Topological chiral crystals with helicoid-arc quantum states

Abstract

The quantum behaviour of electrons in materials is the foundation of modern electronics and information technology1,2,3,4,5,6,7,8,9,10,11, and quantum materials with topological electronic and optical properties are essential for realizing quantized electronic responses that can be used for next generation technology. Here we report the first observation of topological quantum properties of chiral crystals6,7 in the RhSi family. We find that this material class hosts a quantum phase of matter that exhibits nearly ideal topological surface properties originating from the crystals’ structural chirality. Electrons on the surface of these crystals show a highly unusual helicoid fermionic structure that spirals around two high-symmetry momenta, indicating electronic topological chirality. The existence of bulk multiply degenerate band fermions is guaranteed by the crystal symmetries; however, to determine the topological invariant or charge in these chiral crystals, it is essential to identify and study the helicoid topology of the arc states. The helicoid arcs that we observe on the surface characterize the topological charges of ±2, which arise from bulk higher-spin chiral fermions. These topological conductors exhibit giant Fermi arcs of maximum length (π), which are orders of magnitude larger than those found in known chiral Weyl fermion semimetals5,8,9,10,11. Our results demonstrate an electronic topological state of matter on structurally chiral crystals featuring helicoid-arc quantum states. Such exotic multifold chiral fermion semimetal states could be used to detect a quantized photogalvanic optical response, the chiral magnetic effect and other optoelectronic phenomena predicted for this class of materials6.

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Fig. 1: Structural chirality and topological quantum chirality in CoSi and RhSi.
Fig. 2: Fermiology of the (001) surface electronic topology in CoSi and RhSi.
Fig. 3: Observation of topological chiral edge modes in CoSi.
Fig. 4: Observation of helicoid quantum arcs in CoSi.

Data availability

The data supporting the findings of this study are available within the paper, and other findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

Work at Princeton and Princeton-led ARPES measurements and theoretical work reported here were supported by the US Department of Energy under a Basic Energy Sciences grant (number DOE/BES DE-FG-02-05ER46200). M.Z.H. acknowledges a Visiting Scientistship at the Materials Science Division of Lawrence Berkeley National Laboratory. This research used resources of the Advanced Light Source, which is a DOE Office of Science Facility, under contract number DE-AC02-05CH11231. S.J. was supported by the National Natural Science Foundation of China (U1832214, 11774007), the National Key R&D Program of China (2018YFA0305601) and the Key Research Program of the Chinese Academy of Science (grant number XDPB08-1). H.L. acknowledges Academia Sinica, Taiwan for support under the Innovative Materials and Analysis Technology Exploration (AS-iMATE-107-11) programme. T.-R.C. is supported by the Young Scholar Fellowship Program of the Ministry of Science and Technology (MOST) in Taiwan, under the MOST Grant for the Columbus Program (MOST107-2636-M-006-004), National Cheng Kung University, Taiwan, and National Center for Theoretical Sciences (NCTS), Taiwan. C.F. acknowledges financial support by the ERC through Advanced Grant number 291472 (Idea Heusler) and 742068 (TOPMAT). The authors thank S. K. Mo and J. D. Denlinger for beamline support at the Advanced Light Source in Berkeley, California. We thank R. Dhall of the Molecular Foundry at LBL in Berkeley, California for assistance. M.Z.H. acknowledges support from the Miller Institute of Basic Research in Science at the University of California in Berkeley in the form of a Visiting Miller Professorship during the early stages of this work. We also thank G. Bian and H. Verlinde for discussions.

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Contributions

D.S.S. and M.Z.H. conceived the project. D.S.S., I.B., T.A.C. and N.A. conducted the ARPES experiments in consultation with S.-Y.X. and M.Z.H.; X.X., W.X., X.W., G.-Q.W. and S.J. synthesized and characterized the RhSi and CoSi samples; K.M., V.S. and C.F. synthesized the RhSi samples; G.C., C.-Y.H., T.-R.C. and H.L. performed the first-principles/density functional theory calculations; D.S.S., I.B. and T.A.C. performed the analysis, interpretation and figure development in consultation with J.-X.Y., G.C., N.A., D.M., S.S.Z., N.S., S.-Y.X. and M.Z.H.; D.S.S. wrote the manuscript in consultation with I.B., J.-X.Y., G.C., S.-Y.X. and M.Z.H.; M.Z.H. supervised the project.

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Correspondence to M. Zahid Hasan.

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Extended data figures and tables

Extended Data Fig. 1 Topological arcs in the structurally chiral crystal RhSi.

a, ARPES-measured Fermi surface and constant-binding-energy contours with an incident photon energy of 82 eV at 10 K. The BZ boundary is marked in blue. b, Three-dimensional bulk and two-dimensional surface BZ with higher-fold chiral fermions (±). The planes outlined in blue and the red cylinder are two-dimensional manifolds with indicated Chern number33 n. The cylinder enclosing the bulk chiral fermion at Γ has n = +2. c, Fermi arcs (orange) connect the projected chiral fermions. Energy dispersion cuts show that these two chiral edge modes are time-reversed partners propagating in opposite directions. d, Second-derivative Fermi surface with the straight and loop cuts of interest marked. e, ARPES spectrum along a loop \({\mathscr{M}}\), showing two right-moving chiral edge modes, suggesting that the fourfold chiral fermion at R carries Chern number −2. f, ARPES spectrum along cut I, showing a right-moving chiral edge mode. g, ARPES spectrum along cut II on the opposite side of the threefold chiral fermion at \(\bar{{\rm{\Gamma }}}\) (illustrated by the blue sphere), showing a left-moving chiral edge mode.

Extended Data Fig. 2 Helicoid arcs in RhSi.

a, Energy dispersion cut on an inner loop of radius 0.18π/a enclosing \(\bar{{\rm{M}}}\). b, Lorentzian fits (red traces) to the MDCs (blue dots) to track the observed chiral edge modes. c, d, Similar analysis to a and b, but for an outer loop of radius 0.23π/a. Black arrows show the Fermi velocity direction for the chiral edge modes. e, f, Top (e) and perspective (f) view of the helicoid dispersion extracted from the MDCs, plotted on two concentric loops, suggesting a clockwise spiral with decreasing binding energy. g, Constant-energy contours calculated ab initio, showing a consistent helicoid structure.

Extended Data Fig. 3 X-ray diffraction study.

Single-crystal X-ray diffraction precession image of the (0kl) planes in the reciprocal lattice of CoSi. The resolved spots from scans I and III are consistent with space group P213 at 100 K. This is also shown in the reflection intersection, scan II.

Extended Data Fig. 4 Electronic bulk band topology.

a, Band structure of CoSi in the absence of spin–orbit coupling interactions. b, Band structure in the presence of spin–orbit coupling. The highest valence and lowest conduction bands are coloured in blue and red, respectively.

Extended Data Fig. 5 Surveying the \(\bar{{\rm{\Gamma }}}\) and \(\bar{{\rm{X}}}\) pockets on the (001) surface of CoSi.

a, ARPES-measured Fermi surface with the BZ boundary (red dashed line) and the location of the energy–momentum cut of interest (magenta dashed line). b, \(\bar{{\rm{\Gamma }}}-\bar{{\rm{X}}}-\bar{{\rm{\Gamma }}}\) high-symmetry-line energy–momentum cut. c, Second-derivative plot of b. We observe an outer and an inner hole-like band at \(\bar{{\rm{\Gamma }}}\), whereas at \(\bar{{\rm{X}}}\) we observe signatures of a single hole-like band.

Extended Data Fig. 6 Tracking the Fermi arcs by fitting Lorentzians to MDCs.

a, Zoomed-in region of the Fermi surface (Fig. 2d), with the Fermi arcs tracked (blue circles) and the surface BZ marked (red dashed lines). b, Representative fits of Lorentzian functions (red lines) to the MDCs (filled blue circles). The peaks indicate the extracted positions of the Fermi arc (open blue circles) and the \(\bar{{\rm{X}}}\) pocket.

Extended Data Fig. 7 Photon-energy dependence of the Fermi arc in CoSi.

a, Location of the in-plane momentum direction (black arrow) along which the photon-energy dependence was studied, plotted on the Fermi surface (Fig. 2d) with the surface BZ (red dotted lines). b, MDCs at EF along the in-plane momentum direction illustrated in a, obtained for a series of photon energies from 80 eV to 110 eV in steps of 2 eV. The peak associated with the Fermi arc shows negligible variation in photon energy (dashed line) within the experimental resolution, providing further evidence that the observed Fermi arc is indeed a surface state.

Extended Data Fig. 8 Systematics for the helicoid fitting in CoSi.

a, Fermi arc trajectory extracted from Lorentzian fits to the MDCs (blue open circles) near \(\bar{{\rm{M}}}\), with overlaid schematic of the observed features (orange lines). There are two closed contours enclosing \(\bar{{\rm{M}}}\) on which we count chiral edge modes: the outer loop (magenta, Fig. 4) and the inner loop (black). b, Energy–momentum cut along the inner loop (radius 0.14π/a), starting from the green notch in a and winding clockwise (black arrow in a). c, Lorentzian fits (red curves) to the MDCs (blue circles) to extract the helicoid dispersion of the Fermi arcs. We observe two right-moving chiral edge modes dispersing towards EF (dashed orange lines). The corresponding bulk manifold has Chern number13 n = −2.

Extended Data Table 1 Single-crystal crystallographic data for CoSi at 100(2) K and 300(2) K

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Sanchez, D.S., Belopolski, I., Cochran, T.A. et al. Topological chiral crystals with helicoid-arc quantum states. Nature 567, 500–505 (2019). https://doi.org/10.1038/s41586-019-1037-2

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