A nyons with non-Abelian statistics are of great fundamental interest 1 and can be used to realize topologically protected, and therefore intrinsically fault-tolerant qubits. [2][3][4] Non-Abelian anyons are expected to be realized in few fractional quantum Hall (QH) states [5][6][7][8][9] such as the QH states with filling factor ν = 5/2, [10][11][12] and possibly ν = 12/5. 13 However, so far, no unambiguous experimental confirmation exists of the presence of non-Abelian anyons in such QH states.

An alternative route to realize non-Abelian anyons relies on inducing superconducting pairing between counter-propagating edge modes of QH states that intrinsically support only Abelian anyons. [14][15][16][17] These theoretical proposals build on an earlier proposal for creating Majorana zero modes, that is, the anyons with the simplest non-Abelian statistics, using 1D modes at the edge of a 2D topological insulator (TI) in contact with a superconductor (SC). 18 In contrast to TIs, in twodimensional electron gases (2DEGs) in the QH regime, by varying filling factor ν, states can be realized with a variety of topological orders. This allows access to more exotic edge states needed for engineering anyons with richer non-Abelian statistics. Key in all these theoretical proposals is the ability to induce superconducting pairing, via the proximity effect, between the QH edge modes.

The strength of the superconducting correlations that can be induced in a QH-SC heterojunction can be evaluated by obtaining the amplitude of the Andreev reflection of QH edge modes. The early search for Andreev reflection in QH-SC systems focused on InAs and InGaAs semiconductor magnetoresistance oscillations at relatively low magnetic fields 19 followed later by reports of induced superconductivity in QH states. 20 More recently there have been reports of observation of induced superconductivity, 21,22 cross Andreev reflection, 23,24 edge state mediated supercurrent, 25 and interference of chiral Andreev edge states [26][27][28][29] in graphene. To make further progress, it is essential to reliably demonstrate the ability to induce robust superconducting correlations into the edge modes of a QH state.

In this work, we show that in high quality InAs/NbTiN heterostructures, very strong superconducting correlations can be induced in the edge modes of integer QH states realized in the InAs-based quantum wells (QWs). Such correlations appear to be robust, showing no oscillations as a function of doping, for gate voltages within the QH plateaus. We analyze the experimental data in conjunction with a microscopic model to extract the details of the processes determining the transport properties of the QH-SC interface.

Figure 1a shows a cross sectional schematic of the fabricated device used in this work. The QW is formed by a 4 nm layer of In 0.81 Ga 0.19 As layer, a 7 nm layer of InAs, and a 10 nm top layer of In 0.81 Ga 0.19 As. The QW is grown on In x Al 1-x As buffer where the indium content is step-graded from x = 0.52 to 0.81 to help with dislocations originating from lattice mismatch between InP and quantum well in higher levels. A delta-doped Si layer with electron doping n ∼ 1 × 10 12 cm -2 is placed 6 nm below the QW. This epitaxial structure has been used in previous studies on mesoscopic superconductivity [30][31][32][33] in the development of tunable qubits 34 and in studies aimed at realizing and detecting topological superconducting states. [35][36][37] A Hall bar, Figure 1b, is fabricated by electron beam lithography. In order to study the 2DEG/SC interface, a 90 nm thick layer of NbTiN was sputtered as the superconducting contacts with a 150 μm-wide interface after performing wet etch surface cleaning (Device A). We also fabricated a similar device with intentional no surface cleaning step before NbTiN sputtering (Device B). A metallic top gate is created by depositing a layer of Al oxide followed by an Al layer to control the QW electron density. 38 The mobility of the QW is determined to be μ ∼ 12 000 cm 2 /V•s at n ∼ 8.51 × 10 11 cm -2 corresponding to an electron mean free path of l e ∼ 180 nm. All data reported here were taken at T ∼ 30 mK. We have provided more information on transport properties of the sample in the SI. We note that while we focus mainly on one device (Device A) in the main text, we have studied a few other similar devices which their data have been shown in SI.

When the sample is placed in a magnetic field, in the classical picture electrons and holes will alternate their skipping orbits across the interface of the superconductor and 2DEG. 39 In the full quantum-mechanical analysis the electron and hole edge states hybridize due to the proximity of the SC and form a coherent chiral Andreev edge state (CAES) extended along the QH-SC interface. 27,40,41 A schematic of CAES propagation along the QH-SC interface is shown in Figure 1c. In this picture, if more holes than electrons reach the normal lead downstream from the superconducting electrode (lead 5), then a negative potential difference (V 5 -V 4 ′) develops. In Figure 1d, we show the local density of states (LDOS) of a CAES obtained with a tight binding (TB) calculation performed using the python package Kwant. 42 In the TB model, the presence of the magnetic field is taken into account via a Peierls phase, and the superconductivity of the QW proximitized by NbTiN via a mean field an s-wave pairing term of strength Δ ̃. The details of the TB model can be found in the SI.

Figure 1e,f shows the results for the downstream resistance, R D 2L , measured between the voltage contacts 5 and 4′, as a function of gate voltage V g and magnetic field B. Hall resistance data measured between contacts 2 and 5 allow us to determine the filling factor of the different regions of Figure 1e,f. Figure 2a shows the horizontal cut at B = 11 T of Figure 1f,R D 4L , and the corresponding longitudinal resistance R xx . From the R xx measurements, we see that we have well developed integer QH states (IQHS). From Figures 1f and2a, we clearly observe that R D is negative for IQHS, a fact that strongly suggests the presence of Andreev processes at the QH-SC interface for these IQHS. The similarity of R D 2L and R D 4L also is a hint of IQHS being dissipationless states; therefore, going farther from the interface does not affect the magnitude of the negative resistance value. We notice the importance of a clean InAs/ NbTiN interface by comparing the magnitude of negative resistance in Figure 1e,f. The clean interface on device A has been achieved by etching the surface of defined NbTiN pattern area by buffered oxide etchant (BOE) for 2 s immediately followed by loading into sputtering tool's chamber in order to minimize the time for the native oxide growth at the interface. On the other hand, for device B this cleaning step has been skipped and NbTiN sputtered on the defined region after its exposure to air. For the rest of the paper, we focus only on device A results. The upstream resistance R U 2L (measured between contacts 3 and 4) exhibits plateaus corresponding to the R xy plateaus in the magnetic field but with resistance values lower than R xy (Figure 2b). Moreover, R U 2L -R D 2L recovers the quantized Hall value, R xy , as shown in Figure 2c. Note that this difference does not necessarily match the R xy data outside the QH regime.

These results can be understood within the Landauer-Buẗtiker (LB) theory, generalized to allow for the presence of a superconducting lead. 43,44 We start with the six-terminal setup shown in Figure 1b (see also the SI). We assume the terminal 1, 2, 3, 5, 6 to be ideal metallic leads, and contact 4 to be a superconducting lead. We first consider the limit in which no normal reflection or transmission processes take place at the superconducting lead. I i and V i , the currents and voltages, respectively, are allowed at the terminals i = (1, 2, 3, 4, 5, 6). Without loss of generality, we can set V 4 = 0. We can use the charge conservation equation ∑ i I i = 0 to express I 4 in terms of the currents at the other leads. With these considerations, the LB equations reduce to the following system of linear equations

= i k j j j j j j j j j j j j j j j j j j j j j j j j y { z z z z z z z z z z z z z z z z z z z z z z z z i k j j j j j j j j j j j j j j j j j j j y { z z z z z z z z z z z z z z z z z z z i k j j j j j j j j j j j j j j j j j j j j j j j j y { z z z z z z z z z z z z z z z z z z z z z z z z

where ν is the number of edge states, R H is the Hall resistance, and A is the average probability, per edge mode, of Andreev reflection. Considering that no current flows into leads 2, 3, 5, 6, so that

, and setting I ≡ I 1 , it is straightforward to solve eq 1 to obtain

Figure 3a shows the scaling of R D with respect to 1/ν for different values of B. From the slope of the fits to the experimental data shown in Figure 3a, we obtain the value of A exp = 55%, independent, to very good approximation, on the value of V g within the QH plateaus (Figure 3b). It is instructive to compare our work with two other classes of experiments. First, studies based on graphene-MoRe, see ref 26, where the value of R D oscillates significantly with B and gate voltage, and therefore there is no well-defined value of A. Second, studies based on graphene-NbTiN, see ref 23, in which a narrow and long superconducting electrode is placed perpendicular to the vacuum edge of the QH system. This geometry allows for additional processes, cross-Andreev-reflection (CAR) processes that are not present in our setup and the setup in ref 26. From the value of R D at ν = 1 in ref 23, we extract a value of A exp = 50.5% but the comparison to our setup is not straightforward due to the presence of CAR processes. We emphasize that in addition to the large value of A, a unique feature of our devices is the lack of oscillations as a function of V g and B despite a QH-SC interface that is more than an order of magnitude longer than in the experiments in refs 23 and 26, and that this suggests that in QH-SC based on InAs/ SC the CAES might have an extremely long coherence length.

Figure 3c shows the consistency of the measured values of R D 2L and R U 2L with the LB predictions by plotting the ratio (R U

-R U

)/R xy as a function of B that, according to eqs 2 and 3, is expected to be equal to 1.

To understand qualitatively how such values of A can arise, we consider an effective 1D Bogoliubov de Gennes

Here and in the remainder of this Letter, we set ℏ = 1. In the equation for H(x), v d is the drift velocity of the edge modes, τ i values are the Pauli matrices in Nambu space, k F is the edge modes Fermi wave vector number (measured with respect to the QH-SC interface), and Δ ̃is the superconducting superconductor pairing induced via the proximity effect by of the superconducting lead. Using the expression for H, we can obtain the transfer matrix M relating ψ(x) at the two ends of the length L sc of the QH-SC interface of length L sc (see SI), 40,45 and then the expression for the electron-hole conversion probability will be

In eq 4, δϕ is the difference of the phases accumulated by the electron-like and hole-like edge modes along the length of the QH-SC interface. Let k F

, k F (h) be the Fermi wave vector of the electron-like and hole-like edge modes, and δk ≡ |k

We then write δϕ = L sc δk.

Considering that L sc = 150 μm is quite large, any small change of δk induced, for example, by changes in V g , should result in a significant change of A and therefore of R U and R D . However, in the experiment within the QH plateaus, R U and R D do not show any oscillation as a function of V g . It is natural to conclude that this might be due to scattering processes leading to a dephasing of the electron-like and hole-like modes along the QH-SC interface. 28 In this case, the effective A eff can be obtained by averaging over L sc on the right-hand side of eq 4 to obtain

). Considering that (v d k F /Δ ̃)2 > 0, we see that in this case we cannot recover the value of A extracted from the experimental results, A = 0.55 > 0.5.

To explain the large value of A, accompanied by the lack of oscillations as a function of V g , we are led to two possibilities. The first possibility is that δ k does not change appreciably as a function of V g . From eq 4, considering that 0 < sin 2 (δϕ) < 1, we can see that to have A = 0.55 we must have v d k F /Δ ̃< 0.9. the limit when δϕ is such that sin 2 (δϕ) ≈ 0.55, we must have v d k F /Δ ̃≪ 1. In this limit we can write δk ≈ Δ ̃/v d . In good approximation, Δ ̃and v d are independent of V g , and we recover the observed values of R U and R D with no oscillations in the QH plateaus. Notice that the condition v d k F /Δ ̃≪ 1 is equivalent to the condition δkξ ≈ 1, where ξ ≡ v d /Δ can be interpreted as the superconducting coherence length of the edge modes in proximity of the SC. The other possibility is that dephasing processes are accompanied by a finite probability of single electron tunneling into the superconductor and breaking of particle-hole (p-h) symmetry. This would allow to have a situation in which electron-like states are more likely than h-like states to tunnel into the superconductor and therefore contribute less to the downstream current explaining a negative R D even when A ≤ 1/2. If we denote T the probability, per edge mode, of an electronlike state to tunnel in the SC, in eqs 2 and 3 we would replace 2A with 2A + T. In this case, from the measurements of R U and R D we recover 2A + T. Assuming ⟨A⟩ = 1/2 would imply T = 0.2. The smallest value of ⟨A⟩, consistent with particle conservation, is 15% to which it would correspond T = 0.8, a value that implies a very strong breaking of particle-hole symmetry at the QH-SC interface. It is difficult to distinguish between these two possibilities given that we cannot measure separately the quasiparticle and supercurrent contributions to the charge current flowing from the QH region into the superconducting lead.

In conclusion, we have fabricated a quantum Hall-superconductor (QH-SC) epitaxial heterostructure based on InAs and NbTiN and characterized the transport properties of its QH edge modes propagating along a superconducting interface. We have observed negative values for the downstream resistance R D between a normal lead and the superconducting lead and a corresponding suppression of the upstream resistance R U such that in the QH plateaus the difference R U -R D is equal to the Hall resistance R H . The negative values of R D are an unambiguous sign that at the QH-SC interface there is a very large electron-hole conversion probability, A. Using a Landauer-Buẗtiker analysis, we were able to explain the relation between R D and R U and express both resistances in terms of a single effective probability for Andreev reflections at the QH-SC interface. Our analysis led us to the conclusion that either the edge modes propagate along the QH-SC interface with negligible dephasing resulting in an electron-hole conversion close to 55%, or if dephasing processes dominate that a strong breaking of particle-hole symmetry at the QH-SC interface must occur. The interface could be further improved by hybrid methods of epitaxial superconductors and semiconductors. 46 Even the lower bounds' estimates for A that we extract from our measurements are remarkable, larger than any published ■ AUTHOR INFORMATION