Abstract
The search for nontrivial superconductivity in novel quantum materials is currently a most attractive topic in condensed matter physics and material science. The experimental studies have progressed quickly over the past couple of years. In this article, we report systematic studies of superconductivity in Au_{2}Pb single crystals. The bulk superconductivity (onset transition temperature, T_{c}^{onset}=1.3 K) of Au_{2}Pb is characterised by both transport and diamagnetic measurements, where the upper critical field H_{c2} shows unusual quasilinear temperature dependence. The superconducting gap is revealed by pointcontact measurement with gold tip. However, when using tungsten (W) tip, which is much harder, the superconducting gap probed is largely enhanced as demonstrated by the increases of both T_{c}^{onset} and upper critical field (H_{c2}). This can be interpreted as a result of increase in density of states under external anisotropic stress imposed by the tip, as revealed by firstprinciples calculations. Furthermore, novel phase winding of the pseudospin texture along kspace loops around the Fermi energy is uncovered from the calculations, indicating that the observed superconductivity in Au_{2}Pb may have nontrivial topology.
Introduction
The recent discovery of threedimensional (3D) Dirac semimetals,^{1–12} as an intermediate state between a trivial insulator and a topological insulator, has stimulated extensive research on these materials. In 3D Dirac semimetals, the conduction and valence bands contact only at Dirac points in the Brillouin zone, and gap formation is forbidden by crystalline symmetry. Much recent excitement has surrounded the materials like Na_{3}Bi^{2} and Cd_{3}As_{2},^{3} in which the Dirac cone semimetal states were predicted theoretically and soon identified by Angleresolved photoemission spectroscopy (ARPES) experiments.^{4,6,11} Surprisingly, the superconductivity induced by hard point contact (PC) on Cd_{3}As_{2} was recently reported and unconventional superconducting order parameter was suggested by the zerobias conductance peak (ZBCP) and double conductance peaks (DCPs) symmetric around zero bias.^{13,14} In addition, pressureinduced superconductivity in Cd_{3}As_{2} was further confirmed by hydrostatic pressure experiment.^{15}
Another promising material is the cubic Laves phase Au_{2}Pb, which reveals the signature for the symmetryprotected Dirac semimetal state at temperature >100 K. However, after the structural phase transition, the Dirac cone is gapped <100 K.^{16} Here we report systematic studies of lowtemperature transport and diamagnetic properties, PC measurements and firstprinciples calculations of Au_{2}Pb single crystal. The superconductivity of Au_{2}Pb is characterised, and the anomalous quasilinear H_{c2}(T) in Au_{2}Pb suggests the presence of an unconventional superconducting state. The T_{c}^{onset} and H_{c2} revealed by the PC measurement with a gold tip is consistent with that by the bulk transport and magnetisation measurements, but with a hard W tip, the gap is largely enhanced as shown by the increases of both T_{c}^{onset} and H_{c2}. By comparing the computed Fermi surfaces of Au_{2}Pb of relaxed lattice structure and those under 1% uniaxial compression, we find that the Fermi surfaces are enlarged and the density of states near the Fermi energy increases under the applied external pressure, which agrees with the observed increase of T_{c}^{onset} in the PC measurements. More importantly, the possibility of topological superconductivity is suggested by firstprinciples calculations, which show nontrivial topological properties of the projected pseudospin texture corresponding to the p and dorbitals near the Fermi energy, as well as the experimental observations of anomalous quasilinear H_{c2}(T) behaviour and unusual PCS feature detected by W tip. These characteristics make Au_{2}Pb a potential candidate material for topological superconductor.
Results
Sample structure
Highquality single crystal of Au_{2}Pb samples were synthesised by selfflux method in the evacuated quartz ampoule (see Materials and Methods). The powder Xray diffraction (XRD) pattern is given in Figure 1a, indicating a crystal structure of Au_{2}Pb and exhibiting a cubic Laves phase with a_{1}=a_{2}=a_{3}=7.9603 Å at room temperature. The single crystal XRD pattern only shows (111) reflections, suggesting the sample surface of the crystal is (111) plane (inset of Figure 1a). The atomic highresolution transmission electron microscopy (HRTEM) image (Figure 1b), together with the selected area electron diffraction pattern (Figure 1c), further demonstrate the highquality single crystal nature of Au_{2}Pb samples.
Superconductivity
The superconductivity in Au_{2}Pb has been confirmed by our systematic lowtemperature transport and diamagnetic measurements. Figure 2a displays the sample resistivity (ρ) as a function of temperature at zero magnetic field. The behaviour is metallic with an anomaly of ρ(T) at 100 K due to a structural phase transition (cubic Laves phase >100 K).^{16} A steep step at 50 K and hysteresis at 40 K are observed with cooling and heating measurements, owing to structural transition to orthorhombic phase <40 K (the inset of Figure 2a). On cooling down to lower temperature, bulk T_{c}^{onset} ~1.3 K and zero resistance transition temperature T_{c}^{zero} ~1.18 K are observed, comparable to the previous reports.^{16,17} The superconductivity in Au_{2}Pb is further confirmed by Meissner effect measurements. In Figure 2b, the susceptibility (χ) versus temperature curves show quite sharp drops at 1.15 K, in good agreement with T_{c}^{zero} from ρ(T) curve. Magnetic field dependence of the magnetisation (M(H)) curves at various temperatures (inset of Figure 2b) exhibit the expected quasilinear behaviour at low fields but deviate from linearity above the lower critical field H_{c1}. Figure 2c reveals the suppression of the superconducting state of Au_{2}Pb by the magnetic field perpendicular to (111) plane (H_{⊥}). The superconducting transition becomes broader and shifts to lower temperature with increasing fields. Magnetotransport measurements were carried out at various temperatures from 0.18 to 1.4 K (Figure 2d). The onset H_{c2}, defined as the field above which the Au_{2}Pb sample becomes the normal state, is shown in Supplementary Figure S1. H_{c2} linearly increases with decreasing temperature down to T_{c}/6, yields dH_{c2}(T)/dT_{T=Tc}≈0.054 T/K near T_{c}. We calculate the reduced critical field h*=H_{c2}/T_{c}(dH_{c2}/dT_{T=Tc}) (inset of Figure 2d) to compare the data to the known models for swave superconductors (Werthamer–Helfand–Hohenberg theory, WHH, H_{c2}≈0.7T_{c}dH_{c2}/dT_{T=Tc}, h*(0)≈0.7)^{18} and spintriplet pwave superconductors (h*(0)≈0.8).^{19} Obviously, the h* relation is close in the form to that of a polar pwave state, which is suggestive of the finite triplet contribution to the pairing state in Au_{2}Pb.^{19,20} The upper critical field H_{c2}(T) shows little anisotropic property for perpendicular and parallel magnetic field (Supplementary Figure S2). Furthermore, we plotted the normalized magnetoresistivity (MR=(ρ(H)−ρ(0))/ρ(0)×100%) of Au_{2}Pb as the field up to 15 T, from 1 to 200 K (see Figure 2e). In contrast to the classical quadratic MR in metals and semiconductors, the sample presents nonsaturating linearlike MR at lower temperatures and highermagnetic fields. With increasing temperatures, the exponent α (MR∝H^{α}) varies between 1 and 2 and exhibits anomalies when temperatures undergoing structural phase transitions ~40 K, 100 K (Supplementary Figure S3b). In the parallel field (H//(111) plane: H_{//}) configuration, Figure 2f demonstrates how the MR behaviour changes when the direction of the magnetic field is rotated from 0° (H//[2]⊥I) to 90° (H//[10]//I) at 2.5 K. Classically, the resistance has no response to the applied external magnetic field parallel to the excitation current. While in our situation, MR is quasilineardependent on magnetic field (MR∝H^{1.09±0.02}, inset of Figure 2f). The linear magnetoresistance behaviour has usually been observed in semiconductor,^{21} semimetals,^{22} topological insulators^{23–25} and Dirac/Weyl semimetals.^{10,26,27} One general interpretation is inhomogeneity in materials, but the inhomogeneity does not seem to play an important role here, since our Au_{2}Pb crystals show good single crystal quality. Alternatively, it is also tempting to ascribe the linear magnetoresistance to Abrikosov’s quantum magnetoresistance,^{28} however, the estimated carrier density 10^{22} cm^{−3}(see Supplementary Figure S4) is too high for this model to be applicable over entire field range. It is noticed that this longitudinal linear MR (H//I) due to its balanced hole and electron populations, was ever observed in type II Weyl semimetal candidate WTe_{2}.^{22,29} The linearity here may originate from the multiband nature of Au_{2}Pb.
T_{c}^{onset} and H_{c2} enhancements in PC measurement with a W tip
PC Andreev reflection spectroscopy is a powerful tool to probe the order parameters of superconductor.^{30} The temperature dependence of zerobias differential resistance of PC between a gold tip and Au_{2}Pb single crystal is shown in Figure 3a. The presence of a significant resistance drop at the onset temperature T_{c}^{onset}=1.13 K at zero magnetic field and the absence of this drop at 0.03 T applied perpendicular to the (111) surface indicate a superconducting transition, which is similar to that in the bulk measured by standard fourelectrode method (see Figure 2). However, when measured with a W tip as shown in Figure 3b, the onset temperature increases to 2.1 K, higher than the bulk value. And the perpendicular field (H_{⊥}) used to fully suppress the superconductivity transition is 0.755 T, also much higher than the bulk value. The PC spectra (PCS) at different temperatures with a gold tip and a W tip are shown in Figure 3c,d, respectively. For the PC with a gold tip, DCPs are observed at 0.5 K and gradually get smeared with increasing temperature. The resulted broad conductance peak totally diminishes at 1.1 K, consistent with the transition temperature in dV/dI(T) as shown in Figure 3a. For the PC with a W tip, similar DCPs feature is clearly shown at lower temperatures, and the broad conductance enhancement totally diminishes at T=2.1 K, consistent with T_{c}^{onset} in Figure 3b. We note that besides the DCPs feature, there are also conductance dips outside the conductance enhancement regime as shown in Figure 3d for temperatures >1 K, which evolve to broad valleys <1 K (around bulk T_{c} shown in Figure 2b). These conductance dips might be related to the critical current effect.^{31} Alternatively, conductance dips at around the gap energy with a broad ZBCP could be a result of the helical pwave order parameter.^{32} Another unusual feature of the PCS is a small splitting of the conductance peaks at 0.3 K, which can be fitted by the modified Blonder–Tinkham–Klapwijk (BTK) model^{33,34} for two gaps, while the double conductance dips and broad conductance valleys cannot be fitted by conventional swave pairing symmetry considerations (Supplementary Figure S5).
The magnetic field dependence of PCS at 0.5 K is shown in Figure 3e,f for PC with a gold tip and a W tip, respectively. All the PCS show gradual suppression of the superconducting features with increasing magnetic field. The H_{c2} for PC with a gold tip is about 0.02 T at 0.5 K, which is close to the bulk value (~0.05 T) obtained by standard fourelectrode measurements, while the H_{c2} value of PC with a W tip is about 0.60 T, which is >10 times larger than the bulk value.
Analysis of PC results
For PC with a W tip (harder than gold tip), both T_{c}^{onset} and H_{c2} are larger than bulk values (Supplementary Figure S6). The T_{c} enhancement seems related to the pressure applied by the hard W tips, which may cause changes of the band structure or lifting of degeneracy of multiple order parameters.^{35,36} The conventional cause of H_{c2} enhancement in PC is the reduction of the mean free path l in the superconductor as a result of structural defects and/or impurities at interface,^{37,38} which leads to a decrease of the coherence length and increase of H_{c2}. However, this cannot explain the concomitant enhancement of T_{c} and H_{c2} in our situation. In addition, similar superconductivity enhancement for PC with PtIr tips on Au_{2}Pb samples was observed, where PtIr tips are also much harder than gold tips. Both the PC results of W tip and PtIr tip suggest that the superconductivity in Au_{2}Pb is sensitive to pressure and/or possible doping effect. Magnetoresistance measurements with field applied parallel or perpendicular to sample surface show little anisotropy, which rules out the possibility of surface superconductivity (Supplementary Figure S2).
The firstprinciples calculations
Two important clues in the experimental data are worth remarking. First, for the PCS with a W tip, the double conductance dips feature cannot be adequately described by the BTK theory with swave pairing symmetry, and points to possible unconventional superconductivity of this material. Second, for transport experimental results, the quasilinear H_{c2}(T) data imply the expectation for a pwave state and deviate significantly from the WHH theory for an swave superconductor. Therefore, it is important to analyse the electronic structure of Au_{2}Pb in detail to identify possible unusual topological features.
To examine the effects of PC tip indentation on the sample’s local electronic structure, we compare the computed Fermi surfaces of Au_{2}Pb of relaxed structure and those under 1% uniaxial strain along the [111] direction. Although the Fermi surfaces are rather complex and appear to be composed of multiple sectors throughout the Brillouin zone (see Supplementary Figure S7), it is evident that the Fermi surfaces are enlarged under the applied external pressure. The augmentation of Fermi surfaces and the accompanying increase in the density of states at the Fermi energy (shown in Figure 4d) favour higher superconducting T_{c}, which agrees with the observed increase of T_{c} in the PC measurements. It should also be remarked that the size of Fermi surfaces near the Brillouin zone path Γ–R are large and noticeably responsive to external pressure, as shown in Supplementary Figure S7. Thus, in the following, we will focus on the regions near this sector (referred to as the Tsectors hereafter) of the Fermi surfaces.
It is of particular interest to investigate the geometric phase of the bands near the Fermi surfaces. By projecting the Kohn–Sham wavefunctions into various local orbital basis (Supplementary Figure S8), we find that the dominant band components near the Fermi surfaces are parityeven dorbitals of Au and parityodd porbitals of Pb. We choose two closed paths on the valence band around one of the Tsectors, and calculate topological invariants through the phase difference, $\phi \left(k\right)=\text{arg}\u3008dk\u3009\text{arg}\u3008pk\u3009$, of the projection coefficients of Kohn–Sham wavefunctions of the highest occupied onto the Aud and Pbp orbitals along these loops. The two loops are carefully chosen to avoid band degeneracy at the band crossing (see the dark blue curve in Figure 5a). In Figure 5b, a unit vector n=[cos φ, sin φ], is plotted for each kpoint on the loop to visualise the orbital texture.
A few interesting features are observed for the projected orbital textures. In particular, the winding numbers are zero for the orbital textures on the inner loop (see e.g., a typical orbital texture on the inner loop in Figure 5b). In contrast, for the outer loop, the orbital textures corresponding to various Aud and Pbp_{x} orbitals show a nontrivial topology, with the winding number equals −1. On the other hand, from Figure 6 we find that an ‘8’ shape projected nodal line is numerically confirmed on a certain Brillouin zone slice, while on another Brillouin zone slice a circleshaped projected nodal line is observed. These intriguing observations hint at nontrivial band structure and topological properties of the Tsectors.
To capture the main features of the band structure around the Tsectors, we introduce an ad hoc twoband effective Hamiltonian to describe the physics regarding the Aud and Pbp_{x} orbitals:
where k denotes the momentum relative to the central point of the ‘8’ shape nodal line shown in Figure 6a, and M(k) is a polynomial of k, taken as $M(k)={m}_{1}\sqrt{{m}_{2}^{2}+{k}^{2}}$ with m_{1}>m_{2}. The energy spectra read ${E}_{\pm}=\pm \leftM(k)k\right$, exhibiting a nodal point at k=0 and a nodal surface at $k=\sqrt{{m}_{1}^{2}{m}_{2}^{2}}$. One can verify that the above Hamiltonian leads to the band structure and orbital texture consistent with the density functional theory (DFT) calculation results. Actually, the valence band wavefunction of the Hamiltonian is
where θ and ϕ are the polar and azimuthal angles of k, respectively. From the wavefunction, we can see that the orbital texture of the Hamiltonian (1) is determined solely by the $\stackrel{\u2322}{k}\cdot \sigma $ factor. The form of M(k) has been chosen to mimic the spectra from DFT calculation, reproducing the nodal surface near the Fermi level. It is easy to see that the projected orbital texture is dependent only on ϕ evolved along each path. The orbital texture on any loop that encloses the z axis (k_{x}=k_{y}=0) connecting the north and south poles of the Bloch sphere, has a winding number 1 or −1. In contrast, if the loop excludes the z axis, the orbital texture is trivial with the winding number being 0. Accordingly, a loop enclosing the nodal line on the k_{z}=constant plane (i.e., the crossing line between the plane with fixed k_{z} and the nodal surface) always encloses the z axis, giving a nonzero winding number of the orbital texture (the outer loop of Figure 5b). However, a loop inside the nodal line may or may not enclose the z axis. In the latter case, the winding number of the orbital texture is zero, which is understood to be the case of the inner loop of Figure 5b.
Discussion
The nontrivial topology of the orbital texture suggests the possibility of topological superconductivity. Note that the wholeBrillouin zone includes eight different sectors around Fermi energy, with the other seven equivalent copies of the effective Hamiltonian equation (1) obtained by timereversal and symmetry of the orthogonal space group. In particular, the timereversal copy of the Hamiltonian (1) can be described by $\Gamma {H}_{\mathrm{eff}}{\Gamma}^{1}=M(k)\left[{k}_{x}{\sigma}_{x}+{k}_{y}{\sigma}_{y}{k}_{z}{\sigma}_{z}\right]$, where the timereversal operator Γ=iKs_{y} where K is the complex conjugate operator, and s_{y} is a Pauli matrix operating on real spin. We note that the Hamiltonian equation (1) and its timereversed copy have opposite spin orientations, which are not explicitly described here. It is readily verified that the topology of the two timereversed Fermi surfaces is the same. Note that the pairing mechanism cannot annihilate the topological invariants if the Cooper pairs are formed by two Fermi surfaces of the same topology.^{39,40} This implies that the superconductivity due to the pairing between such two timereversed Fermi surfaces is fully gapped and of nontrivial topology inherited from the singleparticle states. Due to the timereversal symmetry, the resulted 3D topological superconductivity is classified by an integer invariant,^{41,42} and thus may be stable in the presence of multiple copies of the above paired Hamiltonians in the Brillouin zone.
The combination of transport, diamagnetic, PC measurements and the firstprinciples calculations are used to study Au_{2}Pb crystals. We characterised the superconductivity of Au_{2}Pb crystals by transport and diamagnetic experiments. Enhancement of superconductivity is found when it is probed by a hard PC, and we attribute this to local pressure, which is consistent with the enlarged Fermi surface by theoretical calculations. The anomalous quasilinear H_{c2} versus T relation, as well as the gap feature (DCPs with double conductance dips) in PCS, imply possible unconventional superconducting properties in Au_{2}Pb crystals. Furthermore, the firstprinciples calculations point to the nontrivial topology of the orbital texture near the dominant Fermi surfaces, which suggests the possibility of topological superconductivity. The results presented in this work indicate the Au_{2}Pb crystal could be a promising platform for the investigation of topological superconductivity. For full understanding of the superconductivity in Au_{2}Pb, further investigations like thermal transport and ARPES experiments are clearly necessary and interesting.
Note added
After the completion of this work, we notice a recent hydrostatic pressure study of bulk Au_{2}Pb,^{43} where T_{c} decreases with pressure. It is opposite to our observation. The major difference between two studies is that their pressure is hydrostatic but ours is nearly local uniaxial pressure on (111) crystal face.
Materials and methods
Sample growth and characterisation
Starting materials of highpurity elemental Au and Pb were prepared to synthesise singlecrystalline Au_{2}Pb, the initial ratio of Au/Pb was 40:60, the extra Pb were used as flux. The materials were sealed in evacuated quartz ampoules, heated to 600 °C, held for 1 day and then slowly cooled down to 300 °C over a period of 30 h. This was then followed by centrifugation to remove the flux. Before transport and diamagnetic measurements, the samples were etched in a hydrogen peroxide solution for several minutes to remove the residual Pb flux.^{16}
The data of powder XRD and single crystal XRD were collected from a Rigaku MiniFlex 600 diffractometer (Rigaku, Tokyo, Japan) and then refined by a Rietica Rietveld programme (http://www.rietica.org/index.html). Crystal purity, structure and lattice constant (a_{1}=a_{2}=a_{3}=7.9603 Å) can be retrieved. It is worth to mention that highquality single crystal of Au_{2}Pb samples are in trapezoid stipe shape. The current is added parallel to the bottom edges for transport measurements, which is in [10] direction for cubic Laves phase.
To obtain HRTEM images, the Au_{2}Pb single crystal was examined by a FEI Tecnai G2 F20 STWIN TEM (FEI, Hillsboro, OR, USA) operating at 200 kV.
Resistivity and diamagnetic measurements
For the transport measurements, the contacts on the single crystals were made by applying the silver paint on the top surface (111) of Au_{2}Pb samples, with contact resistance <1 Ω. The resistance and magnetoresistance were measured in a commercial Physical Property Measurement System (Quantum Design, San Diego, CA, USA, PPMS16, d.c. technique), with the Helium3 option for temperature down to 0.5 K and dilution option down to 0.05 K. The excitation current of 1 mA was used for the measurements in lowtemperature regime. Angular dependence of magnetoresistance was measured by rotating the sample ((111) surface plane) in a Rotator option based on PPMS, within the instrumental resolution of 0.1°.
For diamagnetic measurement, DC magnetisation was studied during the zerofield cooling and field cooling at H=5 Oe, in a Magnetic Property Measurement System (MPMS 7XL SQUID) from Quantum Design Company, with a resolution of 10^{−8} e.m.u. The magnetic field is applied parallel to the (111) face. The χ is estimated by DC magnetisation.
The PC measurements
Our PCS are obtained using a standard lockin technique in quasifour probe configuration. PC measurements are realized in the standard ‘needle–anvil’ configuration. Both W and gold tips are used to make PC on the (111) surface of Au_{2}Pb single crystals. The W tip is prepared by electrochemical etching method with a wire of 0.25 mm diameter, and is hard enough to penetrate through the surface layer and to probe the superconductor underneath. The gold tip is mechanically sharpened from a 0.5 mm wire and is relatively soft.
The firstprinciples calculations
The firstprinciples calculations were performed in the scheme of densityfunctional theory, as implemented by the VASP package.^{44} The projectoraugmented wave pseudopotentials^{45} were used with Perdew–Burke–Ernzerhof exchangecorrelation functional.^{46} The energy cutoff of the plane wave basis set was chosen to be 350 eV. The Brillouin zone was sampled by a 13×19×9 grid in the selfconsistent calculation. During the relaxation procedure, we used a force threshold of 0.01 eV Å^{−1}. Spinorbit coupling was included in the calculation of energy bands and Fermi surfaces, and not included in the calculation of orbital textures to avoid the twofold degeneracy of the bands to obtain a welldefined phase difference.
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Acknowledgements
This work was financially supported by the National Basic Research Programme of China (Grant Nos. 2013CB934600 & 2012CB921300 & 2012CB927400), the Research Fund for the Doctoral Programme of Higher Education (RFDP) of China, the Open Project Programme of the Pulsed High Magnetic Field Facility (Grant No. PHMFF2015002), Huazhong University of Science and Technology and the National Natural Science Foundation of China (Grant No. 11474008).
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J.W. and J.W. conceived and instructed the experiments. X.Z. and S.J. synthesised and characterised bulk single crystals; Y.X., Y.Z. and Z.W. did the transport experiments; H.W. and J.L. performed the point contact measurement; Y.X., L.L. and M.T carried out the diamagnetic experiments; J.L. and Y.W. did the TEM characterisation; X.J.L. contributed to the theoretical interpretation; J.F. and C.K.L. performed the firstprinciples calculations.
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Xing, Y., Wang, H., Li, CK. et al. Superconductivity in topologically nontrivial material Au_{2}Pb. npj Quant Mater 1, 16005 (2016). https://doi.org/10.1038/npjquantmats.2016.5
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