Abstract
Topological nodal-line semimetals (TNLSMs) are materials whose conduction and valence bands cross each other, meeting a topologically protected closed loop rather than discrete points in the Brillouin zone (BZ). The anticipated properties for TNLSMs, include drumhead-like nearly flat surface states, unique Landau energy levels, special collective modes, long-range Coulomb interactions, or the possibility of realizing high-temperature superconductivity. Recently, SrAs3 has been theoretically proposed and then experimentally confirmed to be a TNLSM. Here, we report high-pressure experiments on SrAs3, identifying a Lifshitz transition below 1 GPa and a superconducting transition accompanied by a structural phase transition above 20 GPa. A topological crystalline insulator (TCI) state is revealed by means of density functional theory (DFT) calculations on the emergent high-pressure phase. As the counterpart of topological insulators, TCIs possess metallic boundary states protected by crystal symmetry, rather than time reversal. In consideration of topological surface states (TSSs) and helical spin texture observed in the high-pressure state of SrAs3, the superconducting state may be induced in the surface states, and is most likely topologically nontrivial, making pressurized SrAs3 a strong candidate for topological superconductor.
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Introduction
In recent years, topological semimetals including Dirac, Weyl, and nodal-line semimetals have been theoretically predicted and experimentally verified, opening a new field in condensed-matter physics in which novel properties and new applications can arise from spin-polarized states with unique band dispersion1,2,3. Unlike the discrete points in momentum space in Dirac or Weyl semimetals2,3, the band crossings in nodal-line semimetals can form closed loops inside the BZ4; a nodal chain consisting of several connected loops5; or an extended line traversing the entire BZ6. These one-dimensional nodal curves are topologically protected by certain discrete symmetries, for example mirror reflection, time-reversal, or spin-rotation symmetries2,3. Upon breaking symmetries in a topological nodal-line semimetal (TNLSM), the nodal line is either fully gapped or gapped into several nodal points4. The nodal-line structure is expected to have several intriguing properties3, such as unique Landau energy levels7, special collective modes8, long-range Coulomb interactions9, or drumhead-like nearly flat surface states10,11, which can be considered a higher-dimensional analog of the flat band on the zigzag edge of graphene3. These drumhead states may host interesting correlation effects, and even offer the possibility of realizing high-temperature superconductivity12.
In the search for nodal-line semimetals, several systems have been theoretically proposed since 20112,3. However, only a few candidates including PbTaSe213,14, ZrSiX (X = S, Se, Te)15,16, CaAgX (X = P, As)17,18, and MB2 (M = Ti, Zr)19 have been verified experimentally. More recently, the CaP3 family of materials (MAs3, for M = Ca, Ba, and SrX3 for X = P, As) was proposed as another potential host of TNLSMs20. Among these compounds, only SrAs3 shows a strongly topological nature at ambient pressure, while others need extra compressing20. SrAs3 displays semimetallic behavior with the hole carriers dominating21,22. Previously, unusual galvanomagnetic properties and a first-order longitudinal Hall effect have been found in SrAs323, and quantum oscillation experiments have been applied to map out the shape of the Fermi surface, finding two asymmetric, quasi-ellipsoidal Fermi-bodies as well as light cyclotron effective mass22. Recent magnetotransport measurements on SrAs3 single crystals found a nontrivial Berry phase and a robust negative longitudinal magnetoresistance (MR) induced by the chiral anomaly, which indicates the presence of topological properties in SrAs324,25. Subsequently, Song et al.26 observed the complete nodal-line feature around the Y point by means of angle-resolved photoemission spectroscopy, demonstrating the existence of Dirac nodal-line fermions. In contrast to most TNLSMs, the nodal-line structure in SrAs3 does not coexist with complex topologically trivial Fermi surfaces, which may pave an easy path to potential applications20,26.
Among the topological materials, intense effort has been applied to realizing topological superconductors (TSCs)27,28, one source of Majorana fermions, an effort which suffers from a severe lack of suitable materials to study27,28. Experimentally, applying chemical doping or pressure to search for superconductivity in known topological materials are two common methods to obtain new TSC candidates27,28. While chemical doping introduces chemical complexity and disorder, pressure is a clean and effective approach for tuning the interactions among multiple degrees of freedom, and superconductivity has been found in many materials via this route29,30,31. Among the CaP3 family of materials, CaAs3 was proposed to host a single nodal loop due to time reversal, spatial inversion, and accidental degeneracies32. Li et al.33 reported its transport properties under hydrostatic pressure up to 2.09 GPa, finding a decrease in the resistivity and a possible superconducting transition under pressure. Since SrAs3 has already been demonstrated to be a TNLSM, the lack of any high-pressure report inspired us to explore its pressure dependence.
In this work, we present the results of high-pressure measurements on single-crystalline SrAs3. Upon applying pressure, the topologically protected α pocket and trivial β pocket disappear around 1 GPa, and two higher frequencies denoted as ε and ξ emerge, indicating a Lifshitz transition. More interestingly, a superconducting transition has been observed from 20.6 GPa, with a dome-like pressure dependence. High-pressure X-ray diffraction (XRD) was conducted to investigate the high-pressure structure of SrAs3, and a structural transition was found around 20 GPa. DFT calculations on the high-pressure structure of SrAs3 reveal a TCI state. TCI states have previously been experimentally verified only in narrow-gap IV–VI semiconductors with a rock-salt structure, for example SnTe34 and Pb1−xSnxM (M = Se, Te)35,36. Thus, the observation of a pressure-induced TCI state in SrAs3 offers an alternative route to explore this exotic state. Moreover, the finding of superconductivity in this state makes high-pressure SrAs3 a candidate TSC.
Results
SrAs3 crystallizes in a triclinic (space group \(P\bar 1\)) or monoclinic (space group C2/m) structure; the latter is proposed to possess topological-nodal-line states protected by time-reversal symmetry, spatial-inversion symmetry, and mirror symmetry20. Figure 1a shows the unit cell of monoclinic SrAs3. This crystal structure can be viewed as a stack of two-dimensional (2D) infinite polyanionic layers \({}_\infty ^2\left[ {{\mathrm{P}}_3} \right]^{2 - }\) along the b-axis21. The As layers form channels and the Sr cations are inserted into the channels, as shown in Fig. 1b21. The inset in Fig. 1c shows the XRD rocking curves of SrAs3 single crystals grown from both Bi flux (which we refer to as BF) and self flux (SF). From an X-ray rocking curve of the (002) Bragg peak, a full width at half maximum (FWHM) of 0.04° indicates the high quality of the SrAs3 single crystal grown from Bi flux, while a broader FWHM of 0.15° for the SF sample suggests lower quality. In resistivity measurements (Fig. 1c), the SF sample exhibits semimetallic behavior, while the BF sample exhibits metallic behavior with a residual resistivity ratio RRR = ρ(300 K)/ρ0 of 5, and a residual resistivity ρ0 of 72.7 μΩ cm, which are further indicative of its high quality. For magnetotransport measurements, the higher-quality BF-grown samples were chosen.
Figure 2a shows the temperature-dependent resistivity from 1.8 to 300 K at pressures up to 1.47 GPa on a BF-grown SrAs3 single crystal. Upon increasing the pressure from 0.14 to 1.18 GPa, the low-temperature resistivity becomes increasingly metallic. The MR, shown in Fig. 2b, is non-monotonic in pressure. Upon increasing the pressure from 0.14 to 1.18 GPa, the MR at 9 T and 1.8 K increases from 5000 to 32,500%. However, further increasing the pressure to 1.47 GPa reduces the MR to 22,000%. The oscillatory component ΔRxx, plotted in Fig. 2c, exhibits low-frequency modes for lower pressures and higher-frequency modes at higher pressures, indicating a significant change in the electronic structure at the Fermi surface. Fast Fourier transforms of the MR oscillations, displayed in Fig. 2d contain only the α (1.4 T) and β (5.5 T) pockets from 0.14 to 0.69 GPa, as previously seen at ambient pressure25. At 0.99 GPa, the α and β pockets abruptly disappear, replaced by a single frequency of 21.5 T which we assign to a ξ pocket. Upon increasing the pressure to 1.47 GPa, the ξ-frequency pocket is joined by an even higher frequency of 48.3 T which grows rapidly to 63.2 T, which we assign to an ε pocket.
Figure 2e summarizes the pressure-dependent oscillation frequencies, with shading identifying the transition where the Fermi surface changes. The two obvious scenarios for this abrupt transition are a structural transition or a Lifshitz transition. But high-pressure diffraction, discussed in more detail below, indicates that the ambient-pressure structure persists to ~20 GPa, excluding a structural origin. In topological materials, different types of Lifshitz transitions are possible, involving other types of zeroes in the energy spectrum in addition to or instead of the Fermi surface, such as flat bands, Weyl and Dirac nodes, Dirac nodal lines, zeroes in the spectrum of edge states, Majorana modes, etc.37. For example, in rhombohedral graphite, the Lifshitz transition is related to the change of the shape of the Dirac line37. To study the relationship between the Lifshitz transition and the topologically protected nodal-line structure in SrAs3, a Landau level index fan diagram for the pressure-induced pockets is plotted in Fig. 2f, where no Berry phase has been observed for either the ε or ξ pocket. To produce this diagram, we assign integer indices to the ∆Rxx peak positions in 1/B and half-integer indices to the ∆Rxx valleys. Infinite-field intercepts around 3/8 to 5/8 indicate trivial band topology. In SrAs3, the α pocket is topologically protected, and the nodal line is situated around the Υ point in the BZ24. Hence, we surmise that the Fermi surface of SrAs3 is reconstructed below 1.0 GPa, in a Lifshitz transition which involves the topologically protected nodal-line structure.
As a TNLSM, SrAs3 possesses 2D drumhead-like nearly flat surface states nested inside the closed nodal line, and it may provide an ideal playground for many interaction-induced nontrivial states, such as superconductivity. In the resistivity under higher pressures in Fig. 2a, there is a clear suppression of the low-temperature resistivity. Pressure-induced superconductivity in topological semimetals often appears as the extremely large MR is suppressed31. Given the significant reduction in MR from 1.18 to 1.47 GPa, we suspected that higher pressures may induce superconductivity. Since bismuth displays superconducting transitions from 2.5 to 10.1 GPa with Tc changing from 3.9 to 8.55 K38,39, self-flux-grown SrAs3 crystals were studied to completely eliminate the possibility of Bi flux inclusions (the results of electrical transport measurements on the Bi-flux-grown SrAs3 samples (as shown in Fig. S1) and discussions are only displayed in the Supplementary Information file). Figure 3a shows the resistance curves for SrAs3 under higher pressures. No superconducting transition is observed till 20.6 GPa. At 20.6 GPa, a superconducting transition with \(T_c^{10\% }\) of 3.6 K appears, and the Tc increases to 5.8 K at 54.7 GPa. Upon further pressurization to 63.6 GPa, the Tc decreases slightly to 5.5 K. The pressure dependence of the superconducting transition, summarized in Fig. 3c, is clearly dome-shaped. To verify that this is a superconducting transition, the effect of magnetic field at 39.4 GPa was studied, as plotted in Fig. 3b—the transition is gradually suppressed by magnetic field, as expected for superconductivity. Figure 3d plots the temperature dependence of the upper critical field (μ0Hc2). We used the Ginzburg–Landau (GL) formula μ0Hc2(T) = μ0Hc2(0) (1 − (T/Tc) 2)/(1 + (T/Tc)2) to fit the data at 39.4 GPa; the μ0Hc2(0) values are estimated to be 2.58(2), 2.00(2), and 1.48(3) T for \(T_c^{10\% }\), \(T_c^{50\% }\), and Tczero, respectively, yielding coherence length ξGL(0) of 11.3, 12.9, and 14.9 nm. These fields are much lower than the Pauli limiting fields40,41,42 HP (0) = 1.84Tc ~9.5, 7.9, and 6.9 T, respectively, indicating that Pauli pair breaking is not relevant.
To verify whether the pressure-induced superconductivity arises from a structural phase transition, we performed high-pressure synchrotron XRD measurements on the self-flux-grown samples, shown in Fig. 4a for different pressures. The ambient-pressure (denoted as AP) phase persisted up to ~20 GPa. New diffraction peaks (marked with a dashed line and asterisk) appear at ~23.3 GPa, indicating a pressure-induced structural phase transition, and these continue to strengthen upon further increasing pressure. Figure 4b depicts the crystal structure of SrAs3 under ambient conditions. There are two Wyckoff positions (4i and 8j) for As atoms in each unit cell, and Sr (4i) atoms are coordinated to seven As atoms. The crystal structure of the ambient-pressure phase below 20.4 GPa was refined according to the initial model determined by room temperature single-crystal XRD. Typical GSAS refinements for SrAs3 under 3.6 GPa are illustrated in Fig. 4c. In general, applying pressure tends to increase the coordination number (CN) of cations, the so-called “pressure-coordination” rule, as seen for instance in Bi2Se3, where R-3m (CN = 6) transforms to C2/m (CN = 7) and eventually I4/mmm (CN = 8)41 with pressure. A further effect, the “corresponding static principle”, states that compounds containing light elements tend at higher pressures to adopt the structures observed at lower pressures in analogous compounds with heavy elements of the same group. A typical example is found in the binary antimonides Li3X (X = N, P, As, Sb, and Bi): Li3N crystallizes in the hexagonal P6/mmm structure at ambient pressure and transforms into the hexagonal P63/mmc structure (which is the ambient-pressure structure of Li3P, Li3As, and Li3Sb) around 0.5 GPa. It then undergoes a second structural phase transition to a cubic Fm–3m structure (the structure of Li3Bi at ambient pressure) around 36–45 GPa43.
At ambient conditions, SrP3 and SrAs3 have the same crystal structure (C2/m), while SrBi3 forms in the cubic Cu3Au-type structure (space group: Pm-3m, No. 221)44. Thus the corresponding static principle suggests the Cu3Au structure type as a strong candidate for the high-pressure structure. The crystal structure model for high-pressure SrAs3 was deduced by testing several candidates, and was ultimately refined with a SrBi3-like structure (space group: Pm-3m, No. 221)44. The schematic crystal structure of the high-pressure phase is depicted in Fig. 4d. Sr and As atoms occupy 1a (0, 0, 0) and 3c (0, 0.5, 0.5) Wyckoff positions, respectively, and the CN for Sr has increased to 12. Figure 4e shows the Rietveld refinement of SrAs3 under 49.8 GPa, yielding 70% and 30% for the ambient and high-pressure phases, respectively. The coexistence of high and ambient-pressure phases up to 51.9 GPa indicates that the pressure-induced structural phase transition in SrAs3 is first order and that the two phases have only a minute difference in Gibbs free energy.
Since the pressure-induced superconductivity appears long after the topologically protected α pocket is eliminated, it is important to check whether the high-pressure band structure of SrAs3 is topologically nontrivial, which is a necessary condition for topological superconductivity. To obtain more electronic structure information on the high-pressure phase of SrAs3, we performed DFT calculations for pressurized SrAs3 at 34 GPa, as summarized in Fig. 5. The band structure in the absence of spin–orbit coupling (SOC), shown in Fig. 5a, displays metallic behavior. The electron-dominant valence and hole-dominant conduction bands cross near the Fermi level along the Γ–Μ line. Owing to the cubic symmetry of this material, there are twelve band crossing points at symmetrically equivalent points in the full BZ. Upon turning on SOC, a gap will be opened at these crossing points, resulting in a continuous SOC gap with a curved chemical potential between valence and conduction bands at each Κ point, as shown in Fig. 5b.
To identify the topological nature of this material, the Fu–Kane parity criterion45,46 at eight time-reversal invariant momenta (TRIM) was utilized to determine the Z2 index. We obtain a trivial Z2 of (0;000) from the production of the parities of all occupied bands at the eight TRIM points, as shown in Fig. 5c. However, the surface states on the (001)-projected surface contain two surface Dirac cones located at \({\bar{\Gamma}}\) points, as seen in Fig. 5d. Because of SOC, these Dirac states host a helical spin texture as shown in Fig. 5f. The Wilson loop method was employed to determine the mirror Chern number (MCN), and get MCN = 1, in agreement with the surface state behavior observed on the (001) surface45,46,47. The continuous SOC gap, topologically trivial Z2 index, nontrivial MCN and even number of surface Dirac points indicate that this material is a TCI44,45,46. As a counterpart of topological insulators in which crystalline symmetry replaces time-reversal symmetry to enforce topological protection, TCIs possess topological surface states (TSSs) with an even number of gapless Dirac cones on the surface BZ and host a variety of exotic phenomena47, for example, large-Chern-number quantum anomalous Hall effect48 or strain-induced superconductivity49. The superconductivity in high-pressure SrAs3 may exist in or be induced in these surface states. Owing to their helical spin texture, any superconducting phase in these surface states would most likely be topologically nontrivial34,35,36,47,48,49, making high-pressure SrAs3 a strong candidate for TSC.
Discussion
Searching for Majorana fermions has been fueled by the prospect of using their non-Abelian statistics for robust quantum computation, and they can be realized as a bound state at zero energy, i.e., Majorana bound states, in the vortex core of a TSC27,28. To realize a TSC, two primary routes have been proposed, i.e., bulk spin-triplet superconductivity, and superconductivity in spin-nondegenerate TSSs induced by the proximity effect, for instance through heterostructures stacking conventional s-wave superconductors and topological insulators, quantum anomalous Hall insulators, nanowires, or atomic chains27,28,50. In the former case, due to the breaking of spin degeneracy by asymmetric SOC, the parity-mixed superconducting state in non-centrosymmetric superconductors may also host Majorana fermions if the singlet component is smaller than the triplet component51. Recently, the layered non-centrosymmetric compound PbTaSe2 with strong SOC was reported to possess fully gapped multiband superconductivity52. However, its spin-triplet component is small or absent53,54. Bulk topological nodal-line states and fully spin-polarized TSSs have been also demonstrated in PbTaSe213, which may allow proximity-induced fully-gapped superconducting TSSs, which could pair in px + ipy symmetry and host bound Majorana fermions in the vortices14.
The superconducting phase of SrAs3 is centrosymmetric (Pm-3m), so barring the exceedingly unlikely possibility of bulk spin-triplet superconductivity, the bulk route to Majorana quasiparticles is not available. However, we propose a TCI state hosting TSSs with helical spin texture. Superconductivity in these states or induced in these states by proximity effect from the bulk could potentially be topologically nontrivial. However, unlike in other TCIs such as SnTe or Pb1−xSnxM (M = Se, Te), trivial bulk bands in pressurized SrAs3 cross the Fermi level, and we are unable to distinguish which bands participate in the superconductivity. Further work will be required to elaborate the contributions from TSSs and bulk states to the superconductivity.
In summary, at ambient pressure, SrAs3 is a TNLSM with 2D drumhead-like nearly-flat surface states, which may be strongly correlated and are often associated with the enhancement of superconductivity55, although pressure did not succeed in driving this material superconducting before changing the electronic structure. A Lifshitz transition has been identified below 1.0 GPa, evidencing a topological phase transition, as the quantum oscillations associated with the TNLSM state vanish. Higher-pressure experiments on a powder sample reveal a dome-like superconducting transition accompanying a structural phase transition into a phase which we predict to host a topological-crystalline-insulator state with TSSs and helical spin texture. Besides its intrinsic interest as a TNLSM, SrAs3 offers an alternative route to explore the topological-crystalline-insulator state beyond IV–VI semiconductors and, as a superconducting TCI, high-pressure SrAs3 could serve as a candidate TSC. Doping studies or strain may still be able to induce superconductivity in the low-pressure phase, and should be pursued, and the evolution of the drumhead-like states with pressure remains to be clarified.
Methods
Sample synthesis
Self-flux method
Sr (99.95 %, Alfa Aesar), and As (99.999 %, PrMat) were mixed in a molar ratio of 1:3 and placed into an alumina crucible. The crucible was sealed in a quartz ampoule under vacuum and subsequently heated to 750 °C in 10 h. After reaction at this temperature for 300 h, the ampoule was cooled to 400 °C in 50 h and cooled freely to room temperature. SrAs3 single crystals with black shiny metallic luster were obtained.
BF method
Sr (99.95%, Alfa Aesar), As (99.999%, PrMat) and Bi (99.9999%, Aladdin) blocks were mixed in a molar ratio of 1:3:26 and placed into an alumina crucible. The crucible was sealed in a quartz ampoule under vacuum and subsequently heated to 900 °C in 15 h. After reaction at this temperature for 20 h, the ampoule was cooled to 700 °C over 20 h, and then slowly cooled to 450 °C at 1 °C/h. The excess Bi flux was then removed in a centrifuge, and SrAs3 single crystals with black shiny metallic luster were obtained.
Pressure measurements
Resistance measurements under pressure
For high-pressure experiments, samples were loaded in a piston-cylinder clamp cell made of Be–Cu alloy, with Daphne oil as the pressure medium. The pressure inside the cell was determined from the Tc of a tin wire. A SrAs3 single crystal was cut into a bar shape, and the standard four-probe method was used for resistivity measurements, with contacts made using silver epoxy. Higher-pressure measurements were performed on powder samples comprising crushed single crystals using a diamond anvil cell (DAC). The experimental pressures were determined by the pressure-induced fluorescence shift of ruby56 at room temperature before and after each experiment. A direct-current van der Pauw technique was adopted. Resistance measurements were performed with a physical property measurement system, Quantum Design.
XRD measurements under pressure
SrAs3 single crystals grown by the self-flux method were ground into fine powder in a mortar for use in the high-pressure synchrotron angle dispersive XRD (AD-XRD) measurement. The high-pressure synchrotron XRD experiments were carried out using a symmetric DAC with a 260-micron culet diamond. A rhenium gasket was precompressed to 30 microns in thickness followed by drilling the central part by laser to form a 90-micron diameter hole as the sample chamber. The sample chamber was filled with a mixture of the sample, a ruby chip, and silicone oil as the pressure transmitting medium. The experimental pressures were determined by the pressure-induced fluorescence shift of ruby56. Synchrotron AD-XRD measurements were carried out at beamline BL15U1 of the Shanghai Synchrotron Radiation Facility (SSRF) using a monochromatic beam of 0.6199 Å.
Density functional theory (DFT) calculations
DFT calculations were performed based on the Perdew–Burke–Ernzerhof-type generalized gradient approximation57, and used the projector augmented wave method58, as encoded in the Vienna ab inito simulation package59. The cutoff energy for the plane-wave basis taken was 500 eV. The first Brillouin Zone was sampled, using a Γ-centered 12 × 12 × 12 k-point mesh. The energy convergence criteria were defined as 10−8 eV. The lattice constants were fully relaxed using a conjugate gradient scheme until the Hellmann-Feynman forces on the ions were less than 0.001 eV/Å. We constructed the maximally localized Wannier functions60,61,62 using Sr d and As s and p atomic orbitals. The topological features of surface state spectra were calculated using the iterative Green’s function technique63, as implemented in the Wannier-Tools package64.
Data availability
All data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The computer codes used to carry out the DFT calculations in this work are available upon request from the corresponding author.
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Acknowledgements
This work is supported by the Ministry of Science and Technology of China (Grant nos. 2015CB921401 and 2016YFA0300503), the Natural Science Foundation of China (Grant nos. 11421404, 11674367, and 11874264), the NSAF (Grant no. U1630248), the Natural Science Foundation of Shanghai (Grant no. 17ZR1443300), and the Zhejiang Provincial Natural Science Foundation (Grant no. LZ18A040002). W.W.Z. is supported by the Shenzhen Peacock Team Plan (KQTD20170809110344233) and Bureau of Industry and Information Technology of Shenzhen through the Graphene Manufacturing Innovation Center (201901161514). D.C.P. is supported by the Chinese Academy of Sciences through 2018PM0036. Y.F.G. acknowledges the research fund from the State Key Laboratory of Surface Physics and Department of Physics, Fudan University (Grant no. KF2019_06). Y.F.G. and X.F.K. are also funded by the strategic Priority Research Program of Chinese Academy of Sciences (Grant no. XDA18000000).
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S.Y.L. and Y.F.G. conceived the idea and designed the experiments. E.J.C. was responsible for high-pressure transport experiments. W.X., Y.Z., X.W., Z.Q.Z., H.S., and X.F.K. performed sample synthesis and partial data analysis. Z.H.Y., N.Y., L.W., L.M.Y., and W.G.Y. conducted the high-pressure X-ray diffraction measurements and structure analysis. C.C.Z. and D.Z.D. helped the data collection. S.Y.L., Y.F.G., and W.W.Z. supervised the project. E.J.C., D.C.P., Y.F.G., and S.Y.L. analyzed the data and wrote the paper. E.J.C., W.X., and X.B.S. contributed equally to this work. All authors discussed the results and commented on the paper.
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Cheng, E., Xia, W., Shi, X. et al. Pressure-induced superconductivity and topological phase transitions in the topological nodal-line semimetal SrAs3. npj Quantum Mater. 5, 38 (2020). https://doi.org/10.1038/s41535-020-0240-6
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DOI: https://doi.org/10.1038/s41535-020-0240-6
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