Abstract
Cd_{3}As_{2} is a threedimensional 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 proximityinduced surface superconductivity in Nb/Cd_{3}As_{2} hybrid structures. Our fourterminal transport measurement identifies a pronounced proximityinduced 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/Cd_{3}As_{2}/Nb junctions is also achieved with a Fraunhofer/SQUIDlike pattern under outofplane/inplane 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 twodimensional quantum spin Hall insulators. Our study provides the evidence of surface superconductivity in Dirac semimetals.
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Introduction
The study of topological phases has been one of the central topics in condensed matter physics in the past decade^{1,2}. Among the rich class of topological phases, topological Weyl and Dirac semimetals, which are characterized by discrete gapless nodes in the bulk spectra, have attracted wide attention^{3,4,5}. Referred to as Weyl or Dirac points, these gapless nodes in the bulk can be connected by open strings formed by topologically protected surface states on the boundaries, called Fermiarc states^{6}. Due to their anomalous electromagnetic responses^{7,8,9,10,11,12,13,14}, as well as the interesting interplay between bulk and surface Fermi arcs^{15,16}, topological semimetals have been studied intensively in recent years^{5}. In parallel to the rapid exploration of Dirac properties in topological semimetals, the study of their superconducting states has become an important topic aiming for possible unconventional superconductivity^{17,18,19,20}. It was predicted that Fermiarc states in a Dirac semimetal such as Cd_{3}As_{2} could be used to realize Majorana fermions^{20,21}. This is due to the fact that the Fermiarc states in Cd_{3}As_{2} stem from helical edge states of quantum spin Hall insulators embedded in the timereversal invariant semimetal^{22}. By proximitizing the Fermiarc states with swave superconductors, Majorana flat band can be possibly created at the interface of a πJosephson junction^{20,23}. While the signature of superconductivity has been found in Cd_{3}As_{2}^{24,25,26}, the role of Fermiarc states in these pairing phases remains unclear.
Despite the appealing proposals for creating Majorana fermions, the proposed scheme relies on the assumption^{20,21} that the Fermi arcs can acquire hard superconducting gaps from the parent superconductor. In topological insulator/superconductor (TI/SC) junctions, it is generally believed that the induced superconducting gap in the surface states is small due to high interfacial barriers and Fermi surface mismatch^{27,28,29}. In a Dirac semimetal Cd_{3}As_{2}, the unavoidable coexistence of lowenergy bulk excitations and Fermiarc states^{5} causes further complications to distinguish the roles of bulk states and surface states in the superconducting state. Therefore, a prerequisite experimental evidence of strong superconducting pairing on the surface is clearly desirable. In particular, due to the highly nonuniform spatial distribution of surface states in Cd_{3}As_{2}, the superconducting proximity on the surface and bulk may exhibit distinctive anisotropic signatures.
Here, we report the observation of strong proximityinduced superconductivity on the surfaces of Cd_{3}As_{2} based on two superconducting characteristics: the proximityinduced superconducting gap and the spatial distribution of supercurrents on surfaces. Using fourterminal transport measurements on Nb/Cd_{3}As_{2} hybrid structures, we observe a pronounced proximityinduced gap (Δ_{s}) on the surfaces, with its size comparable to the parent superconducting gap (Δ_{Nb}) of Nb. The sizable surface proximity gap manifests itself as a flat conductance plateau in differential conductance (dI/dV) spectra. In contrast, the proximity gap in bulk states is a few times smaller (Δ_{b} ~ 0.14Δ_{s}) and featured by a zerobias broad peak (ZBBP) in the dI/dV spectra. The relation between the observed conductance plateau/ZBBP and proximityinduced gaps in surface/bulk states are further confirmed by our theoretical calculations with a qualitatively good agreement. Upon changing surfacebulk contributions in Cd_{3}As_{2} samples with different thicknesses, the conductance plateau/ZBBP exhibit evolutionary behaviors consistent with their surface/bulk origins. Furthermore, using superconducting quantum interference (SQI) measurements on Nb/Cd_{3}As_{2}/Nb Josephson junctions (JJs), we observe a SQUID pattern in the surfacestatedominated Cd_{3}As_{2} JJs, which indicates that the supercurrent density is predominantly on top and bottom surfaces.
Results
Proximityinduced superconductivity in Nb/Cd_{3}As_{2}
The proximityinduced superconductivity in Nb/Cd_{3}As_{2} hybrid structure is displayed in Fig. 1a (device #01). The 250 nmthick Nb film, colored in green in the falsecolor scanning electron microscopy (SEM) image, is deposited on top of a ~200 nmthick Cd_{3}As_{2} nanoplate (blue color). A fourterminal measurement across the interface was performed to detect the interface resistance, denoted as R_{Interface} (see Supplementary Note 1, Supplementary Fig. 1 for sample characterizations and Supplementary Note 2, Supplementary Figs. 2 and 3 for measurement details).
Temperaturedependent resistance (R–T) curves at T ≥ 2.0 K across the junction are shown in Fig. 1b (black curve) with three drops at T_{c1} = 8.4 K, T_{c2} ~ 8.2 K, and T_{c3} ~ 4.0 K corresponding to Nb superconducting for T_{c1} and proximityinduced superconductivity for T_{c2} and T_{c3}, respectively, since Nb has been in zeroresistance state at 8.3 K (Fig. 1b inset, also see details in Supplementary Note 3 and Supplementary Fig. 4). Then, we carry out the fourterminal dI/dV measurements to show the induced superconductivity in Cd_{3}As_{2} in Fig. 1b–d. A biasindependent conductance plateau (BICP) appears in the vicinity of ±1.20 mV (Δ_{s}) with a ZBBP around ±0.19 mV (Δ_{b}) and an abovegap dip at high bias (Δ_{d}). For temperatures lower than 4 K, a very sharp zerobias conductance peak (ZBCP) emerges which is superimposed on the broad peak (Fig. 1c). Control experiments exclude the effect from Nb or Cd_{3}As_{2} (Supplementary Note 3 and Supplementary Figs. 5 and 6). Evidently, the ZBBP and ZBCP correspond to the second transition (T_{c2}) and third transition (T_{c3}) in the R–T curve while the first transition (T_{c1}) mostly corresponds to the conductance plateau with a small resistance drop by the abovegap dip in Fig. 1b. The observation of the superconducting proximity effect and conductance plateau suggests that the interface is of high transparency. Fits to the broad peaks in black lines by standard Blonder–Tinkham–Klapwijk (BTK) theory^{30} in Fig. 1c, d confirm the proximityinduced superconductivity in Cd_{3}As_{2}. The BTK fittings yield important parameters of Δ_{b} = 0.19 meV and Z = 0 (Supplementary Note 4 and Supplementary Figs. 7 and 8). Obviously, the amplitudes of BICP, ZBBP and ZBCP decrease with increasing temperature and perpendicular magnetic field. Figure 1e, f, respectively, show temperature and magnetic fielddependent gaps which are fitted to the Bardeen–Cooper–Schrieffer (BCS) theory (Supplementary Note 5). Δ_{s} is extracted to be 1.34 meV at 0 T, comparable to the superconducting gap of Nb estimated from the BCS theory^{31} Δ = 1.76k_{B}T_{c} ~ 1.30 meV, which further verifies the proximity effect in the Nb/Cd_{3}As_{2} system and swave pairing in Cd_{3}As_{2}.
The abovegap dip Δ_{d} and ZBCP are, respectively, attributed to depairing effect near the interface and the proximityinduced superconductivity in another band of Cd_{3}As_{2} such as the band projected onto [112] direction (Supplementary Note 6 and 7 and Supplementary Figs. 9–11).
Next, we probe the relation between BICP, ZBBP and proximityinduced superconductivity. For voltage bias V lower than the Nb superconducting gap, the transport process is dominated by Andreev reflection (AR), in which an incoming electron from the Dirac semimetal is converted to a reflected hole at the Nb/Cd_{3}As_{2} interface. Thus, the presence of both ZBBP and BICP for subgap voltage bias suggests two types of Andreev reflection channels in Cd_{3}As_{2}: one exists only within a narrow energy window given by the width of the broad peak, while the other has a much wider energy range measured by the width of the conductance plateau. Since the energy window for Andreev reflections is directly related to the interfacial coupling between the channels of Cd_{3}As_{2} and Nb, the contrasting features of the broad peak and plateau also suggest very different proximity effects in two types of channels. By placing Cd_{3}As_{2} next to the superconducting Nb, the superconducting proximity effect is expected to occur in both the bulk and the surface region, as shown in Fig. 2a. Importantly, since the wave functions of the surface states are predominantly localized near the interface, the surface states can strongly couple to Nb than the bulk. As a result, it is reasonable to expect that the proximity effect in the surface states can be much stronger than that in the bulk states. Besides, when the Fermi level is close to the Dirac points, AR amplitude for bulkstate channels is expected to be small and the Cooper pair wave function would decay rapidly in the bulk region as shown in Fig. 2b.
To further distinguish the proximity effect on surface and bulk channels, we acquired the continually evoluting features of BICP and ZBBP by tuning the thickness to change the specific surface area with bulk/surface states proportion in Cd_{3}As_{2} (Fig. 2c–f and Supplementary Note 8, Supplementary Figs. 12–15). We measured dI/dV spectra of Cd_{3}As_{2} hybrid structure with different thickness at 2 K as shown in Fig. 2c–f. Devices # 01 and 02 with a thickness of ~200 nm both show BICP and ZBBP while the plateau remains pronounced while the ZBCP becomes strongly suppressed and hardly observable in thinner samples #03 and #04 (<150 nm), as shown in Fig. 2e, f, indicating that the surface channels dominate the proximity effect in thin samples. Thus, the channels responsible for the plateau are indeed from Cd_{3}As_{2} surface states while the broad peak results from lowlying bulk channels in Cd_{3}As_{2}.
Next, we discuss the possible origin of surface superconducting channels. In device #01, the bulk states and Fermiarc surface states can coexist (Table 1). As reported previously^{16,32}, the Fermi surface property in Cd_{3}As_{2} is thicknessdependent. In particular, we note that in samples with thickness >150 nm, the Fermi level is generically well above the Dirac points. Thus, both bulk states and Fermiarc states contribute to electronic transport, while in samples thinner than 150 nm, the Fermi level is closer to the Dirac cone and Fermiarc states dominate. Since both the electronic transport and proximity effects are closely related to the Fermi surface property, we anticipate that the surface channels possibly correspond to Fermiarc surface states.
Both devices # 01 and 02 have shown the bulk states and Fermi arcs in transport from SdH oscillation measurements. Here, we use the relative amplitude of the surface oscillations compared with the bulk oscillation A_{S}/A_{B} to estimate the surface/bulk channels (S/B) domination. In contrast, thinner samples (# 03 and 04) exhibit low bulk domination. Based on the analysis of the SdH oscillations (Supplementary Note 9, Supplementary Fig. 18 and Supplementary Table 1), the Fermi surface property is confirmed to be dominated by the Fermiarc states. Therefore, through the different weight of surfacebulk conduction, we experimentally attribute the flat conductance plateau to the superconducting proximity effect possibly in the Fermiarc states and the broad peak to the relatively weak proximity effect in the bulk states. Besides, the surface and bulk contribution to Andreev reflection is also consistent with the conductance enhancement (Supplementary Note 10 for details).
In the thick limit of Cd_{3}As_{2} where bulk states dominate, weak Andreev reflection occurs with strong tunneling behavior as a result of insufficient electrons from the bulk states to participate in AR^{30} (Supplementary Note 8, Supplementary Figs. 12–14, device #05, >300 nm thick). At the interface, the AR electron density is proportional to \(\frac{{{\mathrm{d}}I}}{{{\mathrm{d}}V}} \propto G\left( {E_{\mathrm{F}}} \right) \propto \frac{{4e^2}}{h}N\left( {E_{\mathrm{F}}} \right) \propto k_{\mathrm{F}}^2 + k_0^2\), where k_{0} is Fermiarc length and can be estimated by^{15}
where F_{s} is surface frequency. The Fermi velocity v_{F} = ℏk_{F}/m* can be extracted from the bulk SdH oscillations. In device #01, k_{F} ~ 0.0249 Å^{−1} and k_{0} ~ 0.8 nm^{−1} (ref. ^{32}), and k_{F} = 0.032 Å^{−1} in device #05 with negligible k_{0}, thus the AR electron density in device #01 is much higher than that in device #05. Furthermore, scattering effects between the bulk and surface states can be significant due to disorders, and the surface states are no longer welllocalized on the surfaces^{33} which reduce the effective coupling between the surface and Nb. Besides, for high Fermi levels, there could possibly exist a significant Fermi surface mismatch between the superconductor and the Cd_{3}As_{2}, which could further suppress the interfacial coupling.
We now analyze the proximityinduced superconducting gaps in Cd_{3}As_{2}. Two superconducting gaps against the relative amplitude of A_{S}/A_{B} are presented in Fig. 2g and Table 1. The wave functions of the surface states are localized at the surface, while these of the bulk states are predominantly in the bulk. Effectively, the surface states couple much more strongly to the superconductor than the bulk states. In thin Cd_{3}As_{2}, the transparent interface with enough electron states from surface channels ensures that the proximity effect arises. Hence, the bulkstate superconducting gap Δ_{b} drops fast while the surface superconducting gap Δ_{s} increases slightly with a larger surface domination. The penetration depth for each surface can be estimated with an assumption that the superconducting gap decreases following^{34}
where ξ_{N} is the superconducting coherence length in Cd_{3}As_{2} (Table 1) and z is the distance in Cd_{3}As_{2} from the interface. At the surface regime, \({\mathrm{\Delta }}_{{\mathrm{Nb}}} = {\mathrm{\Delta }}_{\mathrm{s}}{\mathrm{exp}}\left( {  \frac{{z_{\mathrm{s}}}}{{\xi _{\mathrm{N}}}}} \right)\), we can evaluate the surface penetration depth z_{s} in Cd_{3}As_{2} of 10–50 nm, which is similar to the previous report for Fermi arcs in Cd_{3}As_{2}^{16} and agrees with other numerical simulations based on the lowenergy model of Dirac semimetals^{35} as an evidence of possible Fermiarc superconductivity.
Theoretical calculations on Andreev reflection
Our experimental observations above are further supported by theoretical calculations of the dI/dV spectra for our Nb/Cd_{3}As_{2} hybrid structure. Using numerical Green’s function method based on a fourband tightbinding model of Cd_{3}As_{2}, we calculate the Andreev reflection amplitude of the Nb/Cd_{3}As_{2} junction with the parent superconductor Nb modeled by a usual swave superconductor with the schematic setup in Fig. 3a. Details of the Hamiltonian and the Green’s function are presented in the Method section, Supplementary Note 11, Supplementary Figs. 19 and 20 and Supplementary Table 2.
It is worth noting that due to the lowenergy Dirac spectrum of Cd_{3}As_{2}, the bulk density of states is expected to increase monotonically as a function of E_{F} measured from the Dirac points located at (0, 0, ±k_{0}). In contrast, Fermiarc states with different energies are connected on the surface, thus the number of Fermiarc channels on the junction interface is expected to depend weakly on the Fermi level. Therefore, tuning the chemical potential E_{F} of Cd_{3}As_{2} in our model allows us to theoretically investigate the roles of surface states and bulk states in the Andreev reflection processes.
First, we consider the case that the location of chemical potential is at the Dirac points (E_{F} = 0 meV), where the bulk density of states is very low and the Andreev reflections driven by bulk channels are negligible. This simulates the scenario in thin samples (devices #03 and #04) where the Fermi surface property is dominated by Fermiarc states. As shown in Fig. 3b, in this case only a flat conductance plateau is found in our simulations. Being consistent with our experimental observations, the signatures of ZBBP are hardly observable. Then, by tuning the Fermi level to E_{F} = 70 meV from the Dirac points, the bulk density of states gets enhanced and its contribution to Andreev reflections cannot be ignored. This simulates the case of relative thick samples (devices #01 and #02) where the bulk states and surface Fermi arcs have comparable weights in transport. In this scenario, a ZBBP emerges on top of the plateau in the differential conductance spectrum (Fig. 3c). Therefore, our theoretical calculations of the dI/dV spectrum suggest that the Fermiarc channel can result in the plateau, while the ZBBP arises from Andreev reflections in the bulkstate channel. Here, we briefly note that without Fermiarc states, a welldefined flat conductance plateau cannot be found in the dI/dV spectrum. This further indicates that the plateau most likely originates from Fermiarc states. The detailed results are presented in Supplementary Note 11, Supplementary Fig. 21.
To understand the physical mechanisms for the formation of zerobias peak and flat plateau in the dI/dV curve, it is worth noting that the energy windows for Andreev reflections driven by surface/bulkstate channels are measured by the widths of the plateau/zerobias peak, respectively. This indicates that the superconducting proximity effect in the Fermiarc states is much stronger than that in the bulk states. To demonstrate the proximity effects in Fermiarc and bulkstate channels, we integrate out the superconducting Nb and include its contribution as a selfenergy term on the topmost layer of Cd_{3}As_{2} which interfaces with superconducting Nb. Considering the scenario where the bulk states have a nonnegligible density of states, we use the same parameters in obtaining Fig. 3c to calculate the local spectral density on the topmost layer, as shown in Fig. 3d. The color bar indicates the local density of states on a logarithmic scale, with red (blue) colors indicating high (low) density. Clearly, due to the proximitycoupled swave superconductor, the topmost layer of Cd_{3}As_{2} acquires two different spectral gaps, as indicated by white and yellow double arrows. Notably, the larger pairing gap is induced in states predominantly on the surface (higher local density of states on the surface), with twice of the gap size 2Δ_{s} corresponding to the width of the conductance plateau due to Fermi arcs, which is comparable to twice value of the parent superconducting gap. In contrast, the smaller induced gap is formed in states with lower density on the surface, and its size 2Δ_{b} matches the width of the zerobias peak due to bulk states. This further confirms that the Fermiarc states couple strongly to the superconductor, and thus acquire a sizable superconducting gap (~70% of parent gap Δ). In contrast, the states living in the bulk couple relatively weakly to the superconductor, which results in a smaller proximity gap. Therefore, the flat conductance plateau in the dI/dV curve indicates Fermiarc superconductivity as a possible interpretation, while the zerobias peak arises from relatively small proximity gap in the bulk states, which opens a narrower window near zero bias for Andreev reflections driven by bulk channels.
Surface supercurrent in Nb/Cd_{3}As_{2}/Nb Josephson junctions
Having established the proximityinduced surface superconductivity in Nb/Cd_{3}As_{2}, we next build Nb/Cd_{3}As_{2}/Nb Josephson junctions to directly analyze the spatial distribution of the supercurrent. This is enabled by SQI measurements in different directions.
Figure 4a schematically shows the lateral Cd_{3}As_{2} Josephson junctions with closely spaced superconducting Nb electrodes on the top surface. We choose 120nmthick Cd_{3}As_{2} with 140nmthick Nb electrodes to study the surface dominated supercurrent. The inset of Fig. 4b shows an SEM image of device #06. The length and width of the superconducting channel are L = 500 nm and W = 7 μm, respectively. Figure 4b shows the R–T curve of the junction with two transitions T_{c1} and T_{c2} at zero magnetic field. T_{c1} ~ 7 K originates from the Nb superconducting transition while T_{c2(on)} ~ 3 K comes from the superconducting proximity effect. The resistance continues to decrease as the junction cools down and reaches the zeroresistance state below T_{c2(off)} ~ 1 K. The tail of the resistance drop can be explained in terms of the BKT transition^{36} as shown in green line by Halperin–Nelson equation^{37}
where R_{0} and b are material parameters, which realizes a zeroohmicresistance state driven by the binding of vortexantivortex pairs at the BKT transition temperature T_{BKT} = 1.0 K. On the other hand, junction resistance decreases at the temperature above T_{c2}, leading to a broadened SC onset, which can be well reproduced by the Aslamazov–Larkin^{37} fit in red dashed line for the twodimensional (2D) fluctuation conductivity. These behaviors are consistent with expectations for 2D superconductivity as an evidence for the 2D surface superconductivity in Cd_{3}As_{2} JJ since the thickness of Cd_{3}As_{2} is far away from the 2D condition. Figure 4c and inset display the currentvoltage (I−V) characteristics and differential resistance (dV/dI) of the junction measured at 50 mK, respectively. From the slope of the I−V curve, the normalstate resistance R_{n} ~ 17 Ω is extracted. In the regime I < 1 μA, the voltage across the junction and the dV/dI are zero, indicating a robust Josephson effect. The I_{c}R_{n} product gives a characteristic voltage of ~17 μV which is lower than the transition temperature of 1 K (with superconducting gap Δ = 1.76k_{B}T_{c} ~ 150 μV). It indicates that the junction is in the long junction limit^{38}, where the superconducting coherence length ξ_{N} is smaller than the effective width between two Nb electrodes, i.e., ξ_{N} < L_{eff} (Supplementary Note 12).
The spatial distribution of supercurrent in a Josephson junction can be extracted by SQI measurements, where a magnetic field B perpendicular to the junction induces oscillations in the amplitude of the superconducting current. Measuring the dependence of the critical current \(I_{\mathrm{c}}^{{\mathrm{max}}}\) on B provides a convenient way to extract the distribution of supercurrent which is widely used to probe edgemode superconductivity in quantum spin Hall insulator and quantum Hall systems^{39,40,41}. The particular shape of the critical current interference pattern depends on the phasesensitive summation of the supercurrents traversing the junction. In the case of a symmetric supercurrent distribution, this integral takes the simple form:
where L_{eff} is the effective length of the junction along the direction of the current, accounting for the magnetic flux threading through parts of the superconducting contacts over the London penetration depths. As shown schematically in Fig. 5a, the supercurrent density has an approximately uniform distribution along the zaxis in Cd_{3}As_{2}. Thus, for a magnetic field B_{y} applied along y direction, the uniform current density results in a singleslit pattern yields the singleslit Fraunhofer pattern \(\left {{\mathrm{sin}}\left( {\frac{{\pi L_{{\mathrm{eff}}}BW}}{{{\mathrm{\Phi }}_0}}} \right)/\left( {\frac{{\pi L_{{\mathrm{eff}}}BW}}{{{\mathrm{\Phi }}_0}}} \right)} \right\) shown in the right panel. The slight asymmetry in the Fraunhofer pattern can be due to inversion symmetry breaking in the junction which is predicted in topological semimetals^{42}. In contrast, due to the surfacedominated superconductivity, the supercurrent density along the y direction is predominantly localized on the edges as shown in Fig. 5b. When the magnetic field is applied along z direction, the Fraunhofer pattern has a more sinusoidal oscillation characteristic of a SQUID pattern. Notably, the central lobe width in this case shrinks to Φ_{0} when only the top and bottom surface states dominate. This is due to the destructive interference at half flux quantum in a SQUIDlike geometry.
The corresponding supercurrent distributions along the z/y directions (Fig. 5c, e) are obtained by transforming the singleslit pattern to the realspace current density J_{c}(z) as shown, respectively, in Fig. 5d, f. The full details of the extraction procedure can be found in the Supplementary Note 13 and Supplementary Fig. 22. Clearly, for B applied along the y direction, the supercurrent spreads across the junction (Fig. 5d). On the contrary, when B is applied along z direction, the critical current envelope becomes similar to a sinusoidal oscillation in Fig. 5e. The shift towards a SQUID interference pattern corresponds to the development of sharp peaks in supercurrent density at the mesa edges. The periodicity of the critical current oscillations is contrasted to the observed Fraunhofer pattern in Supplementary Fig. 23 to prove the single period of Φ_{0} at low magnetic field. The deviation from the single slit in Fig. 5c, d may result from the fluctuations in junction length, positiondependent transparency of interface by a granularity of the Nb film and the variation of the surface carrier density. Despite the deviation, the Fraunhofer pattern is obviously different from the SQUID pattern. The first minimum of I_{c} at ~0.2 mT, extracted from the Fraunhofer pattern in Fig. 5c, gives the effective length of L_{eff} ~ 1.4 μm, which is slightly larger than the distance of two Nb electrodes. This can be explained by the triangular shape of Nb along the outofplane direction in the crosssection Transmission Electron Microscopy image, as shown in Supplementary Fig. 3a where the thinner Nb, close to the edge of Nb electrodes, may become nonsuperconducting. Besides, the effective length L_{eff} should also include the London penetration depth λ_{L} ~ 100 nm for Nb^{43} such that L_{eff} = L + 2λ_{L}. These two reasons may contribute to the larger L_{eff}. When the magnetic field is applied along zaxis, the periodicity of oscillations reaches \(\frac{{{\mathrm{\Phi }}_0}}{{L_{{\mathrm{eff}}}t}}\sim 13\,{\mathrm{mT}}\) (t is the thickness of Cd_{3}As_{2}), which accords well with the SQUID pattern in Fig. 5e. We can then estimate the width of the supercurrentcarrying surface channel using a Gaussian line shape. The two surfacefull regime depths z_{s} are extracted to be 13 and 14 nm, respectively, which is consistent with our experimental results on Andreev reflections across the Nb/Cd_{3}As_{2} junctions. As control experiments, thick Cd_{3}As_{2} is found to be hard to achieve Josephson effect while it performs multiple Andreev reflections (MARs) (see Supplementary Note 14 and Supplementary Figs. 24–27 for details).
Discussion
Our experiments on the Nb/Cd_{3}As_{2} and Nb/Cd_{3}As_{2}/Nb hybrid structures demonstrate the surface superconductivity with a large proximity gap from the parent superconductor, and a detailed supercurrent distribution is extracted from the SQI measurements. Here, we would like to discuss the physical origin of the observed surface superconductivity and its potential implications.
First, we discuss alternative explanations of the surface superconductivity, which do not require the existence of surface states, in our Cd_{3}As_{2} samples. These possibilities include surface doping by charge impurities and surface bandbending effects^{44,45}. While our transport and SQI measurements alone cannot directly rule out these possibilities, a rather unusual anisotropy of these effects is generally required to be compatible with the surface supercurrent distribution found in our SQI experiments. Particularly, the charge impurities or surface bandbending should occur primarily on the top and bottom surfaces, which can hardly be met by the realistic conditions of our experimental setup. In contrast, the intrinsic physical properties of surface states (i.e. Fermi arcs), for example, their anisotropic surface distribution and thicknessindependent properties, show naturally consistent characteristic with the signatures of the observed surface superconductivity (see Supplementary Note 15 and Supplementary Fig. 28 for details).
We further contrast our result with previous reports on superconductivity in surface states of a usual 3D topological insulator (TI). Topologically nontrivial surface states also exist in TI and similar BICP due to surface states has been reported in Bi_{2}Se_{3}/NbSe_{2} hybrid structure^{46}. The contribution of surface states and bulk states in TI systems is usually separated by tuning the Fermi level within the bulk gap using electric gating or chemical doping. Generally, sample resistance is a criterion to estimate the bulk states proportion^{47,48}. In Dirac semimetal Cd_{3}As_{2}, an effective tuning of chemical potential^{32} becomes accessible to change the transport property from bulkdominated to surfacedominated, which is also justified by twofrequency oscillations (see Supplementary Note 9 and Supplementary Figs. 1618 for details). Moreover, even eliminating bulk states in TI by chemical doping such as Bi_{1.5}Sb_{0.5}Te_{1.7}Se_{1.3} system, the SQI result is different in Cd_{3}As_{2} and TI because surface states cover all the surfaces of TI. Specifically, in TI, due to the uniform surface states in all surfaces, the supercurrent density is edgedominated along all three directions which results in a mixed Fraunhofer and SQUIDlike pattern rather than a pure SQUID pattern^{49}. Nevertheless, the surface states are generally only on two sidesurfaces along the yaxis of Cd_{3}As_{2}. This enables the observation of SQUID pattern upon magnetic fields applied along the principal zaxis, which is spatially resolved as edge supercurrent along the ydirection as shown in Fig. 5.
As we introduced earlier, the Fermiarc states in Cd_{3}As_{2} originate from edge states of 2D TI embedded in the Dirac semimetal. In particular, Fermi arcs labeled by momentum k_{z} ∈ (−k_{0}, k_{0}) stem from the slice of 2D TI indexed by k_{z}. Thus, proximitizing the surface of Cd_{3}As_{2} is equivalent to inducing pairing in different slices of 2D TI edges as displayed in Fig. 6. Therefore, the Cooper pairs are formed by opposite spin electrons from slices of 2D TIs with opposite k_{z}. In other words, the surface supercurrent observed in our work can be regarded as a higher dimensional analog of edge supercurrent in 2D quantum spin Hall insulators (QSHIs). We note that with certain perturbations that break the C4cymmetry of Cd_{3}As_{2}, Dirac points located along the principal axis can be gapped out. In this case, the system becomes a strong topological insulator^{50}; in particular, the k_{z} = 0 plane is still characterized by a nontrivial Z_{2} invariant of 2D QSHIs^{50}. Thus, this subtlety does not affect our conclusion that the observed surface superconductivity provides signatures of higher dimensional edge supercurrents in 2D QSHIs.
The supercurrent distribution in our surfacedominated provides the first direct evidence of surface superconductivity in topological semimetals. While the observations in the current work cannot rule out alternative physical origins other than Fermi arcs, our results call for future efforts to elucidate the possible Fermiarc origin of the surface superconductivity. Prospectively, when a short linear Josephson junction with π phase difference is formed on the surface of Cd_{3}As_{2}, one pair of Majorana bound states can be created at the junction interface of each slice of 2D TI^{23}. This can result in a large number of nondispersive Majorana fermions, called Majorana flat bands, that connects the separated Dirac points located at (0, 0, ±k_{0})^{20}. The Majorana flat bands created by superconducting Fermi arcs can be signified by a characteristic sudden jump at π phase in the Josephson current–phase relation. Similar to the single Majorana bound state at a πjunction interface on quantum spin Hall edges^{23}, the Majorana flat bands on the surface of Cd_{3}As_{2} are protected by timereversal symmetry and thus remain robust against nonmagnetic disorder. While evidence of single Majorana bound states and chiral Majorana edge states has been spotted in recent experiments^{51,52}, experimental realization of Majorana flat bands still remains unexplored. We believe that our experimental finding of strong proximity effect on surface may establish Cd_{3}As_{2} to be a new promising platform for realizing Majorana flat bands.
Methods
Cd_{3}As_{2} nanostructure growth
The Cd_{3}As_{2} nanoplates were grown using Cd_{3}As_{2} powders as the precursor in a horizontal tube furnace, in which argon was a carrier gas. Before the growth, the furnace was pumped and flushed with argon several times to remove water and oxygen. The temperature was ramped to the growth temperature within 15 min, held constantly for 20 min, and then was cooled down naturally over ~2 h in a constant flow of argon before the substrates were removed at room temperature. The precursor boat was placed in the hot center of the furnace (held at 500 °C), while the smooth quartz substrates were placed in the downstream within a very small temperature range from 200 to 350 °C. The argon flow rate is 50 SCCM (standard cubic centimeters per minute). The smooth quartz substrates then appeared shining to the naked eyes. The largest crystal plane of asgrown Cd_{3}As_{2} nanoplates is [112].
Device fabrication
The Nb/Cd_{3}As_{2} hybrid structures were fabricated by electron beam lithography (EBL) technique and wetetched by standard buffered HF solution for 5 s in the electrode regime. For Nb/Cd_{3}As_{2} device, we first fabricated Cr/Au (10/150 nm) bilayers electrodes on Cd_{3}As_{2} side using magneton sputtering. Then, we use standard EBL method to deposit the Nb layer. For Nb/Cd_{3}As_{2}/Nb Josephson junction, we use the EBL method and magnetic sputtering to deposit 140 nmthick Nb electrodes.
Transport measurements
Fourterminal temperaturedependent transport measurements were carried out in a Physical Property Measurement System (PPMS) system (Quantum Design) (1.9 K) and dilute refrigerator (down to 35 mK) using lockin amplifier (SR830) and Agilent 2912. The differential conductance (dI/dV) spectra were captured by acmodulation technique. For the PPMS measurements, lockin amplifier provides ac input in series with Agilent exporting dc voltage. Through a large resistor (0.05–1 MΩ) the input voltage converts to a constant current. Agilent 2912 and lockin amplifiers with a low frequency (<50 Hz) were used for dI/dV spectra measurements. The R–T curves were measured by dI/dV spectra while the Agilent dc voltage is set to zero.
Tightbinding model of Nb/Cd_{3}As_{2} transport
In the Bloch basis formed by {S_{1/2}, 1/2〉, P_{3/2}, 3/2〉, S_{1/2}, −1/2〉, P_{3/2}, −3/2〉}, the momentumspace tightbinding Hamiltonian of the Dirac semimetal Cd_{3}As_{2} used in the theoretical calculations of Andreev reflections is given by^{22}:
where the matrix elements are defined as:
Here, a = 3 Å, c = 5 Å refer to the lattice constants within the abplane and along the caxis, respectively. For simplicity, we neglect offblock diagonal parts which only account for higher order or bulk inversion asymmetric terms^{22}. The parameters C_{0}, C_{1}, C_{2}, M_{0}, M_{1}, M_{2}, and A_{0} are set in units of eV with their values given in Supplementary Table 2 of the Supplementary Information. E_{F} is the Fermi level measured from the Dirac points in the Cd_{3}As_{2}. More details of the numerical Green’s function method for transport calculations in Fig. 4 are presented in Supplementary Note 11.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported by the National Key Research and Development Program of China (2017YFA0303302 and 2018YFA0305601) and National Natural Science Foundation of China (61322407, 11474058, 61674040, and 11874116). K.T.L. and B.T.Z. acknowledge the support of Croucher Foundation, Dr. TaiChin Lo Foundation and HKRGC through 16309718, 16307117, 16324216, and C602616W. E.Z. acknowledges support from China Postdoctoral Innovative Talents Support Program. Part of the sample fabrication was performed at Fudan Nanofabrication Laboratory. We thank Jinhui Shen from Professor Xiaofeng Jin’s group for help in the Nb metal deposition. We thank Cheng Zhang and Jiwei Ling for helpful discussions. We thank Liyang Qiu from Saijun Wuʼs group for help in Matlab coding.
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F.X. conceived the ideas and supervised the overall research. H.Z., B.Y. and R.L. synthetized Cd_{3}As_{2} nanostructures. C.H., Yimin W., Yihua W. and E.Z. fabricated the nanodevices. C.H., K.H. and X.L. carried out the 1.9 KPPMS, 35 mKPPMS and diluted refrigerator measurement. C.H. and S.L. analyzed the Cd_{3}As_{2} transport data. C.H., H.W. and Z.H. analyzed the differential conductance spectra and used the BTK model to fit the data. B.T.Z. and K.T.L. performed the theoretical calculations and interpreted the results. Z.L., Q.D., Y.C., X.H. and J.Z. performed HRTEM experiments. C.H., B.T.Z. and F.X. wrote the paper with assistance from all other authors.
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Huang, C., Zhou, B.T., Zhang, H. et al. Proximityinduced surface superconductivity in Dirac semimetal Cd_{3}As_{2}. Nat Commun 10, 2217 (2019). https://doi.org/10.1038/s4146701910233w
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DOI: https://doi.org/10.1038/s4146701910233w
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