Introduction of Josephson field effect transistor concept1 sparked active research on proximity effects in semiconductors. Induced superconductivity and electrostatic control of critical current has been demonstrated in two-dimensional gases in InAs, graphene and topological insulators, and in one-dimensional systems including quantum spin Hall edges. Recently, interest in superconductor–semiconductor interfaces was renewed by the search for Majorana fermions, which were predicted to reside at the interface. More exotic non-Abelian excitations, such as parafermions (fractional Majorana fermions) or Fibonacci fermions may be formed when fractional quantum Hall edge states interface with superconductivity. Realization of a long-sought regime of an interplay between superconductivity and strongly correlated states in a two-dimensional electron gas (2DEG) at high magnetic fields requires development of transparent superconducting contacts to high-mobility 2DEG, which remain superconducting at high magnetic fields.
Proximity effects in GaAs quantum wells have been intensively investigated in the past and Andreev reflection has been observed by several groups. Unlike in InAs, where Fermi level (EF) at the surface resides in the conduction band, in GaAs EF is pinned in the middle of the gap, which results in a high Schottky barrier between a 2DEG and a superconductor and low transparency non-ohmic contacts. Heavy doping can move EF into the conduction band and, indeed, superconductivity has been induced in heavily doped bulk n++ GaAs. In quantum wells, similar results were obtained by annealing indium contacts; however, the critical field of indium is ~30 mT, well below the fields required to form quantum Hall effect (QHE) states.
In this article, we report the development of transparent superconducting ohmic contacts to high-mobility 2DEG in GaAs. The superconducting contact is type-II NbN with large critical field >16 T. Induced superconductivity is observed across 1.6 μm of a 2D gas at zero field. From temperature dependence of the critical current and analysis of Andreev reflection, we estimate contact transparency parameter Z 
0.2. Induced superconductivity is observed in magnetic fields up to 0.2 T. At high magnetic fields, we observe deviations of longitudinal and Hall resistances from a similar data obtained with normal contacts, a clear indication of an interplay between superconductivity and QHE edge states.
0.2. Induced superconductivity is observed in magnetic fields up to 0.2 T. At high magnetic fields, we observe deviations of longitudinal and Hall resistances from a similar data obtained with normal contacts, a clear indication of an interplay between superconductivity and QHE edge states.Results
Heterostructures design
In conventional quantum well structures AlGaAs, barrier between 2DEG and the surface of the sample adds an extra 0.3 eV to the Schottky barrier when contacts are defused from the top. We alleviated these problems by growing an inverted heterojunction structures, where a 2DEG resides at the GaAs/AlGaAs interface but the AlGaAs barrier with modulation doping is placed below the 2DEG, see Fig. 1b, where band diagram was calculated using a self-consistent Poisson solver34(The program can be downloaded from http://www3.nd.edu/~gsnider/). Contacts are recessed into the top GaAs layer to bring the superconductor closer to the 2DEG. A thin layer of AuGe and NbN superconductor form low resistance ohmic contacts to the 2DEG after annealing. The inverted heterostructure increases the contact area of side contacts compared with quantum well structures by utilizing all GaAs layer above the heterointerface for carrier injection (130 nm in our inverted heterostructure versus 20–30 nm in typical quantum wells, see Supplementary Fig. 1 andSupplementary Note 1).
(a) Scanning electron microscope images of test devices similar to samples A and B. Enlarged region for sample B is an atomic force microscope image of a real sample. 2D gas regions are false-colour coded with green, superconducting and normal contacts are coded with orange and blue, respectively. Scale bar is 2 μm. (b) Simulation of the conduction band energy profile in the heterostructure. (c) T-dependence of resistance between contacts 3 and 4 in sample B measured with 10 nA a.c. excitation. Superconducting transition is observed at Tc≈290 mK.
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Induced superconductivity
We report induced superconductivity in two devices from different wafers. Sample A has long (70 μm) contacts separated by 1.6 μm of 2DEG, contacts to sample B are formed to the edge of a mesa with 0.6 μm separation, see Fig. 1a. Details of device fabrication are described in Methods. When cooled down to 4 K in the dark, both the samples show resistance in excess of 1 MΩ. After illumination with red light-emitting diode a 2DEG is formed and 2-terminal resistance drops to <500 Ω. As shown in Fig. 1d, sample resistance RB3−4 gradually decreases on cooldown from 4 K to the base temperature and the superconductor–2DEG–superconductor (S–2DEG–S) junction becomes superconducting at Tc~0.3 K.
Voltage–current V(I) characteristics for two S–2DEG–S junctions (between contacts 8 and 9 for sample A, and 3 and 4 for sample B) are shown in Fig. 2. Both the samples show zero-resistance state at small currents with abrupt switching into resistive state at critical currents Ic=0.22 and 0.23 μA for samples A and B, respectively. We attribute hysteresis in V(I) characteristics to Joule heating in the normal state.
Voltage-current characteristics (a,b) and differential resistance (c,d) for samples A (a,c) and B (b,d). The conduction is measured between contacts (8-9) for sample A and (3-4) for sample B. dV/dI is measured with Ia.c.=1 nA. Induced superconductivity with zero voltage is observed with critical currents Ic~220 nA for sample A and Ic~230 for sample B. Red (blue) traces are for current increasing (decreasing).
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The most attractive property of a high-mobility 2DEG is large mean free path l
ξ0, with l=24 μm and the Bardeen-Cooper-Schrieffer (BCS) coherence length ξ0=ℏvF/πΔ=0.72 μm for sample B. Here 
is the Fermi velocity, n is a 2D gas density, m is an effective mass and Δ=1.76kBTc=46 μeV is the induced superconducting gap. Evolution of V(I) with T is shown in Fig. 3a. Experimentally obtained T-dependence of Ic is best described by the Kulik–Omelyanchuk theory for ballistic junctions (L<<l) (ref. 35), the blue curve in Fig. 3b. For comparison, we also plotIc(T) dependence for the dirty limit 
(ref. 36), which exhibits characteristic saturation ofIc at low temperatures.
ξ0, with l=24 μm and the Bardeen-Cooper-Schrieffer (BCS) coherence length ξ0=ℏvF/πΔ=0.72 μm for sample B. Here is the Fermi velocity, n is a 2D gas density, m is an effective mass and Δ=1.76kBTc=46 μeV is the induced superconducting gap. Evolution of V(I) with T is shown in Fig. 3a. Experimentally obtained T-dependence of Ic is best described by the Kulik–Omelyanchuk theory for ballistic junctions (L<<l) (ref. 35), the blue curve in Fig. 3b. For comparison, we also plotIc(T) dependence for the dirty limit (ref. 36), which exhibits characteristic saturation ofIc at low temperatures.
(a) Evolution of the induced superconductivity with T for sample B. The R(I) curves are offset proportional toT for T>50 mK. (b) Temperature dependence of critical current Ic(T) is extracted from (a) and compared with the expected T-dependence for ballistic and diffusive regimes (reduced Ic compared with Fig. 2 is due to larger Ia.c.=10 nA used in this experiment). (c) High-temperature data shows Andreev reflection (excess current and reduced dV/dI around V=0. The curves are not offset. In d, excess current is modelled within the Blonder–Tinkham–Klapwijk theory39 with Z=0.2.
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In short ballistic junctions, L<<ξ0<<l the product Ic(0)RN=πΔ/e does not depend on the junction length L. For L~ξ0 this product is reduced by a factor 2ξ0/(L+2ξ0) (ref. 37). The measuredIcRN=83 μV for sample B is in a good agreement with an estimate πΔ/e·2ξ0/(L+2ξ0)=90 μV. For sample A, the IcRN=19 μV while the estimated product is ≈50 μV. The reduction is consistent with the geometry of sample A, where a region of the 2DEG with induced superconductivity is shunted by a large region of a 2DEG in a normal state.
Transparency of a superconductor/2DEG interface
In one-dimensional junctions, the induced gap 
depends on the broadening of Andreev levels within the semiconductor38 
, where we introduce contacts transparencies D1 and D2. We assume for simplicity that D1=D2=1/(1+Z2), where 0<Z<∞ is a interface barrier strength introduced in ref. 39, and Bagwell’s effective channel length Leff=L+2ξ0. Using NbN superconducting gap, 
(NbN is a strong-coupling superconductor,
) and Tc=0.3 K for 
we obtain Z=0.2. This value is consistent with the fit of the Icversus T-dependence with D as a free parameter (Supplementary Fig. 2; Supplementary Note 2). Similar values of Z can be estimated from the analysis of the shape of dI/dV(V) characteristics at elevated temperatures, as shown in Fig. 3. At 
, Andreev reflection at S–2DEG interfaces results in an excess current flowing through the junction for voltage biases within the superconducting gap Δ0/e and corresponding reduction of a differential resistance dV/dI by a factor of 2. In the presence of a tunnelling barrier, normal reflection competes with Andreev reflection and reduced excess current near zero bias, resulting in a peak in differential resistance. Within the Blonder–Tinkham–Klapwijk theory, a flat dV/dI(V) within Δ0/e, observed in our experiments, is expected only for contacts with very high transparency Z<0.2. For larger Z>0.2, a peak at low biases is expected (Supplementary Fig. 3, Supplementary Note 3). Several features of the experimental I(V) need to be mentioned. First, we observe several sharp peaks in the resistance at high biases (around 2 and 4 mV for T=4 K). Similar sharp resonances has been observed previously, where authors attributed their appearance to the formation of Fabry–Pérot resonances between superconducting contacts. In our devices, the superconducting region is shunted by a low resistance (<100 Ω) 2DEG, thus appearance of >10 kΩ resonances cannot be explained by resonant electron trapping between contacts. These resonances are also observed inI(V) characteristics of a single S–2DEG interface (measured in the S–2DEG–N configuration between contacts 3 and 6, see Supplementary Fig. 3). Differential resistance does not change substantially across resonances, ruling out transport through a localized state. We speculate that in the contacts where these resonances are observed superconductivity is carried out by quasi-one-dimensional channels, and jumps in I/V characteristics are due to flux trapping at high currents. This scenario is consistent with the observation that peaks shift to lower currents at higher fields, seeFig. 4. The second notable feature of our data is reduction of the zero-bias resistance by ≈2.6 times at low temperatures, while Andreev reflection limits the reduction to the factor of 2. We attribute this reduction to the multiple Andreev reflection between two closely spaced contacts, for contacts with larger separation (20 μm) multiple Andreev reflection is suppressed and the reduction of resistance by a factor of 2 is observed (Supplementary Fig. 3).
depends on the broadening of Andreev levels within the semiconductor38 , where we introduce contacts transparencies D1 and D2. We assume for simplicity that D1=D2=1/(1+Z2), where 0<Z<∞ is a interface barrier strength introduced in ref. 39, and Bagwell’s effective channel length Leff=L+2ξ0. Using NbN superconducting gap, (NbN is a strong-coupling superconductor, ) and Tc=0.3 K for we obtain Z=0.2. This value is consistent with the fit of the Icversus T-dependence with D as a free parameter (Supplementary Fig. 2; Supplementary Note 2). Similar values of Z can be estimated from the analysis of the shape of dI/dV(V) characteristics at elevated temperatures, as shown in Fig. 3. At , Andreev reflection at S–2DEG interfaces results in an excess current flowing through the junction for voltage biases within the superconducting gap Δ0/e and corresponding reduction of a differential resistance dV/dI by a factor of 2. In the presence of a tunnelling barrier, normal reflection competes with Andreev reflection and reduced excess current near zero bias, resulting in a peak in differential resistance. Within the Blonder–Tinkham–Klapwijk theory, a flat dV/dI(V) within Δ0/e, observed in our experiments, is expected only for contacts with very high transparency Z<0.2. For larger Z>0.2, a peak at low biases is expected (Supplementary Fig. 3, Supplementary Note 3). Several features of the experimental I(V) need to be mentioned. First, we observe several sharp peaks in the resistance at high biases (around 2 and 4 mV for T=4 K). Similar sharp resonances has been observed previously, where authors attributed their appearance to the formation of Fabry–Pérot resonances between superconducting contacts. In our devices, the superconducting region is shunted by a low resistance (<100 Ω) 2DEG, thus appearance of >10 kΩ resonances cannot be explained by resonant electron trapping between contacts. These resonances are also observed inI(V) characteristics of a single S–2DEG interface (measured in the S–2DEG–N configuration between contacts 3 and 6, see Supplementary Fig. 3). Differential resistance does not change substantially across resonances, ruling out transport through a localized state. We speculate that in the contacts where these resonances are observed superconductivity is carried out by quasi-one-dimensional channels, and jumps in I/V characteristics are due to flux trapping at high currents. This scenario is consistent with the observation that peaks shift to lower currents at higher fields, seeFig. 4. The second notable feature of our data is reduction of the zero-bias resistance by ≈2.6 times at low temperatures, while Andreev reflection limits the reduction to the factor of 2. We attribute this reduction to the multiple Andreev reflection between two closely spaced contacts, for contacts with larger separation (20 μm) multiple Andreev reflection is suppressed and the reduction of resistance by a factor of 2 is observed (Supplementary Fig. 3).
(a,b) Differential resistance is measured as a function of B and Id.c. for two samples at 40 mK. Induced superconductivity (black region) is observed up to 0.2 T in both the samples. (c) 3-terminal resistance for a sample with all normal contacts (red) and between normal and superconducting contacts in sample B (I(2–4) and V (4−1) in Fig. 1) is measured at 70 and 40 mK, respectively. B<0 (B>0) induces clockwise (counterclockwise) chiral edge channels, note resistance scales difference for two field directions.
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Induced superconductivity in low magnetic fields
Finally, we present magnetic field dependence of induced superconductivity. The low-field data is shown in Fig. 4a,b, where black regions correspond to zero-differential resistance. Induced superconductivity is suppressed at ≈0.2 T in both the samples. In sample A, a narrow region of a 2DEG with induced superconductivity is confined between large NbN superconducting leads with rigid phases. Perpendicular magnetic field twists the phase in the 2DEG resulting in Fraunhofer-like oscillations of the critical current. In this sample, although the 2DEG extends beyond the narrow region between the contacts and Ic does not decrease to zero and abrupt jumps in Ic reflect multiple flux jumps. The period of oscillations is ~0.5 mT, which corresponds to an area of 4.1 μm2, much smaller than the area of the 2DEG between the contacts (≈120 μm2). This observation is consistent with the reduced IcRN product measured for this sample as discussed above. In sample B, contacts are fabricated along the edge of the mesa and 2D gas is not enclosed between the contacts. Consequently, Ic is a smooth function of B.
Superconductivity and quantum Hall effect
Competition between superconductivity and chiral quantum Hall edge states is shown in Fig. 4c, where resistance is measured in a 3-terminal configuration over a wide range of magnetic fields. Simple Landauer–Büttiker model of edge states predicts zero resistance for negative and quantized Hall resistance for positive field direction for integer QHE and fractional QHE states, which is clearly seen in a sample with non-superconducting (AuGe) ohmic contacts (red curve). When a superconducting contact serves as a current injector (blue curve), integer ν=1 and fractional ν=2/3 and 3/5 states are well developed for B<0, while the same states are not quantized at proper QHE values for B>0. If we assume that current injection via a superconducting contact results in an extra voltage offset at the contact by Voff≈Δind/e, the measured voltage will be reduced by Voff. The magenta bars for B>0 indicate corrected resistance (V−Voff)/I for Voff=140 μV. While this offset may explain the measured values for fractional states, a twice smaller Voff is needed to reconcile the resistance at ν=1. Note that NbN critical field Bc>16 T. At low fields, states ν=3, 4 and 5 have resistance minima for B<0, indicating a partial equilibration of chiral edge currents with the superconducting contact, while resistance near ν=2 has a maximum. Zero resistance at ν=1 and large resistance at ν=2 are in contrast to the theoretical prediction that ν=2 state should be stronger coupled with a superconducting contact than ν=1 (ref. 23).
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