Two-dimensional superconductivity and magnetotransport from topological surface states in AuSn4 semimetal

Topological materials such as Dirac or Weyl semimetals are new states of matter characterized by symmetry-protected surface states responsible for exotic low-temperature magnetotransport properties. Here, transport measurements on AuSn4 single crystals, a topological nodal-line semimetal candidate, reveal the presence of two-dimensional superconductivity with a transition temperature Tc ~ 2.40 K. The two-dimensional nature of superconductivity is verified by a Berezinsky–Kosterlitz–Thouless transition, Bose-metal phase, and vortex dynamics interpreted in terms of thermally-assisted flux motion in two dimensions. The normal-state magnetoconductivity at low temperatures is found to be well described by the weak-antilocalization transport formula, which has been commonly observed in topological materials, strongly supporting the scenario that normal-state magnetotransport in AuSn4 is dominated by the surface electrons of topological Dirac-cone states. The entire results are summarized in a phase diagram in the temperature–magnetic field plane, which displays different regimes of transport. The combination of two-dimensional superconductivity and surface-driven magnetotransport suggests the topological nature of superconductivity in AuSn4. Surface states of topological semimetals may give rise to unusual transport properties and topological superconductivity. Here, the H-T phase diagram of AuSn4 is experimentally established, displaying 2D superconductivity, Bose metal behavior, and normal-state magnetotransport driven by surface states.

S uperconductivity in reduced dimensions has attracted great attention since it is not only of importance for exploring attractive quantum phenomena at a novel state but also for potential application for developing superconducting electronics 1,2 . Recently, several new two-dimensional (2D) ultrathin superconducting systems have been realized in different materials and hetero-interfaces [3][4][5][6][7][8][9] . Typical transition metal oxide interfaces and interface superconductivity between chalcogenides reported so far include LaAlO 3 /SrTiO 3 (ref. 3 ) and PbTe/PbSe families 4 , respectively. In addition, 2D superconductivity has been particularly observed in some micrometer-size flakes with a fewlayer thickness, such as exfoliated 1T-MoS 2 (ref. 5 ), NbSe 2 (ref. 6 ), and ZrNCl single crystals 7 , or chemical-vapor-depositionsynthesized Mo 2 C 8 and 1T d -MoTe 2 thin films 9 . Otherwise, another novel state of quantum matter, the topological material, has become an important topic in condensed matter physics 10 , in which the robust topological surface states are topologically protected against time reversal-invariant perturbations. In particular, the topological Dirac or Weyl semimetals are new states of matter that exhibit a linear band dispersion in the bulk along momentum directions, whose low-energy quasiparticles are the condensed-matter analogs of Dirac and Weyl fermions in relativistic high-energy physics, constituting one of the most popular topics in condensed matter physics 11 and extending towards a new kind of topological materials, namely, topological superconductors (TSs) 12 . In TSs, the opening of the superconducting gap is associated with the emergence of zero-energy excitations that are their own antiparticles [13][14][15] , where their zero-energy states are generally called Majorana-bound states. It has been pointed out that inducing s wave superconductivity on the surface state of a topological material results in a p wave superconductor that hosts Majorana fermions in its vortex cores 13,16 . Motivated by the prospect of creating Majorana fermions, which have potential applications in quantum computations 15 , several attempts have been made to induce superconductivity on the surfaces of topological materials using bulk superconductors 17,18 or interface superconductivity induced by the superconducting proximity effect 19 . Recently He et al. 20,21 have reported transport measurements on a Bi 2 Te 3 /FeTe heterostructure with both nonsuperconducting materials, which reveal superconductivity at the interface and show the two-dimensional nature of the observed superconductivity with the highest transition temperature around 12 K. More recently, Zhang et al. 22 have further demonstrated that without the fabrication of heterostructure for any proximity effect, the iron-based superconductor FeTe 0.55 Se 0.45 (superconducting transition temperature T c = 14.5 K) hosts Dirac-cone-type spin-helical surface states at the Fermi level; the surface states exhibit an s wave superconducting gap below T c , and reveal 2D topological superconductivity, providing a simple and possibly high-temperature platform for realizing Majorana-bound states.
Recently, a novel topological structure of Dirac node arcs and extremely large magnetoresistance of >10 4 % have been reported in PtSn 4 and PdSn 4 semimetals [23][24][25][26] ; these have been recognized as a new breed of topological materials, namely, topological nodal-line (or nodal loop) semimetals (NLSMs). Meanwhile, AuSn 4 is isostructural with the Dirac nodal arc semimetals PtSn 4 and PdSn 4 (refs. 27,28 ), revealing superconductivity with T c of 2.38 K, as reported in the 1960s 29,30 . Surprisingly few studies have so far been made regarding the superconductivity of AuSn 4 . Inspired by the obtained topological NLSMs PtSn 4 and PdSn 4 , natural superconductivity in AuSn 4 can be regarded as a candidate of topological superconductivity on its surface, as we have noted that the surface states of FeTe 0.55 Se 0.45 are 2D topologically superconducting. Therefore it will be intriguing to probe the dimensionality of superconductivity in AuSn 4 .
In this work, we show that the resulting superconductivity possesses the characteristics of AuSn 4 with a T c of~2.40 K. The superconductivity exhibits 2D nature by showing evidence of a Berezinsky-Kosterlitz-Thouless (BKT) transition, Bose-metal phase, and the vortex dynamics interpreted in terms of thermally assisted flux motion in two dimensions. Moreover, it is found that in the normal state the low-temperature magnetoconductivity of AuSn 4 can be well described by the weakantilocalization (WAL) transport formula. The results strongly support the concept that the surface electrons in Dirac-cone states dominate the normal-state magnetotransport in AuSn 4 single crystals, leading to the observed 2D superconductivity. In contrast to the recently discovered 2D superconductors, which were fabricated in the shape of micrometer-size flakes with a few-layer thickness, the millimeter-size bulk AuSn 4 single crystals peculiarly exhibit 2D superconductivity, and display large normalstate magnetoresistance accompanied by high carrier mobility, revealing the advantages of AuSn 4 as the key to understanding topological 2D superconductivity.

Results
Structure and bulk superconductivity in AuSn 4 single crystals. Figure 1a is a typical picture of the cleaved single crystals, showing millimeter-sized crystals with metallic luster. The crystal structure in real space is schematically shown in Fig. 1b crystallographic c-axis is anticipated to be perpendicular to the mirrored surface shown in Fig. 1a. Figure 1c shows the X-ray θ-2θ diffraction spectrum for a single crystalline specimen, in which only the (00n) (n = 2, 4, 6, and 8) diffraction peaks were observed, indicating that the [001] direction is perpendicular to the plane of the crystals. The lattice constant of the c-axis can be determined precisely to be 11.708 Å and is very close to that of 11.707 Å as reported 27 . The inset of Fig. 1c shows the Laue diffraction pattern of single crystalline AuSn 4 , indicating good crystallization of crystal as judged by the sharp spots in the Laue pattern. All peaks in the diffraction pattern can be well indexed with the orthorhombic Aea2 space group, with the structure similar to PtSn 4 . Figure 2a shows the zero-field-cooling (ZFC) and field-cooling (FC) magnetizations in H = 5 Oe parallel to the crystal c-axis for an AuSn 4 single crystal, demonstrating the observation of superconducting transition temperature T c ≈ 2.40 K, which is consistent with a previous report of 2.38 K 29 . The superconducting shielding fraction is estimated to be over 99%, based on the magnetization at 2 K. The smaller FC signal, compared with the ZFC signal, is caused by vortex pinning. Figure 2b shows the measured magnetization M(H) curves of AuSn 4 at various temperatures in magnetic fields parallel to the crystal c-axis (out-of-plane H). The lower critical field H c1 is determined by the ≤1% deviation of the M(H) curve from the linear fitting in the even lower-field regime. According to the traditional nodeless Bardeen-Cooper-Schrieffer (BCS) and the two-fluid model theories, the temperature dependence of H c1 is proportional to λ −2 and can be expressed by H c1 (T) = H c1 (0) [1 − (T/T c ) 4 ], where λ is the magnetic penetration depth. Figure 2c shows the obtained H c1 as a function of reduced temperature (T/T c ) 4 for AuSn 4 , and also shows a good approximation with adopted H c1 (0) = 39.4 Oe. The temperature dependence of H c1 indicates that the bulk superconductivity of AuSn 4 resembles that of the s wave BCS superconductor as observed for an NbN film 31 . Bulk superconductivity is also confirmed by a large anomaly at 2.418 K in heat capacity. Figure 2d illustrates the low-temperature heat capacity C p /T versus T 2 and the equal-area approximation for the T c determination. The experimental data can fit in the temperature range of 2.5-3.0 K by using the formula C p = γT + βT 3 , where the electronic specific-heat coefficient (Sommerfeld coefficient) γ = 6.73 mJ mol −1 K −2 and phonon specific-heat coefficient β = 2.55 mJ mol −1 K −4 were obtained. Figure 2e illustrates the excess specific heat ΔC p /T of superconductivity by subtracting a smooth normal-state background, estimated by fitting the specific heat through the data above the superconducting transition region. Using γ and the specific-heat-jump value ΔC p /T c at T c , ΔC p /γT c could be calculated and was found to be 1.26, which is slightly smaller than the BCS value of 1.43. Figure 2f shows the whole temperature range of resistivity. The normal-state resistivity for AuSn 4 reveals a metallic-like character (dρ/dT > 0), with the residual resistivity ratio ≈48. The inset of Fig. 2f shows the temperature dependence of resistivity ρ(T) in zero field at temperatures near T c . The estimated T c , as determined by an intersection of two linear fitting lines extrapolated from the normal-state resistivity and resistive transition, respectively, is 2.421 K, which agrees with magnetization and specific heat measurements. It is worth mentioning that samples from different batches of AuSn 4 crystals show almost the same superconductivity, as well as the normal-state transport properties (for details, see Supplementary Note 1). The bulk superconductivity of AuSn 4 , characterized by the magnetization and special heat measurements, reveals a conventional nodeless BCS superconductor. However, the superconductivity explored by Mixed-state magnetotransport properties and 2D superconductivity. Figure 3a, b shows the resistivity as a function of temperature in magnetic fields parallel to the crystal c-axis and ab plane, respectively. Figure 3c shows the upper critical fields H c2,c (T) and H c2,ab (T) for an AuSn 4 crystal in fields parallel to the crystal c-axis and ab plane, respectively, where the corresponding upper critical field H c2 values are derived from the resistive transition in the curves of Fig. 3a, b, like the T c determination in the inset of Fig. 2f. As seen, H c2,c (T) shows the expected linear T dependence, which follows the standard linearized Ginzburg-Landau (GL) theory, where Φ 0 is the flux quantum and ξ GL (0) is the GL coherence length at T = 0 K. However, it is noted that upper critical field for H c2,ab (T) follows the GL form of temperature-dependent behavior for 2D superconductors 32 : where d sc is a corresponding superconducting thickness; the resulting fitting curve is shown in Fig. 3c as well. Using the temperaturedependent H c2 relationships of Eqs. (1) and (2), the mean value of ξ GL (0) could be calculated as 75.58 nm, and the mean value of superconducting thickness d sc was then estimated to be 417.2 nm, which is greatly less than the sample thickness d of~100 μm, but larger than ξ GL (0). Regarding the d > ξ GL (0) case, as reported on the GaN/NbN epitaxial semiconductor/superconductor heterostructures 33 , it has been proposed that samples may behave like a thin film. In view of the 2D-H c2,ab (T) behavior, let us then consider that the 2D electrical transport may originate from the surface conducting channel within a thickness of d sc . When the temperature approaches the critical temperature, the coherence ξ GL would increase to be larger than d sc , thus, the sample would reach the 2D limit. Taking into account that ξ GL (T) = ξ GL (0)·(1 − T/T c ) −1/2 , we can estimate the crossover temperature as d sc = ξ GL and have the crossover temperature of T/T c = 0.967, i.e., at T = 2.32 K, which is in accordance with the apparent 2D behavior of H c2 (T) at T > 2.30 K, as shown in Fig. 3c. This result gives a good account of the surface conductivity that can be explored by electrical transport measurements. The derived H c2 (T) values were used to plot the H c2,ab /H c2,c ratio versus reduced temperature T/T c , as seen in Fig. 3d; it does show a diverging characteristic on approaching T c , characteristic for a 2D nature, as recently observed 5,8,20 . The 2D superconducting behavior is further confirmed by experiments with a tilted magnetic field. Figure 3e shows the magnetic field dependence of the resistivity under different θ at 2.1 K, where θ is the tilted angle between the normal sample plane and the direction of the applied magnetic field (see the inset of Fig. 3f). Clearly, the superconducting transition shifts to a higher field with the external magnetic field rotating from perpendicular θ = 0°(H // c) to parallel θ = 90°(H // ab). The upper critical field H c2 was extracted from Fig. 3e, and plotted in Fig. 3f, as a function of the tilted angle θ; as seen, a cusp-like peak is clearly observed at θ ≈ 90°. The angular dependence of H c2 (θ) was fitted using the 2D Tinkham formula 34  In Fig. 3a, b, the resistivity under magnetic field shows a broadening behavior due to thermally activated flux motion, which has been proposed by Anderson and Kim 36 , and can be described by: ρ(T, H) = ρ 0 exp(−U/k B T). Here U is the activation energy, which is normally both fieldand temperature-dependent. In the Anderson-Kim model, the U value is an indication of the magnitude of effective pinning energy. Figure 4a, b shows the Arrhenius plot of resistivity as a function of 1/T for AuSn 4 in magnetic fields of different magnitudes, parallel to the crystal caxis and ab plane, respectively. The resistive transition just below T c reveals linear behavior in the Arrhenius plot and follows the equation of thermally activated flux motion as highlighted by straight solid lines. In Fig. 4a for resistive transitions in fields parallel to the crystal c-axis, all fitting lines in different fields cross to one point, whose corresponding temperature T m of 2.401 K is close to the T c of AuSn 4 , being in agreement with that reported on the Bi 2 Te 3 /FeTe heterostructures 21 . One may notice the difference that the Arrhenius plots of resistive transitions for H // ab plane in Fig. 4b do not show this result, implying that the simple Arrhenius relation can only be satisfied in a narrower region. It is known that the effects of prefactor ρ 0 and nonlinear relation of U(T, H) may lead to ρ(T, H) deviating from Arrhenius relation for H // ab plane as seen in Fe(Te,S) single crystals 37 . Figure 4c shows the field-dependent U extracted from Anderson and Kim's theory, in which U(H) is plotted as a function of the magnetic field on a semi-logarithmic scale. As seen, U(H) decreases monotonically with increased magnetic field due to an increase of vortex density and interaction between the vortices. It is found that the field dependence of activation energy U can be written as U(H) = U 0 ln(H 0 /H) with U 0 = 30.54 (31.35) K and H 0 = 118.3 (186.9) Oe for H // c-axis (H // ab plane). It has been pointed out that in clean 2D superconductors, U is expected to follow a logarithmic dependence on the magnetic field, based on collective flux creep model 38 . As shown in Fig. 4c, a linear relationship of U(H) ∝ −lnH is observed, in accordance with the model of thermally assisted collective flux motion in two dimensions, as recently observed in other 2D crystalline superconductors [5][6][7][8] . Figure 4d shows a lnρ 0 -lnH plot extracted from the Arrhenius plot of resistivity with H // c-axis, and illustrates the obtained result that ρ 0 is proportional to H −p with p ≈ 12.7, consistent with the calculation of p = U 0 /k B T m derived from the Anderson-Kim resistivity theory (for details, see Supplementary Note 2). In addition, the obtained values of H c2 , U, and H 0 in the H // ab plane are 160-200% larger than those in H // c-axis, showing the anisotropic nature of flux pinning in AuSn 4 , which is commonly observed in layered-structure superconductors, and is much smaller than those of superconducting iron pnictides and cuprates 39,40 . Therefore, the different properties of AuSn 4 between out-of-plane and in-plane magnetic fields should originate from the anisotropic nature, and means that the intrinsic pinning between the Au-Sn layers is dominant for the H // ab plane, and is stronger than extrinsic pinning due to stacking faults or defects for the H // c-axis.
BKT transition. It is well known that the 2D nature of the observed superconductivity can be confirmed by testing whether the electrical transport properties possess a signature of a BKT transition that can be characterized by a BKT temperature T BKT , below which a phase transition leading to a 2D topological order emerges 3,41,42 . Figure 5a displays the I-V isotherms of a AuSn 4 single crystal on a log-log scale. The straight lines in this plot indicate the power-law behaviors, with the powers α equal to the slopes of the lines (i.e., V ∝ I α ). As seen, the I-V curves with a slope of 1 coincides with the high-temperature isotherms at temperatures above 2.40 K, which indicates their ohmic characteristics, while the I-V curve with a slope of 3 marks the initiation of a BKT transition, V ∝ I 3 ; the slopes of the lines then increase drastically at lower temperatures. Notice that, due to the studied bulk single-crystal samples, there is no obviously finitesize-induced linear tail in the I-V data, as is usually observed in thin-film samples, such as one-unit-cell FeSe films 43 or YBa 2 -Cu 3 O 7−δ films 44 . Thus, fitting is usually applied to the upper portion of the dataset, meaning just below the critical currents, as conducted for the Bi 2 Te 3 /FeTe heterostructure 20 and Mo 2 C crystals 8 . As is known, the standard BKT transition is carried out in zero magnetic field, characterized by the thermally driven vortex-anti-vortex pairs unbinding at temperatures above T BKT 41,42 , while the vortex-anti-vortex pairs will break down in the presence of magnetic fields. Figure 5b shows the corresponding I-V isotherms with an applied field of 5 Oe; one can see that the slopes of the lines increase gradually as temperature decreases. Figure 5c shows the temperature dependence of the power-law exponent α deduced from the power-law fits in Fig. 5a, b. In Fig. 5c, the zero-field value of α approaches 3 at the temperature of 2.40 K, which is thus close to T BKT ; it increases rapidly for temperatures lower than 2.40 K, while the 5-Oe value of α increases continuously from 1.0 to 4.6 as the temperature decreases from 2.45 to 2.33 K. These observations are regarded as the hallmark of a BKT transition, which is surprisingly observed on a "bulk" AuSn 4 single crystal. In the infinite-size homogeneous case, a jump is expected from α = 3 to α = 1, as shown in Fig. 5c, whereas for both the one-unit-cell FeSe films 43 and Bi 2 Te 3 /FeTe heterostructure sample 20 , the α value seems to show a smooth transition rather than a sudden sharp jump. Recalling the sharp resistive transition, as shown in Fig. 2f, it is expected that the BKT transition, which occurs at the temperature of 2.40 K, should be close to the resistive transition temperature of 2.42 K, as presented. Moreover, as described by the BKT theory for a 2D superconductor, the temperature-dependent resistivity at zero field is predicted to be in a form of the Halperin-Nelson (HN) relation 45 at temperatures just above T BKT : ρ(T) = ρ n exp(−bt −1/2 ), where ρ n and b are material-specific parameters, and t = T − T BKT is the temperature deviation. As shown in the inset of Fig. 5c, the ρ-T curve near the T BKT regime can be well fitted with HN relation and b = 0.048 K 1/2 , giving evidence of the 2D nature of superconductivity. Furthermore, a scaling ansatz has been proposed by Fisher et al. 46 for a general analysis of a superconducting phase transition in D dimensions using a dynamic scaling argument. This successful ansatz has been applied to a wide variety of systems and transitions, including 2D superconductors, Josephson-junction arrays, and superfluids 47 , using the experimentally determined quantities V and where ξ is the coherence length, F ± are scaling functions above and below the transition, and z = 2 in a dynamic scaling analysis of 2D system I-V data. At this point, all that is required to do dynamic scaling is the temperature dependence of ξ, which can be defined above the transition as the size of a fluctuation or as the average distance between two free vortices and has been given by ξ ∝ exp  present. In Fig. 5d we plot I T I V À Á 1=2 as a function of the scaling function variable Iξ T for data in Fig. 5a and vary the fitting parameter T BKT to achieve the best collapse onto a scaling curve. With T BKT = 2.401 K, which is a little lower than T c of 2.421 K determined by the resistive transition, the scaling data present an acceptable scaling collapse (do not collapse) above (below) the transition T BKT , agreeing well with the expected BKT behavior near the transition. One may notice that the scaling data at T = 2.40 K shows a nearly constant value, which can be understood by taking account of V ∝ I 3 at T = 2.40 K ≈ T BKT, leading to an invariant term of I T I V À Á 1=2 (∝I 1.5 /V 0.5 ) at a fixed temperature. These two resulting T BKT values respectively derived from the HN relation and universal scaling are highly consistent with the value extracted from the power analysis that α ≈ 3 at T = 2.40 K. Our analysis thus provides strong evidences for a 2D nature of the observed superconductivity in AuSn 4 .
Normal-state WAL transport properties. The 2D nature of the observed superconductivity in AuSn 4 will lead us further into considering the origin of 2D superconductivity. Since AuSn 4 is isostructural with the Dirac nodal arc semimetals PtSn 4 and PdSn 4 , the origin of natural 2D superconductivity in AuSn 4 can be inferred from Dirac-like surface band dispersion. Thus, the normal-state transport properties of AuSn 4 are worth studying. Figure 6a shows a result of resistivity with applied fields parallel to the crystal c-axis. As shown, the ρ(H) increases with the applied fields, and a clear non-saturating field-dependent magnetoresistance MR(H), defined as MR(H) = [ρ(H) − ρ(0)]/ ρ(0), can be seen for temperatures up to 50 K and field up to 6 T, as shown in Fig. 6b. A sharp normal-state resistivity dip is clearly observed at lower temperatures, even at temperatures above T c or H > H c2 ; when the temperature increases, the resistivity dip at low fields is broadened and the MR(H) becomes linear at an intermediate field in the whole normal-state region. It is known that a sharp magnetoresistance dip at low temperatures indicates the presence of a WAL effect; when we increase the temperature, the magnetoresistance dip broadens at the low field due to the decrease of the phase coherence length at higher temperatures 48 . Similar types of magnetoresistance behavior without any sign of saturation have been seen in topological materials [48][49][50] . In particular, Shrestha et al. 49 have recently shown extremely large nonsaturating magnetoresistance (540% at T = 2 K under 7 T) and ultrahigh Hall mobility (4.5 × 10 4 cm 2 V −1 s −1 at T = 5 K) due to topological surface states in the metallic Bi 2 Te 3 topological insulator 49 , results which are comparable with those of magnetoresistance ≈ 650% at 2.5 K under 6 T and Hall mobility ≈ 3.5 × 10 3 cm 2 V −1 s −1 at T = 2.5 K observed on our AuSn 4 single crystals (for details, see Supplementary Note 3). We may also note that, due to the non-negligible contribution of transverse resistivity, as arising from the high Hall mobility, the original experimental resistivity data were asymmetric in the field. According to the argument proposed by Segal et al. 51 , the odd longitudinal resistivity can be eliminated via counting the average experimental data in the positive and negative fields, leading to a duplicate result of magnetoresistance in the positive and negative fields, as shown in Fig. 6a, b, respectively. Additionally, in the ρ (H) curves at T < 3.75 K, a crossover of ρ(H) values from a downward-trend field dependence to a nearly linear H dependence can be observed when the magnetic field is beyond a crossover field H*. The crossover field H* can be determined by the plot of an intersection of two linear fitting lines, as shown by the two guide lines in Fig. 6a. One can see that the H* shifts to a lower field with the increase in temperature, as shown in the inset of Fig. 6a. The temperature dependence of the H* suggests the existence of a linear-dispersion electronic band in AuSn 4 , as seen in topological materials [48][49][50] . Now that the normal-state transport property of AuSn 4 indicates a crossover from a WAL-dominant MR(H) to a linear and non-saturating MR(H), the fielddependent transverse magnetoconductivity will be analyzed within the framework of a WAL electrical transport. Figure 7 shows the field-dependent transverse magnetoconductivity change ratio for AuSn 4 at low temperatures, where the magnetoconductivity change ratio ΔMC is defined as ΔMC = [σ(H) − σ (0)]/σ(0). As is known, the transverse magnetoconductivity can be expressed by σ T (H) = σ WAL + σ n , where σ WAL is the surface conductivity from WAL corrections related with intranodal scattering, and σ n is from conventional Fermi surface contributions 52 . The WAL formula is expressed as σ WAL = a ffiffiffiffi H p + σ 0 and σ n = (ρ n + A·H 2 ) −1 , where a, σ 0 , ρ n , and A are determined from the line of best fitting. As seen, the low-field transverse ΔMC can be well described by the WAL transport formula with a negligible value of A ≈ 0. The inset of Fig. 7 shows the temperature dependence of the obtained a values. One can see that parameter a is negative due to a positive magnetoresistance and at higher temperatures; the decrease in |a| values with an increase in Fig. 6 Magnetoresistance of AuSn 4 . a Resistivity with applied fields parallel to the crystal c-axis at various temperatures. The intersection of two dashed lines denotes the determination of the crossover field H*. The inset shows the temperature dependence of H*, as obtained from data in a. b Non-saturating MR(H) for temperature up to 50 K and field of up to 6 T. Note the drastic increase in magnetoresistance at 2.5 K plotted with a contracted scale of MR(H) × 1/6. Also note that the data show a duplicate result of magnetoresistance in positive and negative fields, as described in the text.
temperature indicates a gradual absence of the WAL effect on the field-dependent transverse magnetoconductivity. It is known that in topological materials, strong spin-orbit coupling can induce WAL 53 , and this WAL effect originates from the strong spin-orbit coupling in the band structure, and results in the spinmomentum locking in the topological surface states 54 . Therefore, the WAL phenomenon is always observed in topological materials as an important consequence of spin-momentum locking, as well as the full suppression of backscattering, which is a fingerprint of the surface states 55 . Since the normal-state transport properties of AuSn 4 exhibit the WAL behavior, one can deduce that AuSn 4 also belongs to the topological semimetals, same as its isostructural compounds of PtSn 4 and PdSn 4 . All the results clearly prove that the natural 2D superconductivity in AuSn 4 originates from Dirac-like surface band dispersion, leading to the conclusion that topological superconductivity possibly exists on its surface and exhibits 2D nature. This work also provides important experimental results for further studies on the topological-semimetal candidate of AuSn 4 , such as the theoretical band-structure calculation and angle-resolved photoemission spectroscopy experiments, which are popular topics in condensed matter physics.

Discussion
These presented results enabled us to construct a phase diagram of AuSn 4 crystals in the magnetic field-temperature (H-T) plane. The 2D superconductivity let us consider the emergence of the Bose-metal phase recently identified in 1T-MoS 2 and NbSe 2 (refs. 5,56 ). Returning to the Arrhenius plot of resistivity, as shown in Fig. 8a, this study replotted some selected low-field Arrhenius plots of resistivity with H // c-axis, for a detailed discussion of the Bose-metal phase in AuSn 4 . It has been pointed that a strongly disordered 2D superconductor makes a transition to an insulating state under a perpendicular magnetic field applied 57 , while a system with a low disorder, such as a high-quality single crystal, should show a quantum phase transition to an intermediate 2D metallic state, Bose metal (BM), with a resistance much lower than the normal-state resistivity 6,57,58 . This state is characterized by saturation of the resistance as T approaches 0 K. Additionally, the resistance obeys a power law with the magnetic field, which has been theoretically addressed using a simple scaling of powerlaw BM resistivity 57 , ρ ∝ (H − H c0 ) 2ν , where H c0 and ν are the critical field of~H c1 (0) and the exponent of this step of the superconducting-BM transition, respectively. Figure 8a displays the possible existence of the BM phase in AuSn 4 , where following on a sharp drop in resistivity at the superconducting transition point, the resistivity deviates from thermally activated behavior and gradually decreases to a small value compared to ρ N as the temperature is lowered. The field resistivity does not exhibit a saturated constant value as seen in 1T-MoS 2 (ref. 5 ), which should be due to the presence of bulk superconductivity. Thus, the deviation temperature T* for finite field values as indicated in Fig. 8a can be deduced to define the BM phase in the 2D superconductivity regime. Also shown in Fig. 8b is a log-log scale plot of resistivity versus perpendicular magnetic field for temperatures below T BKT with a fit to the power-law dependence by the use of H c0 = 39.4 Oe and ν = 2.18-2.42 for different temperatures. This power-law resistivity demonstrates the possible existence of the BM phase in AuSn 4 . For further confirming the BM phase of 2D superconductors, we adopted the Ullah-Dorsey (UD) scaling theory 59 to calculate the excess conductance generated by the fluctuation of the superconducting order parameter. The excess conductivity, G fl ≡ 1/ρ (T) − 1/ρ N (T), under different magnetic field is scaled with the universal relation 60 : Here, s = 1 in the case of a 2D system, T c (H) is the mean field transition temperature in a magnetic field and ρ N (T) is the normal-state resistivity. As shown in Fig. 8c, by using ρ(T) data in the presence of a magnetic field and taking T c (H) derived from the data corresponding to the temperature on the H c2 (T) curve as previously obtained in Fig. 3a, it is found that G fl ·(μ 0 H/T) 0.5 curves with a fitting parameter ρ N of~2.564 μΩ cm for scaling in the AuSn 4 at T > T c (H) collapse onto a single curve with the slope of −1 in log-log plots, indeed obeying the 2D UD scaling law of Eq. (3). However, it must be noted that this kind of metallic state can also be interpreted as a result of quantum creep, as seen in a ZrNCl electric-double-layer transistor 7 . In this model, the sheet resistance R s in the limit of the strong dissipation obeys a general form 61 : where R N is the normal-state sheet resistance and C is a dimensionless constant. According to Eq. (4), it can be predicted that the term ln 1 þ 1 R s h 4e 2 should be in proportion to H c2;c ÀH H c2;c , and will show a linear relationship passing through the origin point. However, as seen in Fig. 8d,  In summary, the dimensionality of superconductivity in AuSn 4 single crystals has been explored herein. The bulk superconductivity with the T c of~2.40 K probed by the magnetization and special heat measurements reveals a characteristic of conventional BCS type-II superconductors. However, the superconductivity investigated with electrical transport properties exhibits 2D nature by showing evidence of a BKT transition, BM phase, and the diverging ratio of in-plane to out-plane upper critical field on approaching T c . The vortex dynamics of AuSn 4 further display that the activation energy U(H) and upper critical field H c2 (T) can be interpreted in terms of the TAFF model in two dimensions. Moreover, it is found that in the normal state, the low-temperature magnetoconductivity of AuSn 4 can be well described by the WAL transport formula, which has been commonly observed on topological materials; thus, it strongly supports the scenario that the surface electrons in Dirac-cone states dominate the normal-state magnetotransport in AuSn 4 single crystals. The results clearly prove that the natural 2D superconductivity in AuSn 4 originates from Dirac-like surface band dispersion, leading to the conclusion that topological superconductivity possibly exists on its surface and exhibits 2D nature. Finally, a phase diagram of AuSn 4 crystals in the H-T plane has been constructed, which displays different regimes of transport arising due to the vortex dynamics and surface Dirac linear

Methods
Single-crystal growth. AuSn 4 single crystals were grown from Sn flux with a starting composition of Au:Sn = 0.09:0.91. The mixtures of high-purity Au pieces and Sn ingots were sealed under vacuum in a quartz tube. The quartz ampoule was heated to 630°C for 10 h, cooled to 310°C for 6 h and then slowly cooled down to 240°C at a rate of 1°C per hour. The remaining Sn flux was separated by centrifugation and several platelet-like crystals with a typical size of 3 × 2 × 0.3 mm 3 were mechanically removed from the quartz ampoule. The as-grown crystals are ductile and exhibit a silvery luster. However, they cannot be mechanically exfoliated to be a few-layer thickness like graphene. The AuSn 4 crystals are stable in air over months but corrode in diluted hydrochloric acid. The phase purity and the crystal structure of obtained crystals were characterized by powder X-ray diffraction (Bruker D2 phaser) and Laue diffraction (Photonic Science) measurements with Cu-K α radiation on single crystals.
Transport measurements. For in-plane electrical transport measurements, the cleaved shiny crystals were cut into dimensions of~3.0 × 1.0 × 0.1 mm 3 . Five leads were soldered with indium, and a Hall-measurement geometry was formed to allow simultaneous measurements of both longitudinal (ρ) and transverse (Hall) resistivities (ρ xy ) using the standard dc four-probe technique. Hall voltages were taken in opposing fields parallel to the c-axis up to 6 T and at a dc current density of~30 A cm −2 . The low-temperature specific heat C p measurement was carried out using a 3 He heat-pulsed thermal relaxation calorimeter (PPMS-16 from Quantum Design) in the temperature range of 0.5-3.0 K. The magnetization was measured in a superconducting quantum interference device (SQUID) system (MPMS from Quantum Design).

Data availability
The data that support the findings of this study are available on reasonable request.