Abstract
Nonsymmorphic chiral topological crystals host exotic multifold fermions, and their associated Fermi arcs helically wrap around and expand throughout the Brillouin zone between the highsymmetry center and surfacecorner momenta. However, Fermiarc splitting and realization of the theoretically proposed maximal Chern number rely heavily on the spinorbit coupling (SOC) strength. In the present work, we investigate the topological states of a new chiral crystal, PtGa, which has the strongest SOC among all chiral crystals reported to date. With a comprehensive investigation using highresolution angleresolved photoemission spectroscopy, quantumoscillation measurements, and stateoftheart ab initio calculations, we report a giant SOCinduced splitting of both Fermi arcs and bulk states. Consequently, this study experimentally confirms the realization of a maximal Chern number equal to ±4 in multifold fermionic systems, thereby providing a platform to observe largequantized photogalvanic currents in optical experiments.
Introduction
The discovery of topological insulators reinvigorated the understanding of the electronic band structure and inspired generalization of the topological band theory concerning solid states^{1,2,3,4,5}. This led to the discovery of quasiparticle excitations of the Dirac and Weyl fermions within solidstate materials characterized by a linear band crossing in metals along with the creation of a direct analogy between the said fermions and fundamental particles in highenergy physics^{6,7,8,9,10,11,12,13,14,15}. On the other hand, quasiparticles within electronic band structures need not necessarily follow the Poincare symmetry pertaining to highenergy physics. Instead, they adhere to the crystal symmetry such that new types of fermionic excitations can be realized within solid states without having counterparts in highenergy physics^{16,17,18,19,20}.
Multifold fermions protected by chiral crystal symmetry attracted extensive attentions recently^{16,21,22,23,24,25,26}. In comparison with Dirac fermion with zero topological charge and Weyl fermions with Chern number ±1, multifold fermions in chiral crystals host large Chern numbers and chiral Fermi arcs on their surface states (SSs). Since these symmetryenforced multifold fermions locate at highsymmetry timereversal invariant momenta, realization of longsurface Fermi arcs expanding throughout the Brillouin zone (BZ) becomes topologically guaranteed. These Fermi arcs are orders of magnitude larger and highly robust compared with those in any other Weyl semimetal. This affords a natural advantage over twofold degenerate Weyl fermions with regard to detection of Fermiarc states. Identification of multifold fermions with large Chern numbers has previously been performed via observation of surface chiral Fermi arcs using angleresolved photoemission spectroscopy (ARPES)^{21,22,23,24,25} as well as the remarkable quantized circular photogalvanic effect^{26}.
According to theoretical predictions, the largest topological charge from multifold fermions has a Chern number ±4 hosted by compounds with space groups P2_{1}3 (No. 198), especially in compounds such as CoSi, RhSi, PtAl, CoGe, RhGe, PdGa, etc.^{16,21,22,23,24,26}. Protected by this topological Chern number of ±4, there exist four Fermi arcs crossing surfaceprojected multifold fermions at highsymmetry points. However, given the small spin splitting of Fermi arcs, all ARPES measurements performed to date have only confirmed the existence of chiral Fermi arcs connecting projected multifold fermions. That is, the exact number of Fermi arcs that exist has yet remained unclear. In other words, the theoretical Chern number of ±4 has so far not been experimentally verified by any surfacedetection experiment. Realization of large spin splitting of the Fermi arc requires a very strong spinorbit coupling (SOC) in the chiral crystals. Among all chiral multifold fermionic materials investigated thus far, PtGa demonstrates the strongest SOC. This paper reports results obtained using a combination of highresolution ARPES, quantum oscillations, and stateoftheart ab initio calculations to illustrate the giant spin splitting of topological states within a new chiral topological semimetal PtGa.
Results
Crystal growth and electrical transport measurement
In this study, highquality PtGa singlecrystals were grown using the selfflux technique, as discussed in Methods and Supplementary Note 1. The crystal symmetry was confirmed via rigorous singlecrystal diffraction analysis. The estimated Flack parameter value of 0.03(4) indicates the existence of a single structural chirality in the crystal. The samples demonstrated metallic behavior throughout the measured temperature range (Supplementary Fig. 2). The observed large residual resistivity ratio (RRR = ρ(300 K)/ρ(2 K)) of ~84 and giant magnetoresistance of roughly 1000% at 2 K (Supplementary Fig. 2) reflect the high quality of the crystals. Values of the carrier concentration and mobility were estimated to be 2.1 × 10^{21} cm^{−3} and 4650 cm^{2} V^{−1} s^{−1}, respectively, using fielddependent Hallresistivity data (Supplementary Fig. 3) and the longitudinal resistivity at 2 K.
Crystal structure and electronic structure
PtGa crystallizes in the nonsymmorphic chiral space group P2_{1}3 (no. 198) with lattice parameter a = 4.9114(3) Å (Fig. 1a). The corresponding simple cubic BZ is depicted in Fig. 1c with highsymmetric momenta—Γ, X, M, and R. Results of ab initio calculations and (001) Fermisurface (FS) projection demonstrate the existence of hole and electron pockets at \(\overline \Gamma\) and \(\overline {\mathrm{M}}\), respectively, resulting in Fermi arcs to span over the entire BZ surface. Subsequently, giant spin splitting of the said Fermi arcs was observed post SOC incorporation, as depicted in Fig. 1d. Consequently splitting also occurs for the bulk bands. Owing to the crystal symmetry, a threefold degenerated point, the location of which coincides with that of Γ, follows the spin1 Hamiltonian and hosts a topological charge of Chern number +2. Likewise, band degeneracy at the location of momentum R yields a doubleWeyl fermion with Chern number −2, thereby causing the entire system to follow the “nogo theorem” (Fig. 1c, e, f). Considering SOC, the spin1 fermionic excitation at Γ splits into a doublydegenerated Weyl and fourfold degenerated Rarita–Schwinger–Weyl fermions with topological charges of Chern number +4. Meanwhile, the doubleWeyl fermions at R transform into sixfold fermionic points with Chern numbers −4 as well as a trivial doubledegenerated point (Fig. 1b, d–f). Therefore, it can be inferred that PtGa serves as an ideal platform to visualize the effect of strong SOC on quasiparticle excitations of multifold fermions in chiraltopology semimetals.
Fermisurface topology from quantum oscillation
Quantum oscillations in singlecrystalline PtGa were investigated in this study to detect spinsplit FS pockets due to strong SOC, as illustrated in Fig. 1b, e, and f. Magnetic field up to 7 T was used to perform isothermal magnetization measurements at various temperatures. Results obtained for the case B[001] are depicted in Fig. 2a. Clear de Haas–van Alphen (dHvA) quantum oscillations were observed starting, at 1.8 K, from a field of ~0.5 T, thereby indicating the high quality of the grown single crystal and low effective mass of the associated charge carriers. In addition, a low Dingle temperature of ~2.64 ± 0.19 K was observed (Supplementary Fig. 7). Interestingly, characteristics of quantum oscillations observed for cases B[001], B[110], and B[111] were radically different, thereby reflecting the high anisotropy of the PtGa Fermi surface (Supplementary Figs. 4, 5). A smooth background was subtracted from the measured magnetization data and oscillations periodic in 1/B are clearly resolved up to 15 K (Fig. 2b and Supplementary Fig. 6). We analyzed the corresponding fast Fourier transformation (FFT) to determine the dHvA frequencies and, for simplicity, only the 2K data are depicted in Fig. 2d. Corresponding dHvA oscillations and FFT results obtained for all intermediate temperatures are shown in Supplementary Fig. 6.
The different dHvA frequencies in the FFT results were ascribed to extremal areas of the FS by constructing the full FS with the help of ab initio calculations, details of that are discussed in the Methods section. Evidently, the energy bands of PtGa get spinsplit owing to the strong SOC and noncentrosymmetric crystal structure and this results in the emergence of FS pairs having similar shapes but largely different sizes. The complete FS along with the BZ is illustrated in Fig. 2e, revealing that the FSs are mainly spread around the highsymmetry locations Γ, R, and M. In this study, all dHvA frequencies obtained via FFT could be easily matched with calculated kspace extremal FS crosssection areas with their normal vectors along the zaxis. For simplicity, however, FSs corresponding to only the Γpoint are depicted in the inset of Fig. 2d. The identified FSs for the other FFT frequencies are presented in Supplementary Fig. 8. The fourfold band crossing at the Γpoint generates two electron types nearspherical FSs (with extremal frequencies of 31.9 T for the α_{1} and 93.96 T for the α_{2} orbit) and two squareboxshaped FSs. For each of this latter FS, two extremal crosssections were determined that match well with the experimental FFT results in the frequency range depicted in Fig. 2d (\({\upbeta}_1\) at 113.22 T and \({\upbeta}_2\) at 259.14 T; \({\upgamma}_1\) at 492.6 T and \({\upgamma}_2\) at 679.37 T). The observed large frequency difference of 187 T for the latter pair clearly reveals the giant spin splitting of FSs due to strong SOC. The effective mass m* of the various spinsplit Fermi pockets is estimated from the temperature dependence of the corresponding dHvA frequencies (Fig. 2c) using the thermal damping factor of the Lifshitz–Kosevich formula, \(R_{\mathrm{T}} = X/ \!\sinh (X)\). Here, \(X = 14.69\,m \ast \, T/B\) and B is the magnetic field averaged over 1/B. The extremal crosssectional areas A_{F}, Fermi wave vector k_{F} and Fermi velocity v_{F} of the FS pockets shown in Fig. 2d are estimated using the Onsager relation F = (Φ_{0}/2π^{2})A_{F}, where Φ_{0} = h/2e (=2.068 × 10^{−15} Wb) is the magnetic flux quantum with h the Planck constant; \(k_{\mathrm{F}}={\sqrt{A_{\mathrm{F}}/{\uppi}}}\), v_{F} = k_{F}h/2πm*. The estimated m*, A_{F}, k_{F}, and v_{F} values of various spinsplit Fermi pockets are summarized in Supplementary Table 2. Interestingly, the cyclotron masses of the FS pockets in PtGa are much lighter compared with the chiral sister compound CoSi (~1.2 m_{0}) and are similar to the wellknown Weyl semimetallike TaAs^{27,28}.
Angleresolved photoemission spectroscopy
Using lowenergy highresolution ARPES on the highquality single crystal, we investigate the electronic band structure of PtGa. An FS intensity plot was obtained for the 1st BZ along with ARPES intensity plots along highsymmetry directions on the (001) surface, as depicted in Fig. 3a and c. The calculated FS is presented in Fig. 3b, whereas, the band structure combining surface and bulk states are shown in Fig. 3d. By comparing the ARPES spectrum against results obtained from ab initio calculations—including the electron band at \(\overline \Gamma\) and electronic pocket at \(\overline {\mathrm{M}}\)—it is evident that the ARPES spectrum is well reproduced by calculated SSs. Owing to the relatively low photon energy, no bulk states were observed in the ARPES spectrum. As observed via our calculations, four spinsplit surface bands correspond to Fermiarcrelated states that originate from the \(\overline \Gamma\) point. Two of these bands (green lines) extend to the \(\overline {\mathrm{M}}\) point at the right side, while the other two (blue lines) connect the left \(\overline {\mathrm{M}}\) point. The experimental data are in good agreement with the calculations. In Fig. 3c, four crossing points are observed between \(\overline {\mathrm{M}}\)–\(\overline \Gamma\)–\(\overline {\mathrm{M}}\), as indicated with orange arrows along the white line of Fig. 3a. Each crossing point contains two spinsplit Fermi arcs. However, due to the finite ARPES resolution, the spin splitting of the Fermi arcs are difficult to distinguish along the \(\overline {\mathrm{M}}\)–\(\overline \Gamma\)–\(\overline {\mathrm{M}}\) direction. Compared with the experiments, the calculated Fermi arcs have much more twisted paths, resulting more crossing points.
Spinsplit Fermi arc
We closely look into the PtGa SSs to realize the giant SOCinduced spin splitting of the Fermi arc. On the (001) surface, the Fermi arc was observed to cross the entire BZ that helically wraps around the two \(\overline {\mathrm{M}}\) point with opposite chirality. Figure 4a depicts a ‘z’shaped Fermi arc with 60 eV photon energy, connecting the electron pocket on upper right corner and the one on the left bottom. We present four constant energy maps in Fig. 4b to highlight the split Fermi arcs. The corresponding calculated constant energy maps at different binding energies are presented in Supplementary Fig. 9. In panel (i) of Fig. 4b, two arcs separate near \(\overline {\mathrm{M}}\) at \(\Delta E = E  E_{\mathrm{F}} =  0.1\,{\mathrm{eV}}\). With decreasing binding energy, the inner arc (orange) detaches from the outer Fermi arc (yellow) and the separation becomes larger, as shown in (ii) of Fig. 4b with ΔE = −0.2 eV. Further decrease of the binding energy with ΔE = −0.3 eV, the orange inner Fermi arc disappears as depicted in panel (iii). Finally, the yellow Fermi arc becomes shorter with ΔE = −0.4 eV in panel (iv) of Fig. 4b. To demonstrate the splitting of the Fermi arc, we acquired two ARPES spectra along two cuts, as indicated by the white and green lines in Fig. 4a. Because of the strong SOC in PtGa, we observed a distinct splitting of the Fermi arc from the band dispersion. The largest energy difference of the splitting in Fig. 4c and d is ~0.2 eV. We also calculated the surface spin texture for \(\Delta E = E  E_{\mathrm{F}} =  0.1\,{\mathrm{eV}}\) as shown in Supplementary Fig. 10. It is evident that each pair of neighboring surface states shows different spin textures, with almost opposite spin orientations. This indicates that each pair of neighboring surface states are split from one state.
In Fig. 4e and f, we show the band structures along loop 1 and 2, as indicated in Fig. 4a. The loop 1 enclosing \(\overline {\mathrm{M}}\), presents two rightmoving surface band crossings E_{F}, as indicated with black arrows. The band splittings is clearly observed in Fig. 4c, d and also shown in Fig. 4e. Therefore, each crossing contains two surface bands, suggesting a chiral charge of the \(\overline {\mathrm{M}}\) point as C = 4. The \(\overline \Gamma\) point is enclosed by loop 2, which shows sixband crossings, including four leftmovings and two rightmovings. Here the right and leftmoving crossings denote opposite chiral charge. Therefore, one pair of right and leftmoving crossings cancels out and does not contribute to the chiral charge. Since each crossing contains two spinsplit Fermi arcs, the net crossing count along loop 2 is four; and the resulting chiral charge of the \(\overline \Gamma\) point is C = 4. This is the first experimental observation of SOCinduced spinsplit Fermi arcs and the verification of the maximal Chern number of 4 in topological chiral crystals. Since multifold fermions are a generic feature of all the chiral topological crystals with no. 198 SG, there, indeed, exist four Fermiarc SSs connecting the \(\overline \Gamma\) and \(\overline {\mathrm{M}}\) points.
Discussion
In conclusion, this paper presents a comprehensive investigation performed using highresolution ARPES, quantumoscillation measurements, and stateoftheart ab initio calculations to examine a giant SOCinduced splitting of Fermi arcs and bulk states. Owing to its large SOC, chiral PtGa demonstrates strong spin splitting, as observed via both dHvA quantumoscillation analysis and surface ARPES measurements. The splitting of Fermi arcs connecting the timereversal invariant points of \(\overline \Gamma\) and \(\overline {\mathrm{M}}\) directly confirms a Chern number of ±4 for chiral multifold fermions that exist in this class of topological materials. Thus, the proposed study confirms realization of a maximal Chern number equal to ±4 for the first time in multifold fermionic systems. SOC can be considered as an effective parameter for tuning the sharpness of surface Fermi arcs, thereby paving the way for further observation and exploration of Fermiarcrelated phenomena in multifold chiral fermions.
Methods
PtGa crystal growth and structural refinement
PtGa single crystals were grown from the melt using the selfflux technique. A polycrystalline ingot was first prepared by arc melting stoichiometric amounts of constituent metals with 99.99% purity in an argon atmosphere. Subsequently, the singlephase ingot was crushed, placed in an alumina crucible, and sealed within a quartz tube. The assembly was then melted inside a commercial boxtype furnace at 1150 °C and maintained at that temperature for 10 h to ensure homogeneous mixing of the melt. The sample was then slowly cooled to 1050 °C at a rate of 1 °C/h followed by further cooling to 850 °C at a rate of 50 °C/h. Finally, the sample was annealed at 850 °C for 120 h prior to being cooled to 500 °C at a rate of 5 °C/h. As observed, the annealing process has a major impact on the quality of the grown crystal. In electrical transport, the RRR value significantly increases (up to ~80) with post annealing compared with the corresponding value (~24) for fastcooled single crystals. The obtained single crystal measured ~6 mm in diameter and 17 mm in length, as depicted in Supplementary Fig. 1. Single crystallinity of the grown crystal was first verified at room temperature using a whitebeam backscattering Laue Xray setup. Observation of of single single, sharp Laue spots clearly revealed the excellent quality of the grown crystal sans any twinning or domains. A representative Laue pattern, superposed with a theoretically simulated one is also depicted in Supplementary Fig. 1. Chemical composition of the PtGa crystal was verified via energydispersive Xray (EDX) spectroscopy, results of which demonstrated good agreement with the target composition of PtGa. To analyze structural chirality of the grown crystal, rigorous singlecrystal Xray diffraction experiments were performed, results of which are discussed in Supplementary Information Note 1 structural characterization section. The refined crystal structure confirmed Form A in ref. ^{29} only, as indicated by a Flack parameter value of ~0.03(4), which is indicative of a singlehanded domain.
Magnetic and electrical transport measurements
The magnetization measurements were performed using a commercial MPMS3 from Quantum Design. In electrical transport, the longitudinal and Hall resistivity were measured using a lowfrequency ACT option in a physical property measurement system (PPMS9T, Quantum Design).
Angleresolved photoemission spectroscopy
ARPES experiments were carried out at the Berliner Elektronenspeicherring für Synchrotronstrahlung (BESSY) (beamline UE112PGM1) with a Scienta Omicron R8000 analyzer, and at beamline 13U of the National Synchrotron Radiation Laboratory (NSRL) with a Scienta Omicron DA30 analyzer. The singlecrystalline samples were oriented and finely polished on the (001) surfaces. The samples were Arion sputtered for 30 min in ultrahigh vacuum chamber, and then annealed at 680 °C for 30 min.
Firstprinciples calculations
The electronic structure calculations were performed based on density functional theory (DFT) by using the fullpotential localorbital code^{30} with a localized atomic basis and fullpotential treatment. The exchange and correlation energies were considered in the generalized gradient approximation (GGA) level^{31}. We have projected the Bloch wave functions into the atomicorbitallike Wannier functions, and constructed the tight binding model Hamiltonian. With the tight binding model Hamiltonian, the SSs were calculated in a halfinfinite boundary condition using the Green’s function method^{32,33}.
Data availability
The supporting data that is used to illustrate the findings of this study are available on request from the corresponding authors K.M. and C.F. upon request.
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Acknowledgements
The work in MPICPFS Dresden was financially supported by the European Research Council (ERC) Advanced Grant No. 291472 “Idea Heusler” and 742068 “TOPMAT”; European Union’s Horizon 2020 research and innovation program (grant No. 824123 and 766566) and Deutsche Forschungsgemeinschaft (ProjectID 258499086 and FE 633/301). S.N.G. thanks the Alexander von Humboldt Foundation for fellowship. B.S., J.H., S.C., and J.W. acknowledge the support from the DFG through the WürzburgDresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC 2147, projectid 39085490), the ANRDFG grant ‘FermiNESt’, and by HLD at HZDR, member of the European Magnetic Field Laboratory (EMFL).
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K.M. conceived the idea. All the authors in the paper contributed to the intellectual content of this project. The work was guided by M.Y., K.M., Y.S., and C.F. PtGa single crystal was grown by K.M. and analyzed the crystal structure, chirality etc. with the help of H.B. The ARPES experiments were carried out and analyzed by M.Y. with the help of A.F., V.V., Z.S., and J.F. All the theoretical calculations and analysis were performed by Q.Y. and Y.S. K.M. carried out the quantumoscillation experiment and analyzed the data with the help of B.V.S., J.H., S.C., and J.W. The electrical transport expt. was carried out by K.M. The paper was written by M.Y., K.M., and Y.S. with the help of C.S., N.K., S.N.G., and C.F.
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Yao, M., Manna, K., Yang, Q. et al. Observation of giant spinsplit Fermiarc with maximal Chern number in the chiral topological semimetal PtGa. Nat Commun 11, 2033 (2020). https://doi.org/10.1038/s4146702015865x
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DOI: https://doi.org/10.1038/s4146702015865x
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