Visualizing Higher-Fold Topology in Chiral Crystals

Tyler A. Cochran, Ilya Belopolski, Kaustuv Manna, Mohammad Yahyavi, Yiyuan Liu, Daniel S. Sanchez, Zi-Jia Cheng, Xian P. Yang, Daniel Multer, Jia-Xin Yin, Horst Borrmann, Alla Chikina, Jonas A. Krieger, Jaime Sánchez-Barriga, Patrick Le Fèvre, François Bertran, Vladimir N. Strocov, Jonathan D. Denlinger, Tay-Rong Chang, Shuang Jia, Claudia Felser, Hsin Lin, Guoqing Chang, and M. Zahid Hasan

Phys. Rev. Lett. 130, 066402 – Published 8 February 2023

Abstract

Novel topological phases of matter are fruitful platforms for the discovery of unconventional electromagnetic phenomena. Higher-fold topology is one example, where the low-energy description goes beyond standard model analogs. Despite intensive experimental studies, conclusive evidence remains elusive for the multigap topological nature of higher-fold chiral fermions. In this Letter, we leverage a combination of fine-tuned chemical engineering and photoemission spectroscopy with photon energy contrast to discover the higher-fold topology of a chiral crystal. We identify all bulk branches of a higher-fold chiral fermion for the first time, critically important for allowing us to explore unique Fermi arc surface states in multiple interband gaps, which exhibit an emergent ladder structure. Through designer chemical gating of the samples in combination with our measurements, we uncover an unprecedented multigap bulk boundary correspondence. Our demonstration of multigap electronic topology will propel future research on unconventional topological responses.

Received 19 October 2021
Revised 18 January 2022
Accepted 14 December 2022

DOI:https://doi.org/10.1103/PhysRevLett.130.066402

Physics Subject Headings (PhySH)

Density of states Electronic structure Fermions Quasiparticles & collective excitations Surface states Topological materials Topological phases of matter

Semimetals Single crystal materials

Angle-resolved photoemission spectroscopy Density functional theory Photoemission spectroscopy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Tyler A. Cochran¹, Ilya Belopolski¹, Kaustuv Manna^2,3, Mohammad Yahyavi ^4,5, Yiyuan Liu⁶, Daniel S. Sanchez ¹, Zi-Jia Cheng¹, Xian P. Yang¹, Daniel Multer¹, Jia-Xin Yin¹, Horst Borrmann², Alla Chikina⁷, Jonas A. Krieger^7,8, Jaime Sánchez-Barriga ^9,10, Patrick Le Fèvre¹¹, François Bertran¹¹, Vladimir N. Strocov⁷, Jonathan D. Denlinger¹², Tay-Rong Chang⁴, Shuang Jia⁵, Claudia Felser², Hsin Lin¹³, Guoqing Chang ^5,†, and M. Zahid Hasan^1,14,15,*

¹Laboratory for Topological Quantum Matter and Advanced Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
²Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany
³Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India
⁴Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan
⁵Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link 637371, Singapore
⁶International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
⁷Swiss Light Source, Paul Scherrer Institute, 5232 Villigen, Switzerland
⁸Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, 5232 Villigen, Switzerland
⁹Helmholtz-Zentrum Berlin für Materialien und Energie, Elektronenspeicherring BESSY II, Albert-Einstein Strasse 15, 12489 Berlin, Germany
¹⁰IMDEA Nanoscience, C/ Faraday 9, Campus de Cantoblanco, 28049 Madrid, Spain
¹¹SOLEIL Synchrotron, L’Orme des Merisiers, Départementale 128, F-91190 Saint-Aubin, France
¹²Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
¹³Institute of Physics, Academia Sinica, Taipei 11529, Taiwan
¹⁴Princeton Institute for Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544, USA
¹⁵Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

^*Corresponding author. mzhasan@princeton.edu
^†Corresponding author. guoqing.chang@ntu.edu.sg

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Vol. 130, Iss. 6 — 10 February 2023

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Images

Figure 1
Higher-fold topology beyond Weyl and Dirac. (a) Schematic of a two-band system corresponding to a Weyl fermion in three dimensions. (b) Schematic of bulk-boundary correspondence of a one-gap system with a boundary chiral mode. The momentum path corresponds to the blue plane in Fig. 1. (c) Schematic of a chiral fermion with three bulk branches and two topologically nontrivial energy gaps. (d) Schematic of bulk-boundary correspondence of a two-gap system with boundary chiral modes. The momentum path corresponds to the blue plane in Fig. 1. (e) Crystal structure of the $A_{1 - x} B_{x} Si$ substitutional alloy in the $P 2_{1} 3$ space group (#198), with $A = Rh$ and $B = Ni$ . The bulk and (001) surface Brillouin zone. (f) Ab initio band structure calculation along high-symmetry lines of RhSi.
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Figure 2
Topological chiral fermion at the $Γ$ point. (a) Bulk sensitive SXARPES measured using 550 eV incident photons and ab initio $E_{B}$ vs $k$ spectrum along the $M − Γ − M$ line, showing three bands becoming degenerate at the $Γ$ point. (b) SXARPES and ab initio spectra along the $X − Γ − X$ line, confirming three bands dispersing away from the degeneracy point. (c) SXARPES $E_{B}$ vs $k$ spectrum and ab initio calculation along a path shifted to $k_{y} = 0.1 Å^{- 1}$ . Along this momentum path, gap 1 and gap 2 can be identified. (d) SXARPES spectrum and ab initio calculation along a path further shifted to $k_{y} = 0.22 Å^{- 1}$ . (e) SXARPES measured stack of constant $E_{B}$ contours at the $Γ$ point, showing the band structure evolution through the degeneracy point. (f) Ab initio calculation of constant $E_{B}$ stack.
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Figure 3
Fermi arc switching in momentum space. (a) Surface sensitive UVARPES measured Fermi surface from (001) surface, utiliizing 85 eV incident photons. Long Fermi arc surface states connect pockets at $\bar{Γ}$ and $\bar{M}$ as indicated by the dashed cyan line. (b) UVARPES measured Fermi surface at the $\bar{Γ}$ point with 40 eV incident photons. (c) Second derivative of spectrum in Fig. 3. (d) SXARPES measured Fermi surface at the $Γ$ point. (e) Overlay of Fig. 3 on Fig. 3 showing the Fermi arcs connecting to the bulk band. (f)–(h) $E_{B}$ vs $k$ UVARPES spectra showing chiral Fermi arc surface states along paths defined by the dotted lines in Fig. 3. Arrows indicate chiral modes at the Fermi level, which switch at the threefold chiral fermion at the $\bar{Γ}$ point. (i)–(k) Schematics of chiral Fermi arcs (solid blue lines) and bulk bands (dotted lines) depicting the Fermi arc switching at the $\bar{Γ}$ point.
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Figure 4
Multigap chiral charge and a Fermi arc ladder. (a) and (b) UVARPES measured constant $E_{B}$ contour at $E_{B} = 300 meV$ [Fig. 4] and second-derivative [Fig. 4] at the $\bar{Γ}$ point. Both inner and outer states are indicated by cyan dashed lines in Fig. 4. (c) SXARPES constant $E_{B}$ contour at $E_{B} = 300 meV$ at the $Γ$ point. (d) Overlay of Fig. 4 onto Fig. 4; the inner surface states lie within the bulk projection of band 2. (e) Schematic constant $E_{B}$ contour showing chiral states (thick blue lines) in gap 1 and gap 2 of the bulk threefold chiral fermion (dotted lines). (f) and (g) UVARPES measured $E_{B}$ vs $k$ cut at $k_{y} = 0.22 Å^{- 1}$ [Fig. 4] and second derivative [Fig. 4], along the path indicated by the dark blue dashed line in Fig. 4. Two disconnected chiral modes are observed (dashed cyan lines). (h) SXARPES measured spectrum along the same path as Fig. 4 at $k_{y} = 0.22 Å^{- 1}$ . (i) Overlay of Fig. 4 onto Fig. 4, showing topological surface states propagating in gap 1 and 2 respectively. (j) Schematic $E_{B}$ vs $k$ plot showing the Fermi arc ladder (thick blue lines).
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