Equal-Spin Andreev Reflection on Junctions of Spin-Resolved Quantum Hall Bulk State and Spin-Singlet Superconductor

The recent development of superconducting spintronics has revealed the spin-triplet superconducting proximity effect from a spin-singlet superconductor into a spin-polarized normal metal. In addition recently superconducting junctions using semiconductors are in demand for highly controlled experiments to engineer topological superconductivity. Here we report experimental observation of Andreev reflection in junctions of spin-resolved quantum Hall (QH) states in an InAs quantum well and the spin-singlet superconductor NbTi. The measured conductance indicates a sub-gap feature and two peaks on the outer side of the sub-gap feature in the QH plateau-transition regime increases. The observed structures can be explained by considering transport with Andreev reflection from two channels, one originating from equal-spin Andreev reflection intermediated by spin-flip processes and second arising from normal Andreev reflection. This result indicates the possibility to induce the superconducting proximity gap in the the QH bulk state, and the possibility for the development of superconducting spintronics in semiconductor devices.

shifted for clarity. We measured the resistance in the various carrier density by tuning the gate voltage. We subtract the peak height of the resistance around 4 T.

A. Device fabrication
We used an InAs QW grown by molecular beam epitaxy with the density 3 × 10 11 cm −2 and the mobility 3 × 10 5 cm 2 V −1 s −1 . The 2DEG is formed in the 4 nm-thick InAs layer.
The material stack of the InAs heterostructure is schematically shown in Fig. S1. A mesa was first defined in the substrate by wet etching with an etchant based on H 3 PO 4 . Then, NbTi was sputtered to form the superconducting electrodes on the mesa edges, following a procedure of wet etching to make the clean edge exposed, sulfur passivation to avoid oxidization of the edge, and in-situ Ar plasma cleaning. Finally a gate electrode metal of Titanium and Gold was deposited on top to address the low QH filling regime even under a low magnetic field. The top gate is placed on an insulating layer made from cross-linked PMMA. This fabrication procedure creates no superconducting material on the top surface of the mesa. This is specially devised to control the carrier density not only of the mesa but also near the junction using the top gate voltage (discussed later). The two junctions are separated by 20 µm, so this device is assumed to have two independent contact regions.

B. NbTi superconductivity
To characterize the superconducting properties of the NbTi, we performed a current-bias measurement of the differential resistance dV /dI at various out-of-plane magnetic fields B for a 150 nm-thick NbTi thin film device at 2 K which is lower than the NbTi critical temperature of 6.5 K. The measured data shows a supercurrent branch as dV /dI=0 kΩ in dark purple near the zero current bias in Fig. S2. From this data, we evaluate the critical field of B =7.0 T. Herein, the coexistence of superconducting state and QH states can be realized if the 2DEG is in the spin-resolved QH regime for B < 7.0 T.

C. Superconducting proximity at 0 T
In Fig. 1(c) in the main text, there are dip structures around V sd = ±0.7 mV. These dip structures cannot be expected from the normal BTK model. These dip structures have also been reported in experimental studies of junctions of three dimensional topological insulators and superconductor [1]. Some theoretical works predict existence of spin-triplet superconducting proximity on such junctions which generate the dip structures [2,3]. In our case, strong spin-orbit interaction on the interface can affect the superconducting proximity even at 0 T and invokes finite spin-flip process. Therefore, we suspect that the dip structure may be related to the spin-triplet superconducting proximity even at 0 T. About the superconducting bulk gap, the estimated ∆ bulk of 0.35 meV is smaller than the value of 0.99 meV, predicted from the critical temperature with conventional BCS theory.
We think that this is due to degrade of NbTi in fabrication process of the top gate structure.

D. Transconductance in the QH regime
We measured the conductance as functions of magnetic field and top gate voltage in order to estimate quality of our InAs quantum Hall effect and Zeeman effect. Figure S3 shows the conductance as functions of B and V tg . As seen in this figure, quantized conductance plateaus appear at B > 0.6 T and the Zeeman splitting is found at B >∼ 1 T. 2.4 and 4 T is large enough to study the coexistence between the spin-resolved QH state and superconductor.
Furthermore, we measured the transconductance defined by the deviation of dI/dV at 2.4 T. The results are plotted in Fig. S4. The diamond-shaped structure can be found. We focus on the bright areas, namely the plateau-transition regime in the main text. We now discuss why there is huge difference between the edge and the bulk in the AR signals. In the QH edge transport, the Andreev edge state is formed on the interface according to the theoretical prediction [4]. This Andreev edge state can classically be understood as sequential ARs with skipping orbit along the interface. (In our case, this sequential ARs should be the sequential equal-spin ARs.) Therefore, the reflected particle into the QH edge state depends on the number of ARs and can be an electron. However, AR occurs only once and reflects the hole in the QH bulk transport so it is expected the conductance enhancement due to single AR in the bulk transport.

F. Numerical calculation
We executed the numerical calculation using the model in which we assume the two channels, α and β. The fitting parameters are the superconducting gaps ∆ α and ∆ β , barrier strength Z α (Z β = 1), normal state interface conductance G n , effective temperature ω, the relative contribution of channel α, P and offset conductance G offset . We define G αβ int (V sd , Z α , T, ∆ α , ∆ β ) as eqn. (2) in the main text. Then the fitting function can be written as ( 1 We take care to confirm that our device has two independent superconducting-QH bulk state junctions. Therefore, the sub-gap features and the position in V sd of the side peaks are consistent with 2∆ α and 2∆ β . To account for the effective temperature, we approximated the deviation of the Fermi-Dirac distribution function in eqn. (1) in the main text as the Gaussian where ω is ideally equal to T but now ω includes the broadening due to inelastic scattering, inhomogeneity of the gap and the local heating [7,8].
To execute the fitting, we constrict the fitting ranges for all the parameters, and especially we tightly constrict the ∆ α and ∆ β from the curve shapes. In order to reproduce the curve shape around the zero bias voltage, we changed the fitting range for each of the curves because the differential conductance of the 2DEG appears as background and the conductance has a large dependence on the bias voltage near the plateau regime. Due to this background dependence, we could not reproduce the curve shapes in the two lower curves of the left panel and middle panel, and the lowest curve of the right panel in Fig.3(b) of the main text.
In these cases, we evaluated only ∆ α and ∆ β from the sub-gap peak features (2∆ α and 2∆ β are indicated on the panels in Fig.3(b) as open and closed hexagons). Our fitting scheme includes many free parameters and results are sensitive to the constriction of the variable range. Additionally, the obtained errors for the parameters also depends on the constriction.
However, the estimated gap energies and P produce relatively constant results with different fitting ranges, so we think it is valuable to discuss these parameters. All fits are executed with a genetic algorithm (GenCurvefit package for Igor Pro).

G. Calculation of the proximity gap energy
As written in the main text, we analyzed our experimental data with the model to evaluate ∆ α and ∆ β , the superconducting gap energies. However, these values are enlarged from the true bulk and proximity superconducting gap energies due to dissipation induced from the bulk state of the mesa. In the plateau regime, the transport is non-dissipative in the mesa, while the transport is dissipative in the plateau-transition regime due to the QH bulk state. Herein, in the plateau-transition regime, applied V sd between two superconductors is divided into the voltage on the junctions and on the mesa, then the deduced ∆ β gives a larger gap energy as the contribution of the QH bulk state in the transport becomes larger. The equivalent circuit is represented in Fig. 1 Fig.S6.

H. The maximum position of the bulk contribution
Our results for P and Z α have a maximum and minimum, respectively, at ∆g ∼ 0.7.
In this section, we estimate how large ρ xx should be to make the bulk contribution have a maximum at ∆g ∼ 0.7.
As the shape of our device is square, the two-terminal conductance is written by G 2t = √ σ xx + σ xy . σ xx and σ xy are the longitudinal conductivity and Hall conductivity [9][10][11], respectively. σ xx has a maximum when the bulk contribution is maximum and the situation is given by σ xy = 0.5e 2 /h + ne 2 /h (n = 0, 1, 2, 3...), at which change in the filling factor ∆ν is equal to 0.5. Herein, if the bulk contribution is maximum at ∆g ∼ 0.7 (namely G 2t = 0.7e 2 /h + ne 2 /h), σ xx (σ xy ) should be 0.49e 2 /h (0.5e 2 /h), 0.80e 2 /h (1.5e 2 /h), and 1.02e 2 /h (2.5e 2 /h) in the n = 0, 1, and 2 cases, respectively. From these conductivity, we can calculate the longitudinal resistance,ρ xx = σ xx /(σ 2 xx + σ 2 xy ), resulting in 1.0h/e 2 (∼ 26kΩ), 0.28h/e 2 (∼ 7.0kΩ), and 0.14h/e 2 (∼ 3.6kΩ) in the n = 0, 1, and 2 cases, respectively. The calculated longitudinal resistance should be obtained when the bulk contribution is maximum, meaning ρ xx is maximum with the calculated resistance in the region corresponding to the plateautransition regime. We measured a Hall bar device at 2 K, fabricated from the same InAs quantum well wafer as we used for the superconducting devices and the results are shown in Fig. S7. ρ xx has some peaks consistent with the finite bulk contribution. The maximum of ρ xx at 4 T is at 4 kΩ and 1.5 kΩ corresponding to the n = 1and 2 cases, respectively. The measured resistances are comparable to the estimated resistance based on the assumption that the bulk contribution is maximum at ∆g ∼ 0.7 (G 2t = 0.7e 2 /h + ne 2 /h).