- Letter
- Open Access
Structural spillage: An efficient method to identify noncrystalline topological materials
Phys. Rev. Research 5, L042011 – Published 12 October, 2023
DOI: https://doi.org/10.1103/PhysRevResearch.5.L042011
Abstract
While topological materials are not restricted to crystals, there is no efficient method to diagnose topology in noncrystalline solids such as amorphous materials. Here we introduce the structural spillage, a new indicator that predicts the unknown topological phase of a noncrystalline solid, which is compatible with first-principles calculations. We illustrate its potential with tight-binding and first-principles calculations of amorphous bismuth, predicting a bilayer to be a new topologically nontrivial material. Our work opens up the efficient prediction of noncrystalline solids via first-principles and high-throughput searches.
Physics Subject Headings (PhySH)
Corrections
15 November, 2023
Correction: The footnote indicating equal contribution for the first two authors was missing at publication and has been inserted.
Article Text
Supplemental Material
References (91)
- J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.-J. Slager, Topological classification of crystalline insulators through band structure combinatorics, Phys. Rev. X 7, 041069 (2017).
- H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry-based indicators of band topology in the 230 space groups, Nat. Commun. 8, 50 (2017).
- Z. Song, T. Zhang, Z. Fang, and C. Fang, Quantitative mappings between symmetry and topology in solids, Nat. Commun. 9, 3530 (2018).
- N. C. Frey, M. K. Horton, J. M. Munro, S. M. Griffin, K. A. Persson, and V. B. Shenoy, High-throughput search for magnetic and topological order in transition metal oxides, Sci. Adv. 6, eabd1076 (2020).
- B. J. Wieder, B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, L. Elcoro, A. A. Soluyanov, C. Felser, T. Neupert, N. Regnault, and B. A. Bernevig, Topological materials discovery from crystal symmetry, Nat. Rev. Mater. 7, 196 (2022).
- E. Prodan, Quantum transport in disordered systems under magnetic fields: A study based on operator algebras, Appl. Math. Res. eXpress 2013, 176 (2012).
- A. Agarwala and V. B. Shenoy, Topological insulators in amorphous systems, Phys. Rev. Lett. 118, 236402 (2017).
- N. P. Mitchell, L. M. Nash, D. Hexner, A. M. Turner, and W. T. M. Irvine, Amorphous topological insulators constructed from random point sets, Nat. Phys. 14, 380 (2018).
- K. Pöyhönen, I. Sahlberg, A. Westström, and T. Ojanen, Amorphous topological superconductivity in a Shiba glass, Nat. Commun. 9, 2103 (2018).
- C. Bourne and E. Prodan, Non-commutative Chern numbers for generic aperiodic discrete systems, J. Phys. A: Math. Theor. 51, 235202 (2018).
- P. Corbae, S. Ciocys, D. Varjas, E. Kennedy, S. Zeltmann, M. Molina-Ruiz, S. M. Griffin, C. Jozwiak, Z. Chen, L.-W. Wang, A. M. Minor, M. Scott, A. G. Grushin, A. Lanzara, and F. Hellman, Observation of spin-momentum locked surface states in amorphous , Nat. Mater. 22, 200 (2023).
- Y.-B. Yang, T. Qin, D.-L. Deng, L.-M. Duan, and Y. Xu, Topological amorphous metals, Phys. Rev. Lett. 123, 076401 (2019).
- M. Costa, G. R. Schleder, M. Buongiorno Nardelli, C. Lewenkopf, and A. Fazzio, Toward realistic amorphous topological insulators, Nano Lett. 19, 8941 (2019).
- Q. Marsal, D. Varjas, and A. G. Grushin, Topological weaire–thorpe models of amorphous matter, Proc. Natl. Acad. Sci. USA 117, 30260 (2020).
- I. Sahlberg, A. Westström, K. Pöyhönen, and T. Ojanen, Topological phase transitions in glassy quantum matter, Phys. Rev. Res. 2, 013053 (2020).
- A. G. Grushin, Topological phases of amorphous matter, in Low-Temperature Thermal and Vibrational Properties of Disordered Solids, edited by M. A. Ramos (World Scientific, Europe, 2022).
- A. Agarwala, V. Juričić, and B. Roy, Higher-order topological insulators in amorphous solids, Phys. Rev. Res. 2, 012067(R) (2020).
- P. Mukati, A. Agarwala, and S. Bhattacharjee, Topological and conventional phases of a three-dimensional electronic glass, Phys. Rev. B 101, 035142 (2020).
- P. Corbae, F. Hellman, and S. M. Griffin, Structural disorder-driven topological phase transition in noncentrosymmetric BiTeI, Phys. Rev. B 103, 214203 (2021).
- H. Spring, A. R. Akhmerov, and D. Varjas, Amorphous topological phases protected by continuous rotation symmetry, SciPost Phys. 11, 022 (2021).
- Q. Marsal, D. Varjas, and A. G. Grushin, Obstructed insulators and flat bands in topological phase-change materials, Phys. Rev. B 107, 045119 (2023).
- J.-H. Wang, Y.-B. Yang, N. Dai, and Y. Xu, Structural-disorder-induced second-order topological insulators in three dimensions, Phys. Rev. Lett. 126, 206404 (2021).
- C. Wang, T. Cheng, Z. Liu, F. Liu, and H. Huang, Structural amorphization-induced topological order, Phys. Rev. Lett. 128, 056401 (2022).
- J. Ma and H. Huang, Amorphous Kane-Mele model in disordered hyperuniform two-dimensional networks, Phys. Rev. B 106, 195150 (2022).
- S. Manna, S. K. Das, and B. Roy, Noncrystalline topological superconductors, arXiv:2207.02203.
- P. Corbae, J. D. Hannukainen, Q. Marsal, D. Muñoz-Segovia, and A. G. Grushin, Amorphous topological matter: Theory and experiment, Europhys. Lett. 142, 16001 (2023).
- I. Sahlberg, M. N. Ivaki, K. Pöyhönen, and T. Ojanen, Quantum Hall effect and Landau levels without spatial long-range correlations, Phys. Rev. Res. 5, 033218 (2023)
- Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Topological states and adiabatic pumping in quasicrystals, Phys. Rev. Lett. 109, 106402 (2012).
- F. Mei, S.-L. Zhu, Z.-M. Zhang, C. H. Oh, and N. Goldman, Simulating topological insulators with cold atoms in a one-dimensional optical lattice, Phys. Rev. A 85, 013638 (2012).
- Y. E. Kraus, Z. Ringel, and O. Zilberberg, Four-dimensional quantum Hall effect in a two-dimensional quasicrystal, Phys. Rev. Lett. 111, 226401 (2013).
- K. A. Madsen, E. J. Bergholtz, and P. W. Brouwer, Topological equivalence of crystal and quasicrystal band structures, Phys. Rev. B 88, 125118 (2013).
- M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Observation of topological phase transitions in photonic quasicrystals, Phys. Rev. Lett. 110, 076403 (2013).
- X. Deng and L. Santos, Topological transitions of interacting bosons in one-dimensional bichromatic optical lattices, Phys. Rev. A 89, 033632 (2014).
- D.-T. Tran, A. Dauphin, N. Goldman, and P. Gaspard, Topological Hofstadter insulators in a two-dimensional quasicrystal, Phys. Rev. B 91, 085125 (2015).
- M. A. Bandres, M. C. Rechtsman, and M. Segev, Topological photonic quasicrystals: Fractal topological spectrum and protected transport, Phys. Rev. X 6, 011016 (2016).
- I. C. Fulga, D. I. Pikulin, and T. A. Loring, Aperiodic weak topological superconductors, Phys. Rev. Lett. 116, 257002 (2016).
- A. Lau, J. van den Brink, and C. Ortix, Topological mirror insulators in one dimension, Phys. Rev. B 94, 165164 (2016).
- D. Varjas, A. Lau, K. Pöyhönen, A. R. Akhmerov, D. I. Pikulin, and I. C. Fulga, Topological phases without crystalline counterparts, Phys. Rev. Lett. 123, 196401 (2019).
- O. Zilberberg, Topology in quasicrystals [Invited], Opt. Mater. Express 11, 1143 (2021).
- R. Zallen, The Physics of Amorphous Solids (Wiley-VCH, Weinheim, 1998).
- M. Le Gallo and A. Sebastian, An overview of phase-change memory device physics, J. Phys. D 53, 213002 (2020).
- A. J. Uría-Álvarez, D. Molpeceres-Mingo, and J. J. Palacios, Deep learning for disordered topological insulators through entanglement spectrum, arXiv:2201.13306.
- A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. January Special Issue, 321, 2 (2006).
- E. Prodan, Non-commutative tools for topological insulators, New J. Phys. 12, 065003 (2010).
- R. Bianco and R. Resta, Mapping topological order in coordinate space, Phys. Rev. B 84, 241106(R) (2011).
- A. Marrazzo and R. Resta, Locality of the anomalous Hall conductivity, Phys. Rev. B 95, 121114(R) (2017).
- Z. Li and R. S. K. Mong, Local formula for the invariant of topological insulators, Phys. Rev. B 100, 205101 (2019).
- T. A. Loring and H. Schulz-Baldes, The spectral localizer for even index pairings, J. Noncommutative Geom. 14, 1 (2020).
- H. Schulz-Baldes and T. Stoiber, The spectral localizer for semifinite spectral triples, Proc. Am. Math. Soc. 149, 121 (2021).
- M. Guzmán, D. Bartolo, and D. Carpentier, Geometry and topology tango in ordered and amorphous chiral matter, SciPost Phys. 12, 038 (2022).
- A. Cerjan and T. A. Loring, Local invariants identify topology in metals and gapless systems, Phys. Rev. B 106, 064109 (2022).
- P. d'Ornellas, R. Barnett, and D. K. K. Lee, Quantized bulk conductivity as a local Chern marker, Phys. Rev. B 106, 155124 (2022).
- A. Cerjan and T. A. Loring, An operator-based approach to topological photonics, Nanophotonics 11, 4765 (2022).
- J. D. Hannukainen, M. F. Martínez, J. H. Bardarson, and T. K. Kvorning, Local topological markers in odd spatial dimensions and their application to amorphous topological matter, Phys. Rev. Lett. 129, 277601 (2022).
- W. Chen, Universal topological marker, Phys. Rev. B 107, 045111 (2023).
- D. Ceresoli and R. Resta, Orbital magnetization and Chern number in a supercell framework: Single -point formula, Phys. Rev. B 76, 012405 (2007).
- R. Favata and A. Marrazzo, Single-point spin Chern number in a supercell framework, Electron. Struct. 5, 014005 (2023).
- S. M. Griffin and N. A. Spaldin, On the relationship between topological and geometric defects, J. Phys.: Condens. Matter 29, 343001 (2017).
- J. Liu and D. Vanderbilt, Spin-orbit spillage as a measure of band inversion in insulators, Phys. Rev. B 90, 125133 (2014).
- Y.-Z. You, Z. Bi, A. Rasmussen, K. Slagle, and C. Xu, Wave function and strange correlator of short-range entangled states, Phys. Rev. Lett. 112, 247202 (2014).
- J. Y. Lee, Y.-Z. You, and C. Xu, Symmetry protected topological phases under decoherence, arXiv:2210.16323.
- B. Focassio, G. R. Schleder, M. Costa, A. Fazzio, and C. Lewenkopf, Structural and electronic properties of realistic two-dimensional amorphous topological insulators, 2D Mater. 8, 025032 (2021).
- K. Choudhary, K. F. Garrity, and F. Tavazza, High-throughput discovery of topologically non-trivial materials using spin-orbit spillage, Sci. Rep. 9, 8534 (2019).
- K. Choudhary, K. F. Garrity, N. J. Ghimire, N. Anand, and F. Tavazza, High-throughput search for magnetic topological materials using spin-orbit spillage, machine learning, and experiments, Phys. Rev. B 103, 155131 (2021).
- W. S. Jung, C. S. Leem, C. Kim, S. R. Park, S. Y. Park, B. J. Kim, E. Rotenberg, and C. Kim, Imaging the electron density in solids by using multi-Brillouin-zone angle resolved photoelectron spectroscopy, Phys. Rev. B 82, 235105 (2010).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.5.L042011 for a discussion of the approximations entering Eq. (3), definition of the tight-binding models, and further benchmarks on their phase diagrams as well as details regarding our DFT calculations, which includes Refs. [67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80].
- W. A. Harrison, Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond (Dover Publications, New York, 1989).
- E. H. Lieb, Two theorems on the Hubbard model, Phys. Rev. Lett. 62, 1201 (1989).
- Q. Li, J. S. Smith, Y. Yin, C. Wang, M. V. Klymenko, J. H. Cole, and N. V. Medhekar, Localized Wannier function based tight-binding models for two-dimensional allotropes of bismuth, New J. Phys. 23, 063042 (2021).
- S. Singh, Z. Zanolli, M. Amsler, B. Belhadji, J. O. Sofo, M. J. Verstraete, and A. H. Romero, Low-energy phases of Bi monolayer predicted by structure search in two dimensions, J. Phys. Chem. Lett. 10, 7324 (2019).
- P. Ares, J. J. Palacios, G. Abellán, J. Gómez-Herrero, and F. Zamora, Recent progress on antimonene: A new bidimensional material, Adv. Mater. 30, 1703771 (2018).
- C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, Kwant: a software package for quantum transport, New J. Phys. 16, 063065 (2014).
- A. Weiße, G. Wellein, A. Alvermann, and H. Fehske, The kernel polynomial method, Rev. Mod. Phys. 78, 275 (2006).
- D. Varjas, M. Fruchart, A. R. Akhmerov, and P. M. Perez-Piskunow, Computation of topological phase diagram of disordered using the kernel polynomial method, Phys. Rev. Res. 2, 013229 (2020).
- G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
- G. Kresse and J. Hafner, Ab initio molecular dynamics for liquid metals, Phys. Rev. B 48, 13115 (1993).
- J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
- G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999).
- S. P. Ong, W. D. Richards, A. Jain, G. Hautier, M. Kocher, S. Cholia, D. Gunter, V. L. Chevrier, K. A. Persson, and G. Ceder, Python materials genomics (pymatgen): A robust, open-source python library for materials analysis, Comput. Mater. Sci. 68, 314 (2013).
- M. Soriano and J. J. Palacios, Theory of projections with nonorthogonal basis sets: Partitioning techniques and effective Hamiltonians, Phys. Rev. B 90, 075128 (2014).
- F. Reis, G. Li, L. Dudy, M. Bauernfeind, S. Glass, W. Hanke, R. Thomale, J. Schäfer, and R. Claessen, Bismuthene on a SiC substrate: A candidate for a high-temperature quantum spin Hall material, Science 357, 287 (2017).
- S. Sun, J.-Y. You, S. Duan, J. Gou, Y. Z. Luo, W. Lin, X. Lian, T. Jin, J. Liu, Y. Huang, Y. Wang, A. T. S. Wee, Y. P. Feng, L. Shen, J. L. Zhang, J. Chen, and W. Chen, Epitaxial growth of ultraflat bismuthene with large topological band inversion enabled by substrate-orbital-filtering effect, ACS Nano 16, 1436 (2022).
- A. G. Grushin and C. Repellin, Amorphous and polycrystalline routes toward a chiral spin liquid, Phys. Rev. Lett. 130, 186702 (2023).
- C. L. Kane and E. J. Mele, topological order and the quantum spin Hall effect, Phys. Rev. Lett. 95, 146802 (2005).
- S. Murakami, Quantum spin Hall effect and enhanced magnetic response by spin-orbit coupling, Phys. Rev. Lett. 97, 236805 (2006).
- M. Wada, S. Murakami, F. Freimuth, and G. Bihlmayer, Localized edge states in two-dimensional topological insulators: Ultrathin Bi films, Phys. Rev. B 83, 121310(R) (2011).
- Z. Liu, C.-X. Liu, Y.-S. Wu, W.-H. Duan, F. Liu, and J. Wu, Stable nontrivial topology in ultrathin Bi (111) films: A first-principles study, Phys. Rev. Lett. 107, 136805 (2011).
- Z.-Q. Huang, F.-C. Chuang, C.-H. Hsu, Y.-T. Liu, H.-R. Chang, H. Lin, and A. Bansil, Nontrivial topological electronic structures in a single Bi(111) bilayer on different substrates: A first-principles study, Phys. Rev. B 88, 165301 (2013).
- T. D. Kühne et al., Cp2k: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations, J. Chem. Phys. 152, 194103 (2020).
- B. Focassio, G. R. Schleder, F. Crasto de Lima, C. Lewenkopf, and A. Fazzio, Amorphous structural, electronic, and topological nature from first principles, Phys. Rev. B 104, 214206 (2021).
- T. F. Harrelson, E. Sheridan, E. Kennedy, J. Vinson, A. T. N'Diaye, M. V. P. Altoé, A. Schwartzberg, I. Siddiqi, D. F. Ogletree, M. C. Scott et al., Elucidating the local atomic and electronic structure of amorphous oxidized superconducting niobium films, Appl. Phys. Lett. 119, 244004 (2021).