- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Sixfold excitations in electrides

Simin Nie1,2, B. Andrei Bernevig3,*, and Zhijun Wang1,4,†

  • 1Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
  • 2Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, USA
  • 3Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
  • 4University of Chinese Academy of Sciences, Beijing 100049, China

  • *bernevig@princeton.edu
  • †wzj@iphy.ac.cn
PDF

Phys. Rev. Research 3, L012028 – Published 22 March, 2021

DOI: https://doi.org/10.1103/PhysRevResearch.3.L012028

Abstract

Due to the lack of full rotational symmetry in condensed matter physics, solids exhibit new excitations beyond Dirac and Weyl fermions, of which the sixfold excitations have attracted considerable interest owing to the presence of maximum degeneracy in bosonic systems. Here, we propose that a single linear dispersive sixfold excitation can be found in the electride Li12Mg3Si4 and its derivatives. The sixfold excitation is formed by the floating bands of elementary band representation A@12a originating from the excess electrons centered at the vacancies (i.e., the 12a Wyckoff sites). There exists a unique topological bulk-surface-edge correspondence for the spinless sixfold excitation, resulting in trivial surface “Fermi arcs” but topological hinge arcs. All gapped kz slices belong to a two-dimensional higher-order topological insulating phase, which is protected by a combined symmetry TS̃4z and characterized by a quantized fractional corner charge Qcorner=3|e|4. Consequently, the hinge arcs are obtained in the hinge spectra of the S̃4z-symmetric rod structure. The state with a single sixfold excitation, stabilized by both nonsymmorphic crystalline symmetries and time-reversal symmetry, is located at the phase boundary and can be driven into various topologically distinct phases by explicit breaking of symmetries, making these electrides promising platforms for the systematic studies of different topological phases.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (61)

  1. P. B. Pal, Dirac, Majorana, and Weyl fermions, Am. J. Phys. 79, 485 (2011).
  2. Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, Three-dimensional Dirac semimetal and quantum transport in Cd3As2, Phys. Rev. B 85, 195320 (2012).
  3. Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, Three-dimensional Dirac semimetal and quantum transport in Cd3As2, Phys. Rev. B 88, 125427 (2013).
  4. S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, Dirac Semimetal in Three Dimensions, Phys. Rev. Lett. 108, 140405 (2012).
  5. P. Tang, Q. Zhou, G. Xu, and S.-C. Zhang, Dirac fermions in an antiferromagnetic semimetal, Nat. Phys. 12, 1100 (2016).
  6. G. Hua, S. Nie, Z. Song, R. Yu, G. Xu, and K. Yao, Dirac semimetal in type-IV magnetic space groups, Phys. Rev. B 98, 201116(R) (2018).
  7. X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates, Phys. Rev. B 83, 205101 (2011).
  8. G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Chern Semimetal and the Quantized Anomalous Hall Effect in HgCr2Se4, Phys. Rev. Lett. 107, 186806 (2011).
  9. H. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai, Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides, Phys. Rev. X 5, 011029 (2015).
  10. S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang et al., A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class, Nat. Commun. 6, 7373 (2015).
  11. Z. Wang, M. G. Vergniory, S. Kushwaha, M. Hirschberger, E. V. Chulkov, A. Ernst, N. P. Ong, R. J. Cava, and B. A. Bernevig, Time-Reversal-Breaking Weyl Fermions in Magnetic Heusler Alloys, Phys. Rev. Lett. 117, 236401 (2016).
  12. S. Nie, G. Xu, F. B. Prinz, and S.-c. Zhang, Topological semimetal in honeycomb lattice LnSI, Proc. Natl. Acad. Sci. USA 114, 10596 (2017).
  13. Q. Wang, Y. Xu, R. Lou, Z. Liu, M. Li, Y. Huang, D. Shen, H. Weng, S. Wang, and H. Lei, Large intrinsic anomalous Hall effect in half-metallic ferromagnet Co3Sn2S2 with magnetic Weyl fermions, Nat. Commun. 9, 1 (2018).
  14. S. Nie, Y. Sun, F. B. Prinz, Z. Wang, H. Weng, Z. Fang, and X. Dai, Magnetic Semimetals and Quantized Anomalous Hall Effect in EuB6, Phys. Rev. Lett. 124, 076403 (2020).
  15. A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, Type-II Weyl semimetals, Nature (London) 527, 495 (2015).
  16. S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor, Science 346, 602 (2014).
  17. Q. L. He, L. Pan, A. L. Stern, E. C. Burks, X. Che, G. Yin, J. Wang, B. Lian, Q. Zhou, E. S. Choi et al., Chiral Majorana fermion modes in a quantum anomalous Hall insulator–superconductor structure, Science 357, 294 (2017).
  18. P. Zhang, K. Yaji, T. Hashimoto, Y. Ota, T. Kondo, K. Okazaki, Z. Wang, J. Wen, G. Gu, H. Ding et al., Observation of topological superconductivity on the surface of an iron-based superconductor, Science 360, 182 (2018).
  19. D. Wang, L. Kong, P. Fan, H. Chen, S. Zhu, W. Liu, L. Cao, Y. Sun, S. Du, J. Schneeloch et al., Evidence for Majorana bound states in an iron-based superconductor, Science 362, 333 (2018).
  20. J. Ma, C. Yi, B. Lv, Z. Wang, S. Nie, L. Wang, L. Kong, Y. Huang, P. Richard, P. Zhang et al., Experimental evidence of hourglass fermion in the candidate nonsymmorphic topological insulator KHgSb, Sci. Adv. 3, e1602415 (2017).
  21. B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser, R. J. Cava, and B. A. Bernevig, Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals, Science 353, aaf5037 (2016).
  22. P. Tang, Q. Zhou, and S.-C. Zhang, Multiple Types of Topological Fermions in Transition Metal Silicides, Phys. Rev. Lett. 119, 206402 (2017).
  23. G. Chang, S.-Y. Xu, B. J. Wieder, D. S. Sanchez, S.-M. Huang, I. Belopolski, T.-R. Chang, S. Zhang, A. Bansil, H. Lin et al., Unconventional Chiral Fermions and Large Topological Fermi Arcs in RhSi, Phys. Rev. Lett. 119, 206401 (2017).
  24. T. Zhang, Z. Song, A. Alexandradinata, H. Weng, C. Fang, L. Lu, and Z. Fang, Double-Weyl Phonons in Transition-Metal Monosilicides, Phys. Rev. Lett. 120, 016401 (2018).
  25. W. Rarita and J. Schwinger, On a Theory of Particles with Half-Integral Spin, Phys. Rev. 60, 61 (1941).
  26. B. J. Wieder, Z. Wang, J. Cano, X. Dai, L. M. Schoop, B. Bradlyn, and B. A. Bernevig, Strong and fragile topological Dirac semimetals with higher-order Fermi arcs, Nat. Commun. 11, 627 (2020).
  27. Z. Rao, H. Li, T. Zhang, S. Tian, C. Li, B. Fu, C. Tang, L. Wang, Z. Li, W. Fan et al., Observation of unconventional chiral fermions with long Fermi arcs in CoSi, Nature (London) 567, 496 (2019).
  28. D. Takane, Z. Wang, S. Souma, K. Nakayama, T. Nakamura, H. Oinuma, Y. Nakata, H. Iwasawa, C. Cacho, T. Kim et al., Observation of Chiral Fermions with a Large Topological Charge and Associated Fermi-Arc Surface States in CoSi, Phys. Rev. Lett. 122, 076402 (2019).
  29. N. Kumar, M. Yao, J. Nayak et al., Signatures of sixfold degenerate exotic fermions in a superconducting metal PdSb2, Adv. Mater. 32, 1906046 (2020).
  30. Z. P. Sun, C. Q. Hua, X. L. Liu, Z. T. Liu, M. Ye, S. Qiao, Z. H. Liu, J. S. Liu, Y. F. Guo, Y. H. Lu et al., Direct observation of sixfold exotic fermions in the pyrite-structured topological semimetal PdSb2, Phys. Rev. B 101, 155114 (2020).
  31. X. Yáng, T. A. Cochran, R. Chapai, D. Tristant, J.-X. Yin, I. Belopolski, Z. Chéng, D. Multer, S. S. Zhang, N. Shumiya, M. Litskevich, Y. Jiang, G. Chang, Q. Zhang, I. Vekhter, W. A. Shelton, R. Jin, S.-Y. Xu, and M. Z. Hasan, Observation of sixfold degenerate fermions in PdSb2, Phys. Rev. B 101, 201105(R) (2020).
  32. R. Chapai, Y. Jia, W. A. Shelton, R. Nepal, M. Saghayezhian, J. F. DiTusa, E. W. Plummer, C. Jin, and R. Jin, Fermions and bosons in nonsymmorphic PdSb2 with sixfold degeneracy, Phys. Rev. B 99, 161110(R) (2019).
  33. S. B. Dawes, D. L. Ward, R. H. Huang, and J. L. Dye, First electride crystal structure, J. Am. Chem. Soc. 108, 3534 (1986).
  34. J. L. Dye and M. G. DeBacker, Physical and chemical properties of alkalides and electrides, Annu. Rev. Phys. Chem. 38, 271 (1987).
  35. S. Matsuishi, Y. Toda, M. Miyakawa, K. Hayashi, T. Kamiya, M. Hirano, I. Tanaka, and H. Hosono, High-density electron anions in a nanoporous single crystal: [Ca24Al28O64]4+(4e−), Science 301, 626 (2003).
  36. K. Lee, S. W. Kim, Y. Toda, S. Matsuishi, and H. Hosono, Dicalcium nitride as a two-dimensional electride with an anionic electron layer, Nature (London) 494, 336 (2013).
  37. S. Nie, Y. Qian, J. Gao, Z. Fang, H. Weng, and Z. Wang, The application of topological quantum chemistry in electrides, arXiv:2012.02203.
  38. D. J. Singh, H. Krakauer, C. Haas, and W. E. Pickett, Theoretical determination that electrons act as anions in the electride Cs+(15−crown−5)2e−, Nature (London) 365, 39 (1993).
  39. R. C. Phillips, W. P. Pratt, and J. L. Dye, Thermionic emission from cold electride films, Chem. Mater. 12, 3642 (2000).
  40. R. H. Huang and J. L. Dye, Low temperature (−80 ∘C) thermionic electron emission from alkalides and electrides, Chem. Phys. Lett. 166, 133 (1990).
  41. M. Kitano, Y. Inoue, Y. Yamazaki, F. Hayashi, S. Kanbara, S. Matsuishi, T. Yokoyama, S.-W. Kim, M. Hara, and H. Hosono, Ammonia synthesis using a stable electride as an electron donor and reversible hydrogen store, Nat. Chem. 4, 934 (2012).
  42. B. Bradlyn, L. Elcoro, J. Cano, M. Vergniory, Z. Wang, C. Felser, M. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature (London) 547, 298 (2017).
  43. H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry-based indicators of band topology in the 230 space groups, Nat. Commun. 8, 1 (2017).
  44. H. Huang, K.-H. Jin, S. Zhang, and F. Liu, Symmetry-based indicators of band topology in the 230 space groups, Nano Lett. 18, 1972 (2018).
  45. M. Hirayama, S. Matsuishi, H. Hosono, and S. Murakami, Electrides as a New Platform of Topological Materials, Phys. Rev. X 8, 031067 (2018).
  46. S.-C. Zhu, L. Wang, J.-Y. Qu, J.-J. Wang, T. Frolov, X.-Q. Chen, and Q. Zhu, Computational design of flexible electrides with nontrivial band topology, Phys. Rev. Materials 3, 024205 (2019).
  47. X. Zhang, B. Fu, L. Jin, X. Dai, G. Liu, and Y. Yao, Topological nodal line electrides: Realization of an ideal nodal line state nearly immune from spin-orbit coupling, J. Phys. Chem. C 123, 25871 (2019).
  48. S. Nie, H. Weng, and F. B. Prinz, Topological nodal-line semimetals in ferromagnetic rare-earth-metal monohalides, Phys. Rev. B 99, 035125 (2019).
  49. J. Wang, X. Sui, S. Gao, W. Duan, F. Liu, and B. Huang, Anomalous Dirac Plasmons in 1D Topological Electrides, Phys. Rev. Lett. 123, 206402 (2019).
  50. B. Bradlyn, L. Elcoro, M. G. Vergniory, J. Cano, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Band connectivity for topological quantum chemistry: Band structures as a graph theory problem, Phys. Rev. B 97, 035138 (2018).
  51. D. Kevorkov, R. Schmid-Fetzer, and F. Zhang, Phase equilibria and thermodynamics of the Mg-Si-Li system and remodeling of the Mg-Si system, J. Phase Equilib. Diffus. 25, 140 (2004).
  52. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.3.L012028 for the details on the vectors of SG 220 and structure information of Li12Mg3Si4, calculation method, introduction to TQC and its application in electrides, representation matrices of the generators at H, band dispersions of the low-energy effective k·p model, Wilson-loop technique, the TB Hamiltonian, surface states and strain effect, hinge states, and band structure of Li12Mg3Si4 with SOC and A12A3′B4 without SOC.
  53. Bilbao crystallographic server, https://www.cryst.ehu.es/cgi-bin/cryst/programs/bandrep.pl.
  54. J. Gao, Q. Wu, C. Persson, and Z. Wang, Irvsp: To obtain irreducible representations of electronic states in the VASP, Comput. Phys. Commun. 261, 107760 (2021).
  55. J. Wang, Q. Zhu, Z. Wang, and H. Hosono, Ternary inorganic electrides with mixed bonding, Phys. Rev. B 99, 064104 (2019).
  56. H. T. Stokes, B. J. Campbell, and R. Cordes, Tabulation of irreducible representations of the crystallographic space groups and their superspace extensions, Acta Crystallogr., Sect. A: Found. Crystallogr. 69, 388 (2013).
  57. W. A. Benalcazar, T. Li, and T. L. Hughes, Quantization of fractional corner charge in Cn-symmetric higher-order topological crystalline insulators, Phys. Rev. B 99, 245151 (2019).
  58. C. W. Peterson, W. A. Benalcazar, T. L. Hughes, and G. Bahl, A quantized microwave quadrupole insulator with topologically protected corner states, Nature (London) 555, 346 (2018).
  59. S. Franca, J. van den Brink, and I. C. Fulga, An anomalous higher-order topological insulator, Phys. Rev. B 98, 201114(R) (2018).
  60. H. Xue, Y. Yang, F. Gao, Y. Chong, and B. Zhang, Acoustic higher-order topological insulator on a kagome lattice, Nat. Mater. 18, 108 (2019).
  61. F. Schindler, M. Brzezińska, W. A. Benalcazar, M. Iraola, A. Bouhon, S. S. Tsirkin, M. G. Vergniory, and T. Neupert, Fractional corner charges in spin-orbit coupled crystals, Phys. Rev. Research 1, 033074 (2019).

Outline

Information

Phys. Rev. Research 3, L012028– Published 22 March, 2021
  • Received 24 September 2020
  • Accepted 9 February 2021
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation