combined with the weak coupling to the substrate, makes them extremely difficult to thermalize below 10 mK and imposes in situ thermometers. Here we demonstrate electronic quantum transport at 6 mK in micrometrescale mesoscopic circuits. The thermometry methods are established by the comparison of three in situ primary thermometers, each involving a different underlying physics. The employed combination of quantum shot noise, quantum back action of a resistive circuit and conductance oscillations of a singleelectron transistor covers a remarkably broad spectrum of mesoscopic phenomena. The experiment, performed in vacuum using a standard cryogenfree dilution refrigerator, paves the way towards the submillikelvin range with additional thermalization and refrigeration techniques.
Introduction
Advances towards lower temperatures are instrumental in the fundamental exploration of quantum phenomena. In the context of quantum electronics, typical examples are the exploration of the correlated fractional quantum Hall physics^{1,2,3,4,5}, of the quantum criticality, for example, with multichannel Kondo nanostructures^{6,7,8}, or of the quantum aspects of heat^{9,10,11,12}. Although commercial dilution refrigerators readily achieve temperatures in the 5–10 mK range at the mixing chamber, the pertinent value is the temperature of the electrons within the cooled quantum circuits. Owing to microwave heating, insufficient thermal contacts and electrical noise transmitted through the measurement lines, this electronic temperature is usually well above the refrigerator base temperature. Consequently, only rare examples demonstrate electronic temperatures significantly below 10 mK in quantum circuits. Moreover, the concept of temperature pervades the laws of physics, and its accurate knowledge is generally imperative whenever comparing experimental measurements with theoretical predictions; however, establishing the validity of the thermometry is particularly challenging already below 50 mK. Because of the thermal decoupling between electrons and substrate, it requires a comparison of the electronic temperature determined in situ, in the same device, by different methods.
The lowest electronic temperatures in solidstate quantum circuits were obtained in large, millimetrescale, devices that are thereby weakly sensitive to heating through the measurement lines. The lowest reported value of 3.7 mK, to our knowledge, was obtained in a large array of 600 metallic islands, each ∼100 μm wide and interconnected by tunnel junctions^{13}. Comparably low temperatures, of possibly ∼4 mK, were inferred in twodimensional (2D) electron gas (2DEG) chips in the quantum Hall regime by two different teams^{2,14,15}. For the more broadly pertinent micrometrescale mesoscopic circuits, the reported electronic temperatures are significantly higher. We note the remarkably low value of 9 mK determined with current fluctuation measurements across a quantum point contact (QPC) in a 2DEG^{16}. Although singleelectron devices are particularly challenging, because of their high charge sensitivity, comparably low electronic temperatures, down to ∼10 mK, were recently demonstrated in 2DEG quantum dots^{6,8,17}.
Here we investigate three primary electronic thermometers, and demonstrate quantum electronic transport at 6 mK in micrometrescale mesoscopic circuits. For this purpose, the experiment is performed on a highly tunable 2DEG nanostructure, that can be set by field effect to different circuit configurations. The complementary underlying physics of the thermometry methods give us access to different facets of the electronic temperature, and cover a broad spectrum of mesoscopic quantum phenomena. Whereas quantum shot noise thermometry measures the temperature of the electronic Fermi quasiparticles, through their energy distribution^{18}, the quantum back action of a resistive circuit also probes the temperature of the electromagnetic environment^{19}. In contrast, the temperature inferred from the conductance oscillations of a singleelectron transistor (SET) is very sensitive to charge fluctuations induced by nonthermal highenergy photons^{20}. At the applied magnetic field B=1.4 T, we find with the quantum shot noise measured across a voltagebiased QPC T_{N}≃6.0±0.1 mK. From the conductance peaks across the device set to a SET configuration, we obtain T_{CB}≃6.3±0.3 mK. From the dynamical Coulomb blockade conductance dip across two separate realizations of a QPC in series with a resistance, we find T_{DCBL}≃6±1 mK and T_{DCBR}≃6.5±1 mK. The observed agreement between the three primary thermometers establishes their validity on an extended temperature range.
Results
Cooled tunable mesoscopic circuit
A colourized electron micrograph of the measured device is shown in Fig. 1a, with the corresponding circuit schematic displayed Fig. 1b. A highmobility 2DEG is located 105 nm below the surface of a Ga(Al)As heterojunction. It is confined by etching within the darker grey areas delimited by bright lines, and can be tuned in situ, by field effect, with the bias voltages applied to metallic gates deposited at the surface and capacitively coupled to the 2DEG (colourized green, yellow and blue in Fig. 1a). The metallic split gates at the topleft (QPC_{L}) and bottomright (QPC_{R}) of Fig. 1a (colourized green) are used to form QPCs in the 2DEG. Note that the split gates at the topright of Fig. 1a (colourized yellow) is here set to fully deplete the 2DEG underneath, thereby closing the gate, and can be ignored. The buried 2DEG is galvanically connected, with a negligible interface resistance, to the central micrometresized metallic island (colourized red). For this purpose, the metallic island was diffused into the Ga(Al)As heterojunction by thermal annealing. The lateral continuous gates at the surface (colourized blue) implement the equivalent of shortcircuit switches in parallel with the island (blue switches in Fig. 1b). The experiments are performed with a magnetic field B applied perpendicular to the 2DEG, which corresponds to the quantum Hall regime at integer filling factors ν=6, 3 and 2 for B=1.4, 2.7 and 3.8 T, respectively. In this regime the current flows along ν chiral edge channels, represented as a single red line with the propagation direction indicated by arrows in Fig. 1a. Note that the quantum Hall effect is not necessary for the investigated primary thermometers (although it allows eliminating possible heating artefacts in the quantum shot noise thermometry, see Discussion). An important device parameter is the singleelectron charging energy E_{C}≡e^{2}/2C of the central metallic island, with C its overall geometrical capacitance and e the elementary electron charge. In particular, E_{C} sets the temperature scale extracted from Coulomb blockade thermometry. The charging energy is most straightforwardly determined by setting the device in the SET configuration, with the shortcircuit switches open (as shown in Fig. 1a,b) and QPC_{L,R} tuned to tunnel contacts. The SET conductance is plotted in Fig. 1c (higher values shown brighter) versus the capacitively coupled gate voltage V_{g} and the applied drainsource dc voltage V_{dc}. The charging energy is directly related to the periodic ‘Coulomb diamond’ patterns in Fig. 1c: , with the diamonds’ maximum dc voltage.
Electronic current fluctuations
The current across a voltagebiased quantum coherent conductor fluctuates because of the thermal agitation (the JohnsonNyquist noise) and the granularity of charge transfers (the shot noise)^{18}. These fluctuations give information on the charge of the carriers, for example, in the fractional quantum Hall regimes^{3,16,21,22}, as well as on the statistics of the charge transfers^{23,24,25}, and also provide a very robust primary thermometer for the electronic temperature^{26}.
We have measured the current fluctuations across the device tuned into a voltagebiased QPC (schematic shown in top panel of Fig. 2a, see Supplementary Note 2 for details on the current fluctuations measurement setup). For this purpose, the right shortcircuit switch in Fig. 1b was effectively closed, by applying V_{g}=0 to the continuous gate adjacent to QPC_{R}. Thereby, the 2DEG is not depleted and the edge current flows underneath the gate without backscattering, implementing an ideal closed switch (see Supplementary Fig. 1).
The dependence with bias voltage V_{dc} of the current fluctuations’ spectral density, S_{I}(V_{dc}), is directly related to the electrons’ energy distribution (including in outofequilibrium situations^{27,28}). For a short quantum conductor, the excess spectral density ΔS_{I}(V_{dc})≡S_{I}(V_{dc})−S_{I}(0) can be calculated in the standard framework of the scattering approach^{29,30,31}. It reads^{18}:
where the quantum conductor is described as a set of independent conduction channels, indexed by the label n, each characterized by a transmission probability τ_{n}, and with k_{B} (h) the Boltzmann (Planck) constant. Note that the noise added by the amplification chain is cancelled out by considering the excess spectral density ΔS_{I}. Importantly, the product between the gain of the amplification chain and ∑τ_{n}(1−τ_{n}) is given by the temperatureindependent linear slope predicted at eV_{dc}≫k_{B}T. Fitting the raw spectral density data based on equation 1 therefore allows a selfcalibrated determination of the electronic temperature, without requiring the knowledge of {τ_{n}} or of the amplification gain (see Supplementary Note 2 for further details).
The symbols in the top panel of Fig. 2a display the excess current spectral density measured at B=1.4 T versus the dc bias voltage applied across the QPC, which is tuned into the advantageous configuration of a single halftransmitted conduction channel (τ≃0.55). Note that in order to display the current fluctuations data in A^{2}/Hz, and, although it is not necessary for extracting the electronic temperature, the effective amplification chain gain is calibrated by matching the linear bias voltage increase in the raw spectral density at large eV_{dc}≫k_{B}T with the prediction of equation 1 for the measured τ=0.55. The continuous (dashed) line shows ΔS_{I} calculated using equation 1 with τ=0.55 and T=6.0 mK (T=0, with a negative vertical offset to match the T=6.0 mK calculation at eV_{dc}≫k_{B}T). Experimentally, the main difficulty is to reach a sufficient resolution to accurately extract the electronic temperature. To this aim, we developed a fully homemade cryogenic noise amplification scheme, based on high electron mobility transistors grown and nanostructured in the laboratory^{32,33}. Despite the unfavourable current–voltage conversion at ν=6 because of the low quantum Hall resistance h/6e^{2}≃4.3 kΩ, we resolve ΔS_{I} with an extremely high statistical precision of ±9 × 10^{−32}A^{2} Hz^{−1}, slightly smaller than the symbols’ size.
Most directly, we have determined the electronic temperature and experimental uncertainty T_{N}=6.0±0.1 mK from the mean value (red horizontal line in bottom panel of Fig. 2a) and statistical uncertainty of an ensemble of 131 values (symbols in bottom panel of Fig. 2a) independently obtained by separately fitting successive noise measurement sweeps ΔS_{I}(V_{dc}). Note that the ΔS_{I} data shown in the top panel of Fig. 2a was obtained by averaging these successive sweeps (each resolved with an individual noise precision of ±10^{−30 }A^{2} Hz^{−1}).
Coulomb blockade oscillations
At low temperatures, T≪E_{C}/k_{B}, the charge of a mesoscopic island connected through tunnel contacts is quantized in units of the elementary electron charge e. This allows for the manipulation of single electrons in circuits, which has led to the field of ‘single electronics’^{19}. Setting the device in the SET configuration (see schematic in bottom panel of Fig. 2b), charge quantization results in periodic peaks of the SET conductance G_{SET} when sweeping the capacitively coupled gate voltage V_{g}. In the presence of dc bias voltage, the peaks develop into periodic ‘Coulomb diamond’ patterns as shown in Fig. 1c. The width of these conductance peaks at zero dc bias voltage constitutes a wellknown primary thermometer, frequently used in the context of mesoscopic physics. For a metallic island, with a continuous density of states and connected through tunnel contact, the SET conductance reads^{34}:
with G_{∞} the classical (high temperature) conductance of the SET, Δ≃712±2 μV the gate voltage period and δV_{g} the gate voltage difference to charge degeneracy. Note that the Coulomb blockade thermometry is possible only with tunnel contacts. In the presence of connected conduction channels with large transmission probabilities, the quantum fluctuations of the island’s charge would average out Coulomb oscillations and thereby impede the Coulomb blockade thermometry (see ref. 35 for a characterization of charge quantization versus transmission probability on the same device).
The symbols in the top panel of Fig. 2b represent G_{SET} measured at B=1.4 T versus δV_{g}. The continuous line shows the SET conductance calculated using equation 2 with T=6.3 mK, G_{∞}=0.088e^{2}/h, Δ=711 μV and E_{C}=25 μeV.
Similarly to quantum shot noise thermometry, we determined the electronic temperature and statistical precision T_{CB}=6.3±0.05 mK from an ensemble of 222 values (symbols in the bottom panel of Fig. 2b) obtained by separately fitting individual sweeps of G_{SET}(δV_{g}). The 222 sweeps are distributed among 14 adjacent Coulomb peaks, spreading over 10 mV in gate voltage. We find the same electronic temperature, at experimental accuracy, for the different Coulomb peaks and also for the 15 or 16 measurements of each peak. Note that our experimental accuracy T_{CB}≃6.3±0.3 mK is limited by our resolution of the charging energy, E_{C}≃25±1 μeV. The uncertainty is consequently much larger than the statistical precision. Note also that, despite a relatively low ac voltage of 0.35 μV_{rms} applied to probe G_{SET}, we estimate (using the master equation generalizing equation 2 to finite voltages^{19}) that it is responsible for an effective increase of 0.1 mK in T_{CB} (we have not corrected for this small effect). Finally, we point out that the G_{SET} data shown in the top panel of Fig. 2b were obtained by averaging the 222 individual sweeps.
Dynamical Coulomb blockade conductance renormalization
The conductance of a quantum coherent conductor is progressively reduced upon cooling by the quantum back action of the circuit in which it is embedded^{19}. This phenomenon, called dynamical Coulomb blockade, results from the granularity of charge transfers combined with Coulomb interactions. It has been extensively studied, and the theory is now well established in the simplest limit of a small tunnel conductor inserted into a linear circuit (see ref. 19 and references therein; for recent developments beyond the tunnel limit see refs 36, 37, 38, 39).
We consider here the case of a tunnel contact in series with a linear resistance R, as shown in the schematics of Fig. 2c. In this configuration, the conductance at zerobias voltage (zero temperature) vanishes with temperature T (bias voltage V_{dc}) as (as ). Similarly to quantum shot noise thermometry, the equilibrium (V_{dc}≪k_{B}T/e) to nonequilibrium (V_{dc}≫k_{B}T/e) crossover provides a primary electron thermometer. In general, the electronic temperature can be extracted by fitting the conductance versus dc voltage with the full quantitative numerical prediction of the dynamical Coulomb blockade theory (see ref. 40 for a formulation involving a single numerical integration). Note that the extracted electronic temperature reflects equally the thermal energy distributions of the Fermi electron quasiparticules, and of the bosonic electromagnetic modes of the quantum circuit. The dynamical Coulomb blockade was previously used to probe the nonFermi energy distribution of electrons driven outofequilibrium in the presence of a thermalized RC circuit^{41}.
In the lowtemperature and lowbias voltage regime (k_{B}T, eV_{dc}≪E_{C}), the primary dynamical Coulomb blockade thermometry reduces to the simple procedure described below. The QPC conductance at low temperature (T≪E_{C}/k_{B}) and at zerobias voltage V_{dc}=0 reads^{42}:
where G_{∞} is the tunnel conductance in the absence of dynamical Coulomb blockade renormalization and Γ(x) is the gamma function. Extracting the temperature from the zerobias conductance apparently requires a precise knowledge of both G_{∞} and the circuit parameters (R, C). However, the necessary information is provided by the bias voltage dependence. In the nonequilibrium regime k_{B}T<<eV_{dc} and at low energy compared with the singleelectron charging energy eV_{dc}<<E_{C}, the QPC conductance reads^{19}:
with γ≃exp(0.5772). Consequently, the bias voltage exponent gives the series resistance R, and one can rewrite the zerobias voltage conductance as:
with calibrated from the conductance measured in the lowenergy nonequilibrium regime, where equation 4 applies, and A(R) a known function, straightforwardly obtained from equations 3 and 4.
Here we determined the electronic temperature by setting one QPC in the tunnel regime (G_{∞}∼0.1e^{2}/h), while the other QPC was tuned to fully transmit two electronic channels, thereby implementing a linear series resistance R=h/2e^{2} (which is not renormalized by dynamical Coulomb blockade^{37,38,39,43}; obtained from a very broad and flat conductance plateau owing to the quantum Hall effect^{11,39}). Symbols in the top (bottom) panels of Fig. 2c represent the conductance measured with the left (right) QPC in the tunnel regime, versus dc bias voltage, at B=1.4 T. The continuous lines display the quantitative numerical calculations of the dynamical Coulomb blockade prediction using the separately characterized E_{C}=25 μeV and R=h/2e^{2} (also corresponding to the linear bias voltage dependence), and with G_{∞}=0.123e^{2}/h, T=6 mK for the top panel (G_{∞}=0.139e^{2}/h, T=6.5 mK for the bottom panel). The grey areas represent a temperature uncertainty of ±1 mK. The dashed lines are the T=0 predictions of equation 4 for the same device parameters. Note that for the present circuit implementation A(R=h/2e^{2})≃0.40 and the nonequilibrium conductance increases linearly with bias voltage, as can be directly verified on the conductance data. See Supplementary Fig. 2 for a comparison between the numerically calculated dynamical Coulomb blockade predictions and the data up to larger bias voltages.
Electronic temperature versus experimental conditions
Information on the limiting factors towards lower electronic temperatures T in our cryogenfree dilution refrigerator is obtained by measuring T for different magnetic fields B and for different additional Joule powers P_{J} dissipated directly on the mixing chamber plate. Note that T is here obtained from quantum shot noise thermometry up to 35 mK, and from the identical but faster readings of our standard RuO_{2} thermometer at higher temperatures. Each set of symbols in Fig. 3 corresponds to a different applied B∈{1.41,2.74,3.76} T. For P_{J}≳5 μW, we observe the usual quadratic dependence with temperature (T^{2}∝P_{J}), independently of the applied B. However, we find that the electronic temperature at zero Joule power is higher for larger magnetic fields (top left inset). Assuming that the observed relationship (straight black line in the main panel) holds at all temperatures when substituting the additional Joule power by the full dissipated power P=P_{0}+P_{J}, we extract the refrigeratordissipated power P_{0} versus magnetic field. The corresponding P_{0} values are shown as symbols in the bottom right inset. We find that the increase of P_{0} with B is compatible with a quadratic magnetic field dependence (continuous black line: P_{0}=0.5+0.1(B/1 T)^{2} μW), which is a typical signature of eddy current dissipation.
Discussion
A 6 mK electronic temperature was obtained in micrometrescale quantum circuits using a mediumsized cryogenfree dilution refrigerator (Oxford instruments Triton, with 200 μW of cooling power at 100 mK), with the sample in vacuum and in the presence of a 1.4 T magnetic field. At larger magnetic fields B, we observe a temperature increase that corresponds to an additional dissipated power quadratic in B, as typically expected for eddy currents. In our cryogenfree refrigerator, the underlying vibrations originate from the pulse tube.
The sample environment and wiring shown Fig. 4 offers a proven guideline to ultralow electronic temperatures with an allpurpose setup, including 35 measurement lines and a toploaded sample holder. Although additional details are provided in the Supplementary Note 1, we here briefly point out several key ingredients. The sample is strongly protected from spurious highenergy photons, by two shields at base temperature. The most important thermal anchoring of the measurement lines at base temperature is performed by dipping insulated copper wires into silver epoxy very close to the sample, inside the inner stainless steel shield. The measurement lines are all individually shielded in a coaxial cable geometry (except for the abovementioned copper wires and for a short distance inside the shielded sample holder, between the input connector and the RC filters). The highfrequency filtering and initial thermalization to the mixing chamber plate of the electrical lines are performed with homemade resistive microcoaxes (μCoax NiCr in Fig. 4a)^{20}. Because the electrical noise integrated over the full bandwidth needs to be smaller than a fraction of μV, we only keep the bandwidth used for the measurements with personalized RC filters directly located inside the sample holder. This is most particularly important with a cryogenfree dilution refrigerator in the presence of a magnetic field because of the electrical noise induced by vibrations. We now compare the three investigated primary electronic thermometers.
Quantum shot noise thermometry stands out as the most robust and straightforward approach. It is based on simple physics, directly probes the temperature of the electrons through their energy distribution^{18} and does not require a separate calibration of the noise measurement setup. The main possible artefact is local heating induced by the dissipated Joule power at finite dc bias. Such a heating typically scales linearly with V_{dc} (refs 44, 45). It is therefore difficult to distinguish from a slight increase in the shot noise^{24,25}. The present implementation in the quantum Hall regime, however, provides a strong protection against heating artefacts, owing to the spatial separation between incoming and outgoing currents. Although it is not necessary to determine the factor ∑ τ_{n}(1−τ_{n}) for the voltagebiased quantum conductor, it is important to make sure that it does not depend on V_{dc}. For a singlechannel quantum conductor, the dependence of τ(1−τ) with voltage bias is minimized at τ∼0.5, and τ(V_{dc}) can be monitored simultaneously with the noise measurements. The main challenge with quantum shot noise thermometry is in the sensitivity of the noise measurement setup; however, the associated temperature uncertainty can be statistically quantified. Note that the achieved resolution of 6.0±0.1 mK is comparable to the accuracy of the provisional lowtemperaturescale standard (PLTS2000)^{46}.
Coulomb blockade thermometry is also very straightforward and has the advantage of being less demanding on the measurement sensitivity. It is consequently widespread in the field of mesoscopic physics. However, the extracted temperature is easily/often artificially increased by charge fluctuations in the device vicinity, or by the electrical noise on the capacitively coupled gates. Such an artefact could be detected as a gate voltagedependent increase in the noise level, proportional to ∂G_{SET}/∂V_{g}, provided that a significant part of the charge fluctuations is within the measurement bandwidth. In general, the corresponding temperature increase is difficult to establish, except by comparing with another electronic thermometer. Here the agreement obtained with both the quantum shot noise and dynamical Coulomb blockade thermometers demonstrates a negligible artificial increase in the electronic temperature.
Dynamical Coulomb blockade thermometry can be difficult to use in general, if the surrounding circuit is not known a priori at relevant GHz frequencies. As in the case of quantum shot noise, a possible artefact is heating at a finite dc bias. This can be minimized by using a tunnel contact of very large impedance compared with the circuit. In contrast to quantum shot noise, the quantum Hall regime does not provide a protection against heating (in the central metallic island, for the dynamical Coulomb blockade experimental configurations). However, the very large renormalized tunnel resistance, 100 larger than the series resistance, ascertains negligible heating effects. Moreover, the dynamical Coulomb blockade thermometry is here particularly straightforward to implement because of the precise knowledge of the circuit.
With the consistent temperatures obtained by three primary thermometers, each relying on different physical mechanisms, we firmly established electronic thermometry standards in the regime of ultralow temperatures. The achievement of 6 mK electronic temperature, with the mesoscopic circuit in vacuum and using a mediumsized dilution refrigerator, provides a platform for further reduction of the temperature, using additional thermalization and cooling techniques^{15,47,48} towards the submillikelvin range.
Methods
Sample
The sample was nanostructured by standard ebeam lithography in a Ga(Al)As 2DEG of density 2.5 × 10^{11} cm^{−2} and mobility 10^{6} cm^{2 }V^{−1 }s^{−1}. The AuGeNi metallic island was diffused by thermal annealing into the semiconductor heterojunction to make an electrical contact of negligible resistance with the 2DEG (see Methods in ref. 8 for the electrical characterization of the contact in the same sample).
Measurement techniques
The differential conductance measurements were performed using standard lockin techniques at frequencies below 200 Hz and using rms ac excitation voltages smaller than k_{B}T/e. The sample was currentbiased by a voltage source in series with a 100 MΩ polarization resistance at room temperature. The applied current was converted onchip into a voltage independent of the device configuration by taking advantage of the welldefined quantum Hall resistance to an adjacent grounded electrode (h/νe^{2} at filling factor ν). Similarly, the current transmitted across (reflected from) the device was converted into a voltage with the h/νe^{2} quantum Hall resistance. The noise measurement setup includes a homemade cryogenic preamplifier and an L–C tank circuit of resonant frequency 0.84 MHz; see the online Supplementary Information and also the Supplementary Material of ref. 11 for a more detailed description.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Additional information
How to cite this article: Iftikhar, Z. et al. Primary thermometry triad at 6 mK in mesoscopic circuits. Nat. Commun. 7: 12908 doi: 10.1038/ncomms12908 (2016).
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Acknowledgements
This work was supported by the European Research Council (ERC2010StG20091028, no. 259033), the French RENATECH network, the national French programme ‘Investissements d’Avenir’ (Labex NanoSaclay, ANR10LABX0035) and the European Seventh Framework Program (EU FP7, no. 263455).
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Affiliations
Centre de Nanosciences et de Nanotechnologies, CNRS, Univ Paris SudUniversité ParisSaclay, Université Paris DiderotSorbonne Paris Cité, 91120 Palaiseau, France
 Z. Iftikhar
 , A. Anthore
 , S. Jezouin
 , F. D. Parmentier
 , Y. Jin
 , A. Cavanna
 , A. Ouerghi
 , U. Gennser
 & F. Pierre
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Contributions
Measurements and analysis: F.P. with inputs from Z.I.; low temperature setup: F.P. with inputs from A.A. and S.J.; noise measurement setup: F.D.P., S.J., A.A. and F.P.; HEMT nanofabrication: Y.J.; heterojunction growth: A.C., A.O. and U.G.; sample nanofabrication: F.D.P. and A.A; manuscript preparation: F.P. with inputs from A.A. and U.G.; project planning and supervision: F.P.
Competing interests
The authors declare no competing financial interests.
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Correspondence to F. Pierre.
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