Proximity-induced surface superconductivity in Dirac semimetal Cd3As2

Cd3As2 is a three-dimensional Dirac semimetal with separated Dirac points in momentum space. In spite of extensive transport and spectroscopic studies on its exotic properties, the evidence of superconductivity in its surface states remains elusive. Here, we report the observation of proximity-induced surface superconductivity in Nb/Cd3As2 hybrid structures. Our four-terminal transport measurement identifies a pronounced proximity-induced pairing gap (gap size comparable to Nb) on the surfaces, which exhibits a flat conductance plateau in differential conductance spectra, consistent with our theoretical simulations. The surface supercurrent from Nb/Cd3As2/Nb junctions is also achieved with a Fraunhofer/SQUID-like pattern under out-of-plane/in-plane magnetic fields, respectively. The resultant mapping shows a predominant distribution on the top and bottom surfaces as the bulk carriers are depleted, which can be regarded as a higher dimensional analog of edge supercurrent in two-dimensional quantum spin Hall insulators. Our study provides the evidence of surface superconductivity in Dirac semimetals.

RInterface. (c) Normalized magnetic field dependent RInterface -T curves. The resistance is normalized to a normal-state value at 9 K. The curves are shifted vertically for clarity. The transition temperature is denoted by triangles. (d) H-T phase diagram of Nb and proximity-induced superconductivity in Cd3As2. To make a visual contrast, we use the on-temperature as the transition temperature. The red and blue dashed lines show the GL fitting of Nb and Cd3As2, respectively. Inset displays a schematic drawing with the magnetic field direction perpendicular to the Nb/Cd3As2 plane.

Supplementary Figure 5 | Supplementary R-T data for device #01 at 0 T. (a)
Interface resistance-temperature curves measured from the other two electrodes (4)(5)(6). (b) The contrast of Interface and Nb at 7-9 K. The off-transition temperature of Nb is 8.3 K and it is higher than the on-temperature of the second transition Interface . (c)-(d) Cd  (c) Zero-field temperature-dependent dI/dV of RInterface, normalized by the normal-state conductance at 9 K. The curves are vertically shifted for clarity. Four obvious features are observed as the abovegap dip (Δ d ), bias-independent conductance plateau (BICP) (Δ s ), broad peak (Δ b ) from proximityinduced superconductivity in Cd3As2 bulk states and zero-bias conductance peak (ZBCP). (d) Temperature-dependent normalized differential conductance spectra of Cd3As2. Figure 7 | BTK fits of differential conductance spectra in Nb/Cd3As2 device #01. (a)-(b) Normalized dI/dV-V characteristics from experimental data (red hollow circles) and fitting results (black solid line) at 2 K and 0 T for RInterface from different electrodes.

Supplementary
-500 0 500 1000 -1000 -500 0 500 1000   Supplementary Fig. 1a is an optic image of Nb/Cd3As2 hybrid structure after transferring onto SiO2/Si substrate (device #05 for thick Cd3As2). Nanoplates with a typical thickness around 100-250 nm were chosen to study the proximity-induced superconductivity in Fermi arcs while the samples with a thickness exceeding 300 nm are for bulk-dominated study. A thickness less than 150 nm can enhance the phase coherence of Fermi arcs and surface contribution while avoiding the gap opening due to the quantum confinement effect. To identify the composition of the sample, energy dispersive X-ray spectroscopy (EDX) has been carried out as shown in Supplementary  Fig. 1b and the result is reproducible. The atomic ratio of Cd and As is 1.54: 1. The slight composition deviation might be due to arsenic vacancies 1 .

Supplementary Note 2. Device fabrication and measurement setup
The Nb/Cd3As2 hybrid structures were fabricated as shown in Supplementary Fig.  2b. We first transfer the as-grown Cd3As2 nanoplates onto SiO2/Si substrate. We used a standard electron beam lithography (EBL) technique to deposit Cr/Au (10/150 nm) bilayer electrodes on Cd3As2 using magneton sputtering. Then, another EBL process is performed to fabricate the Nb film on top of Cd3As2.
The differential conductance spectra were captured by an ac-modulation technique using a lock-in amplifier (SR830) and Agilent 2912 as shown in Supplementary Fig.  2a. A small ac voltage coupled with a dc input was fed through a large standard resistor (0.05-1.0 MΩ) in series to deliver a constant dc current, which was superimposed by a small ac current. This current is passed through the device. The lock-in amplifier working at 17 Hz measures the ac voltage as the differential voltage d ; the d is estimated to be a constant since the change of device resistance is four orders of magnitude smaller than the resistor. By scanning the dc current , we can acquire is plotted against to produce the differential conductance spectrum.
All the measurements are deployed by the four-terminal method to prevent contact resistance. For example, the measurement setup for device #01 is presented in Supplementary Fig. 2b and c. We apply a constant current from electrode 1 to 8, and meanwhile, measure the voltage difference between electrode 2 and 3 as the Nb resistance (by dividing the constant current). Such a four-terminal method can exclude the effect of current redistribution in the 2-terminal method. To make the measurement more accurate, all the resistance data ( 23 , 35 , 46 , 56 and 67 ) are measured at the same time.
We can estimate the interface resistance Interface as shown at the bottom of Supplementary Fig. 2b is a valid estimation.
Then, we performed the cross-section TEM experiments of Nb/Cd3As2 interface as shown in Supplementary Fig. 3a, the thickness of each layer can be estimated to be 140 nm and 120 nm for Nb and Cd3As2, respectively. Moreover, the EDS mapping result displayed in Supplementary Fig. 3b indicates low oxidization and high interface quality.

Supplementary Note 3. R-T measurement for device #01
Temperature-dependent resistance (R-T) curves at ≥ 2.0 K across the junction are shown in Supplementary Fig. 4b. The resistance drop region can be broadened as the applied magnetic field increases ( Supplementary Fig. 4c) which makes it easier to distinguish the two transitions. When the temperature is lower than 4 K ( c3 ), the resistance drops again and a small magnetic field of 0.5 T can destroy the transition. The upper critical fields c2 (Nb) and c2 (Interface), as a function of temperature, are plotted in Supplementary Fig. 4d. The coherence length of Nb is extracted to be From Supplementary Fig. 2b, we can simultaneously measure Interface of electrodes 3-5 and electrodes 4-6. Similarly, three transitions are observed in Supplementary Fig. 5a-b. The difference of the drop for the first transition is due to the different Nb resistance between Interface (4−6) and Interface (3−5) . Besides, from the resistance measurements of two sides Cd3As2 as shown in Supplementary Fig. 5c-d, we can exclude the possibility of Cd3As2 intrinsic effect to the three transitions 3 .
Differential conductance curves were acquired as shown in Supplementary Fig. 6. The critical current of Nb is c~1 700 μA at 8 K in Supplementary Fig. 6a which is larger than our measurement limit at ≤ 7 K, indicating that Nb is always in the superconducting state in our dI/dV measurement. Supplementary Fig. 6b shows the original data of interface differential conductance. Since the normal state conductance does not change with bias at 8.5 K, we use the conductance (~0.4 S) at 8.5 K to normalize the dI/dV. Supplementary Fig. 6c shows the dI/dV-I curve of Interface (3−5) . Supplementary Fig. 6d demonstrates the dI/dV curves of Cd3As2 without AR-related signals. Thus, we can conclude that the behavior including ZBCP, BICP, ZBBP and above-gap dip in Interface is due to the interface superconductivity. Now, we need to rule out other possibilities of the ZBBP. Our measurements show a proximity-induced gap appearing immediately below c with 0.14 times the normal conductance, different from previous reports on phase conjugation 4 . The device with high interface barrier Z shows the lack of the proximity effect and only the superconducting gap of Nb is observed, thus dismissing other possible processes occurring in Nb alone ( Supplementary Fig. 12).

Supplementary Note 4. BTK fits of Nb/Cd3As2 differential conductance spectra
This section mainly focuses on the theoretical calculations to understand the density of states (DOS) features at the Nb/Cd3As2 interface with the superconducting proximity effect at an energy scale within the intrinsic superconducting gap. We used BTK theory to simulate the differential conductance at finite temperature with respect to the bias voltage , that is given by 5 where ( , ) is the Fermi distribution function. ( ) demonstrates the BTK conductance at = 0 as follows, where N is transparency of the barrier in the BTK approximation of current injection totally perpendicular to the Nb/Cd3As2 interface: = ℏ F is a dimensionless parameter modeled with a -function barrier = ( ).
In addition, we consider the quasiparticle lifetime. The AR structures in the experimental spectra are not only depressed in amplitude but also spread in energy which is attributed to the reduction of the quasiparticle lifetime, resulting from the imaginary part of the quasiparticle self-energy, as discussed in the tunneling regime by Dynes et al 6 , and inelastic quasiparticle scattering processes 7 . It is possible to globally take these effects into account by including in the BTK model a single broadening parameter Γ in the form of an imaginary part of the energy → + iΓ which is called modified BTK model.
We summarize the BTK fits in the broad peak region in Supplementary Fig. 7. Interface differential conductance spectra measured by two different electrodes, Interface (3−5) and Interface (4−6) , are shown in Supplementary Fig. 7a and b with fitting parameters Δ = 0.19 mV , Z = 0 and Γ = 0.13 mV and 0.10 mV , respectively. The data is fitted well and zero Z value indicates a high interface quality. Corroborated with our theoretical simulations in Supplementary section 7, we attribute it to the proximity effect on the bulk Cd3As2 states.
The BTK fits of dI/dV in three devices (#02, #03, #04) are plotted in Supplementary Fig. 8. We use the broad peak region to prove that the behavior is attributed to the proximity-induced superconductivity.

Supplementary Note 5. BCS fits of the superconducting gap
Here, we use an approximate equation of BCS theory to fit the data 8,9 Δ( ) = Δ 0 tanh(1.74√ c − 1), The temperature-dependent above-gap dip is shown in Supplementary Fig. 9.

Supplementary Note 6. Above-gap dip analysis
The above-gap dip shows a good BCS relation (Supplementary Fig. 9) with temperature and it disappears above c , indicating its physical origin from the superconductivity of Nb. The above-gap dip has been observed in many systems 10,11,12,13 and still under debate. Applying a high bias during the differential conductance measurement may produce Joule heating which causes a break-down of superconductivity in Nb and drives the resistance to the normal state superimposed by irregular structure 11 . The above-gap dips at Δ d are caused by a breakdown of a small excess current observed in the superconducting state. Δ d has a temperature dependence given by √ c − corresponding to an energy balance condition, where the dissipated power is proportional to the temperature shift 10 .
We plot the Δ d~√ c − relation in Supplementary Fig. 10 and use the Joule heating model to fit. The fitting is not good, where Δ d has saturated around 4 K in experiments rather than increased with a large value in the fitting curve. Such a saturation behavior is more suitable for the local destruction of superconductivity. We note that the above-gap dip occurs at a current of ~2 mA, corresponding to a current density of ~5 × 10 3 A/cm 2 . Such a large could destroy superconducting pairing near the Nb/Cd3As2 interface and affect the local contact resistance 14 near the interface 12 . Indeed, device #03 (no proximity effect) also exhibits an above-gap dip at large bias ~20 meV, corresponding to a large current density of ~4 × 10 3 A/cm 2 , similar to device #01. Thus, we attribute Δ d to depairing near the Nb/Cd3As2 interface which affects the local contact resistance 14 or the local destruction of superconductivity of the Nb near the interface 12 . Figure S11a shows the measured differential conductance spectra at different temperatures within a smaller bias range of ±0.15 mV. The ZBCP is pronounced at 3.5 K and becomes stronger and sharper as temperature decreases. The emergence of the ZBCP coincides with the third resistance drop observed in R-T curves as explained in the main text. This means that the ZBCP is associated with the proximity-induced superconductivity or AR in Cd3As2 nanoplate. The temperature dependence of the ZBCP intensity and width is plotted in Supplementary Fig. 11c. The ZBCP intensity is increased from 3.5 to 2 K. More interestingly, the ZBCP width is almost constant below 2.5 K, even decreases a little at low temperatures. With increasing the magnetic field, the width of the ZBCP is a bit broadened and the intensity is reduced. When the magnetic field is above 0.3 T, the ZBCP cannot be observed.

Supplementary Note 7. ZBCP analysis
The ZBCP has been reported in various superconductor-normal metal hybrid structures 15,16,17,18,19 , but its physical origin is still under debate. The ZBCP can be induced by the proximity-induced pair current across Schottky barrier at superconductor-semiconductor interfaces 4 . But the ZBCP width is expected to increase with decreasing temperatures, which is contrary to the behavior from our experiments and no barrier is observed in Nb/Cd3As2.
The second possible mechanism is related to incoherent accumulation of AR, which happens when there is a large probability of backscattering due to the involvement of the other surface of the normal-metal thin film 13,16 . ZBCP of this kind usually increases immediately below c which contradicts to our results.
The third possibility is related to coherent scattering of carriers near the interface due to a phase conjugating between the electron's and the hole's trajectories, leading to an enhanced AR probability 15 . A ZBCP caused by this mechanism is sensitive to both temperature and magnetic field, since it involves a coherent loop. However, it should begin to appear in differential conductance spectra right below c while in our devices the ZBCP can only be observed at 3.5 K. Besides, these theories do not take into account the strong spin-orbit coupling (SOC) and its resulted Berry phase 12 . In the presence of strong SOC, the phase accumulated by the incident electron along its path cannot be canceled by the retro-reflected hole. Furthermore, the theory 20 suggests that this kind of ZBCP often appears in junctions with a relatively strong scattering rate, and that the value of the conductance peak will not exceed the conductance of the normal state, whereas in our experiments the ZBCP can be 1.5 times larger than the normal state conductance. Therefore, we believe that the ZBCP is not caused by the aforementioned constructive interference.
The fourth scenario is to attribute ZBCP to a pair current flowing between the superconducting Nb and the proximity-induced superconducting Cd3As2 phase. Its behavior will resemble the critical supercurrent of a Josephson junction. As the temperature decreases, the critical current of a Josephson junction will first increase and then gets saturated. For a ZBCP of this type, therefore, its peak width is expected to increase with decreasing T. However, The ZBCP width does not change much in Supplementary Fig. 9c and the pair current picture seems inapplicable to our results.
The fifth probability is Andreev bound state of an anisotropy superconductor (s-wave or d-wave) 21,22,23 or the Majorana zero mode in the core of the vortices of topological superconductor 19 . ZBCP of these kinds usually shows a weak dependence on the magnetic field. However, this is apparently not our case. The ZBCP is quenched quickly by 0.4 T magnetic field in Supplementary Fig. 11b. Some situations like Kondo correlations and weak anti-localization are also not suitable to explain our results because the ZBCP we observed is related to superconductivity.
The sixth case is the proximity-induced superconductivity in another Cd3As2 band. Note that the projection onto [112] direction can make the Fermi surface anisotropic. Another Cd3As2 band projected onto [112] direction may account for the observed ZBCP with a lower transition temperature ( c3 ) and gap (Δ b ′ ).
The seventh mechanism of ZBCP involves unconventional superconductivity with an asymmetric orbital order parameter 24,25 . This mechanism has been proved in p-wave superconductor Sr2RuO4 24 , superconductor/topological insulator 12 and proximityinduced p-wave superconductivity in graphene 26 . Since Cd3As2 has topological nontrivial order in bulk states, by taking into account the unconventional phase-diagram in Fig. 1d, a possible theoretical mechanism has been reported in Cd3As2 that quasi-2D helical p-wave superconducting states exist in the bulk 27 with specific z , which is explainable for the appearance of the ZBCP in this experiment.

Supplementary Note 8. Supplementary data for Nb/Cd3As2 device #03 and #05
Here we discuss the transport properties in bulk-dominated Cd3As2 device #05. The SEM image in Supplementary Fig. 12a illustrates clear white color Cd3As2 boundary, indicating its large thickness. Figure S12b shows R-T curves of Nb, interface, and Cd3As2, respectively 3 . Superconducting transition begins at 8.4 K and finishes at 8.2 K similar to device #01. Two transitions occur at c1 = 8.4 K and c2 = 8.0 K. The first corresponds to Nb superconducting. The second is unclear at this moment, because the resistance increases at low temperature, indicating superconducting tunneling behavior. As a contrast, Cd 3 As 2 does not show any transition. Figure S12c-d display dI/dV spectra of Nb, interface and Cd3As2, respectively. Nb is in the superconducting state with the applied current and Cd3As2 shows no particular behavior. Two signatures can be easily observed in Interface which are above-gap dips at Δ d~± 20 mV and a conductance dip around zero bias with two pronounced conductance peaks at Δ d~± 1.3 mV, consistent with the superconducting gap of Nb. Since the AR can enhance the conductance below the superconducting gap and the tunneling effect happens in a nontransparent interface, the conductance dip with two peaks is attributed to AR and superconducting tunneling. Unlike normal metals in which various bands cross the Fermi level, only two bands 28,29 take part in transport in Cd3As2 and such a low density of electrons cannot sustain a BICP.
Then, we discuss the temperature and magnetic field dependence of dI/dV spectra. Both two features are temperature dependent (Supplementary Fig. 13a) and can be well fitted by BCS theory (Supplementary Fig. 13c). Moreover, the conductance dip with two peaks region can be well fitted by BTK model in Supplementary Fig. 13, showing the evidence of AR with a parameter = 1.25 . Furthermore, the magnetic field 23 / 34 dependent dI/dV spectra in Supplementary Fig. 14 confirm that both two features are attributed to superconductivity-related effect. As discussed in Supplementary section 4.3, we think that the above-gap dip is due to the local destruction of superconductivity of Nb near the interface because it does not change with the different bulk or surface contribution in Cd3As2. The supplementary data for surface dominated Cd3As2 device #03 is displayed in Supplementary Fig. 15. Suplementary Fig. 15b shows R-T curves of Cd 3 As 2 . Only one transition at c1 = 8.4 K is observed. Contrasting with dI/dV spectra in Supplementary  Fig. 15c, the BICP corresponds to the transition in R-T curve. Besides, a small ZBCP emerges at c1 = 3.0 K, due to the proximity-induced superconductivity in bulk states. The ZBCP is too small to be detected in the R-T curve and a small resistance drop at ~3 K is on the same order as the background noise. The resistance saturates at low temperature and the Andreev reflection amplitude from bulk states is much weaker than surface channels. The gap from BTK fits at 2K in Supplementary Fig. 8b-c is also quite small. Even we estimate the proximity-induced bulk gap at zero temperature by BCS theory that Δ(2 K) = Δ 0 tanh(1.74√ 3 2 − 1) = 0.85Δ 0 , Δ 0 does not change much.
Therefore, we can conclude the weak proximity effect in bulk states.

Supplementary Note 9. Formation of Weyl orbits and 2D Fermi surface in Cd3As2 nanoplates
The simple bulk band structure and the controllable thickness growth of nanoplates with different Fermi levels make Cd3As2 a good candidate for studying the Fermi-arc states through SdH oscillations 1, 30, 31, 32 . As shown in Supplementary Fig. 16a, we apply a constant current in Cd3As2 nanoplates along [11 ̅ 0] direction. Perpendicular and parallel magnetic fields correspond to [112] and [11 ̅ 0] direction, respectively. Figure S16b illustrates the formation of Weyl orbits and Fermi arcs in thin Cd3As2. Figure S16c shows a typical angle dependence of MR curves in bulk-dominated and bulk-surface-mixed Cd3As2. Clear beating patterns are observed in the SdH oscillations in device #02, indicating possible multiple cyclotron orbits.
The behavior is in stark contrast with the single-frequency SdH oscillations observed in bulk-states dominated device #05. The angle-dependence of FFT spectra is shown in Supplementary Fig. 16d. Two peaks are found in device #02 while only one distinct peak exists in device #05. Both bulk peaks do not change much at different angles. However, the other peak position increases with a larger angle. The angle dependence of the surface frequency can be well fitted by 1/ cos function in Supplementary Fig. 16e, corresponding to the 2D Fermi surface. Nevertheless, the second frequency has also been suggested due to the Fermi surface nesting 33 or band curvature near the Lifshitz transition 34 in bulk states. Notably, the bulk states in both samples show a similar oscillation frequency, indicating very close Fermi levels in these devices, which excludes the influence of band structure difference at different Fermi levels 1 .
To estimate the superconducting coherence length in device #01, we need to 24 / 34 calculate the diffusion coefficient , which constitutes two parameters F , F that can be extracted from the SdH oscillations. We show the SdH oscillations in device #01 in Supplementary Fig. 17. Following the Lifshitz-Kosevich formula 35 , the oscillation component Δ xx can be described by where T , D and S are three reduction factors accounting for the phase smearing effect of temperature, scattering and spin splitting, respectively. Figure  Temperature dependent FFT spectra is displayed in Supplemetnary Fig. 17b. By performing the best fit of the thermal damping oscillation to the equation by FFT amplitude in Supplementary Fig. 17c, the effective mass is extracted to be * = 0.058 e , where is electron mass. The Fermi velocity F can be extracted by Since the thickness of Cd3As2 is large enough to be treated as a 3D material, the electron diffusion coefficient is given by = F MFP 3 , where F is Fermi velocity.
In device #01, = 3.15 × 10 −2 m 2 s −1 and the coherence length in the clean limit is 452 nm at 8.4 K, the superconducting transition temperature of Nb. The obtained N is large than MFP in all low temperature regime ( < 8.4 K), thus we use the dirty limit expression N is 207 nm at 8.4 K, seemingly not to accord with the dirty limit. However, when the temperature is lower, the N can be larger than MFP (at 2 K, N = 424 nm). We can then use this value as a good estimation.
The SdH oscillations in device #03 are illustrated in Supplementary Fig. 18. The analysis is similar, and the parameters are summarized in Supplementary Table 1.

Supplementary Note 10. Bulk and surface contribution to Andreev reflection
We note that the 1.14 times conductance enhancement in Fig. 1b cannot be used to judge the interfacial Z for surface channels. This is because the normal-state conductance ( > ) of our hybrid junction always has contributions from the bulk states, not just from the surface channels alone.
Suppose the conductance from the bulk states and surface states are B and S at normal states, respectively. The conductance at non-superconducting regime can be expressed as N = B + S . At 8.2 K where only the surface states participate in AR, the whole conductance becomes, where ~1.5 μm is the distance between the electrode and the interface. S~0 .5 μm is solved from 2 K . Such a superconducting layer is comparable to the coherence length N which also verifies our semi-quantification model. Therefore, the large conductance enhancement at low temperatures is due to the enhancement of proximityinduced superconducting layer.

Supplementary Note 11. Theoretical simulations
In this section, we present the detailed tight-binding Hamiltonian used in our theoretical calculations. The total Hamiltonian for the Nb/Cd3As2 junction can be written as: where DS / SC refers to the Hamiltonians for the Dirac semimetal Cd3As2, and the superconductor (Nb) respectively. c is the coupling Hamiltonian at the Dirac semimetal/superconductor interface.
The tight-binding Hamiltonian HDS for Cd3As2 is defined in the Method section of the main text. The model parameters are listed in Supplementary Table 2.
To construct a numerically solvable model for the junction formed by Cd3As2 and Nb, in our simulations we consider the junction geometry shown schematically in Supplementary Fig. 19, which is physically equivalent to the junction geometry defined in the main text. For simplicity, we choose the [010] surface of Cd3As2 to interface with the top surface of the superconductor. Since the Dirac points are located at (0,0, ± 0 ) along the [001]-axis in the Brillouin zone of Cd3As2, with this choice there exists Fermi arc states on the [010] surface which can couple directly to the superconductor. In addition, the periodic boundary condition is assumed along the x-direction, and momentum x can serve as a good quantum number.
With the junction geometry defined in Supplementary Fig. 19b, the coupling Hamiltonian at the junction interface can be written as: Here, ,sσ ( x ) annihilates an electron at site R in Cd3As2 with orbital s(= S, P), spin σ(=↑, ↓) and momentum x . In Fig.4, we set the coupling strength to be c ≅ . To calculate the differential conductance c = d /d in the transport measurements, we consider a semi-infinite Cd3As2 with a section of its bottom layer attached to the top layer of the superconductor as shown schematically in Supplementary Fig. 20.
We note that for voltage bias < Δ/e, the only scattering process contributing to current flow is the Andreev reflection, in which an incoming electron is converted into a reflected hole. While this process occurs at the physical interface (blue solid line in Supplementary Fig. 20) formed by Cd3As2 and Nb, current conservation guarantees that the current flow driven by Andreev reflections can be obtained from scattering matrices defined at any interface in the Cd3As2. To calculate the conductance in a convenient way, we choose the interface indicated by the red dashed-line in Supplementary Fig. 20 in our numerical calculations. The d /d based on scattering formalism is given by: Here, ee ( eh ) refers to the reflection coefficient for an incoming electron to be reflected as an electron (hole) at the artificially chosen interface (red dashed-line). Note that all the Andreev reflection processes at the physical interface (blue solid line) are essentially captured by eh at the artificial interface indicated by the red dashed-line (see schematic drawing in Supplementary Fig. 20).
With translational invariance in the x-direction, the total scattering matrix can be brought into sub-blocks characterized by momentum kx. The reflection coefficients for a fixed kx and energy E are given by the Lee-Fisher formula 37 Here, α,β ∈{e,h} are the electron/hole indices, σ0 is the identity matrix in the spin space, and I is the identity matrix in the rest of the Hilbert space.
ii 1 is the retarded Green's function at the interface (red dashed line). Γ is the broadening function defined as Γ(E,kx) = i[Σ(E,kx)−Σ † (E,kx)], where Σ(E,kx) is the selfenergy of the semi-infinite Cd3As2. The total differential conductance is thus obtained by summing over all kx in the neighborhood of the Fermi surface: Here, FD ( , ) = 1/[ B + 1] is the Fermi-Dirac distribution function.
In the main text, we pointed out that a flat conductance plateau cannot be found when Fermi arc states are absent. Here, we demonstrate this result by artificially closing both the top and the bottom surfaces of Cd3As2 in our simulations, and calculate the dI/dV for F = 70 meV with the same parameters used in Fig.3c of the main text. With the boundaries of the Dirac semimetal being closed, there exist no boundaries at the top/bottom surfaces to host Fermi arcs at the interface. As a result, the Andreev reflections in this case are solely driven by bulk state channels. As shown in Supplementary Fig. 21, in this case, a ZBBP stands in the midst of two separated coherence peaks, and the conductance plateau is absent. We note that, while this scenario without boundaries at top/bottom surfaces cannot be realized in real experimental settings, the theoretical result shown in Supplementary Fig. 21 identifies Fermi-arc states as the origin of the conductance plateau.

Supplementary Note 12. Josephson junctions in long junction limit
The c n product is a characteristic junction parameter that provides useful information about superconducting transport through the Josephson junction. The c n product is usually around ∆/ for short junctions, while it is much smaller for long junctions 39 . For our junction, we obtain c n~1 7 μV at the lowest temperature, which is about 9 times smaller than ∆/~150 μV. This indicates that the junction is in the long junction limit, where the superconducting coherence length N is smaller than width between two Nb electrodes. The N can be evaluated by N = ℏ F Δ in the system.
The Fermi velocity of thin Cd3As2 is 2.0 − 2.8 × 10 5 m/s from Supplementary Table  1. Then, we get N~0 .3 − 0.4 μm which is smaller than the effective width eff~1 .4 μm estimated from the Fraunhofer pattern in SQI. In the diffusive junctions, the c n product yields c n ∝ 1/ at low temperature which can explain the low c n product observed.

Supplementary Note 13. Analysis of current density profile in Cd3As2 Josephson junctions
In a Josephson junction immersed in a perpendicular magnetic field B, the magnitude of the maximum critical current c max ( ) depends strongly on the supercurrent density between the leads. Here we convert our measured interference patterns to their originating supercurrent density profiles. Our method follows the approach developed by Dynes and Fulton. 40 At a fixed magnetic field, the total critical current through the Josephson junction is a phase-sensitive summation of supercurrent over the width of the junction. Suppose the supercurrent density profile S ( ), its complex Fourier transform yields a complex critical current function C ( ) where the normalized magnetic field unit = 2 eff Φ 0 , and the magnetic flux quantum Φ 0 = ℎ/2 . The experimentally observed c max ( ) is the magnitude of this 28 / 34 summation: C max ( ) = | C ( )| . We use the even ( E ( ) ) and odd part ( O ( ) ) extracted from the C max ( ). Then the C ( ) can be expressed as The observed critical current C max ( ) = √ E 2 ( ) + O 2 ( ) is therefore dominated by E ( ) except at its minima points. Approximately, E ( ) is obtained by multiplying C max ( ) with a flipping function that switches sign between adjacent lobes of the envelope function ( Supplementary Fig. 22a-b). When E ( ) is minimal, the odd part O ( ) dominates the critical current. O ( ) can then be approximated by interpolating between the minima of C max ( ), and flipping sign between lobes ( Supplementary Fig.   22c). A Fourier transform of the resulting complex C ( ), over the sampling range of , yields the current density profile (Supplementary Fig. 22d): In Supplementary Fig. 23, we show the magnetic field dependence of the critical current. For the out-of-plane ( y ) direction, the critical current modulation resembles the Fraunhofer pattern, in which the critical current amplitude oscillates as a sine function with B. For a magnetic field z , the modulation is qualitatively different.

Supplementary Note 14. Control experiments for thick Cd3As2 Josephson junctions
Despite the successful demonstration of thin Cd3As2 Josephson junction dominated by surface as shown in the main text of device #06, we purposely use a thick Cd3As2 Josephson junction as control experiments. As shown in Supplementary Fig.  24a, the junction has 12 μm width and 400 nm length channel which is similar to the device #06. We deposited 200 nm Nb on top. Such a thick Cd3As2 (clear boundary at the sample region) is dominated by bulk states as demonstrated before. The temperature-dependent resistances of 12 and 23 are displayed in Supplementary  Fig. 24b-c, respectively. Neither zero resistance nor Josephson effect is realized. Supplementary Fig. 25b shows a plot of dI/dV as a function of the V measured by a four-probe method. A family of peak features is observed in both 12 and 23 channels, symmetrically around = 0. The conductivity at = 0 is finite which is different from the Josephson effect in device #06. The peaks correspond to the subharmonic energy-gap structure caused by multiple Andreev reflections (MARs) 41 , with peak positions given by n = 2Δ/ ( = 1, 2, 3, ⋯ ). MARs allow for Andreev channels to open up in the S-N-S junction at bias voltages below the superconducting energy gap 2Δ. These Andreev channels arise from a progressive increase of the incident carrier energy as the carrier reflects between the two interfaces. From a fit of the MARs peak positions in Supplementary Fig. 25c, we can determine that 2Δ~2.2 meV corresponds to the Nb superconducting gap and the observed peaks correspond to = 3, 4, 5, 6, 8, 10, 11, 13, 19, 23, ⋯ and = 1, 2, 3, 4, 6, 9, 13, 19, ⋯ for 12 and 23 , respectively. we note that the position of the = 1 of 23 at high bias does not agree with the energy-gap value, likely due to the heating of the junction at high bias voltages 42 that would reduce Δ. These observed sub-harmonic peaks are