Chapter 5
Ab Initio Calculations of Two-Dimensional Topological Insulators*
Gustav Bihlmayer,
Yu. M. Koroteev,
T. V. Menshchikova,
Evgueni V. Chulkov,
Stefan Blügel,
Gustav Bihlmayer
Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
Search for more papers by this authorYu. M. Koroteev
Siberian Branch, Russian Academy of Sciences, Institute of Strength Physics and Materials Science, pr. Akademicheskii 2/4, Tomsk, 634021, Russia
Search for more papers by this authorT. V. Menshchikova
Tomsk State University, pr. Lenina 36, 634050, Tomsk, Russia
Search for more papers by this authorEvgueni V. Chulkov
Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 San Sebastián/Donostia, Basque Country, Spain
Search for more papers by this authorStefan Blügel
Forschungszentrum Jülich and JARA, Peter Grünberg Institut and Institute for Advanced Simulation, 52425 Jülich, Germany
Search for more papers by this authorGustav Bihlmayer,
Yu. M. Koroteev,
T. V. Menshchikova,
Evgueni V. Chulkov,
Stefan Blügel,
Gustav Bihlmayer
Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
Search for more papers by this authorYu. M. Koroteev
Siberian Branch, Russian Academy of Sciences, Institute of Strength Physics and Materials Science, pr. Akademicheskii 2/4, Tomsk, 634021, Russia
Search for more papers by this authorT. V. Menshchikova
Tomsk State University, pr. Lenina 36, 634050, Tomsk, Russia
Search for more papers by this authorEvgueni V. Chulkov
Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 San Sebastián/Donostia, Basque Country, Spain
Search for more papers by this authorStefan Blügel
Forschungszentrum Jülich and JARA, Peter Grünberg Institut and Institute for Advanced Simulation, 52425 Jülich, Germany
Search for more papers by this authorBook Editor(s):Frank Ortmann,
Stephan Roche,
Sergio O. Valenzuela,
Frank Ortmann
Dr.
Technische Universität Dresden, Institute for Materials Science and Dresden Center for Computational Materials Science, 01062 Dresden, Germany
Search for more papers by this authorStephan Roche
Prof.
ICREA, Institució Catalana de Recerca i Estudis Avançats, 08070 Barcelona, Spain
ICN2—Institut Catala de Nanociencia i Nanotecnologia, Campus UAB, 08193 Bellaterra (Barcelona), Spain
Search for more papers by this authorSergio O. Valenzuela
Prof.
ICREA, Institució Catalana de Recerca i Estudis Avançats, 08070 Barcelona, Spain
ICN2—Institut Catala de Nanociencia i Nanotecnologia, Campus UAB, 08193 Bellaterra (Barcelona), Spain
Search for more papers by this authorSummary
This chapter focuses on different two-dimensional (2D) topological insulators (TIs), and introduces the basic concepts through the example of graphene and discusses the HgTe/CdTe quantum wells. It shows that the actual material parameters that allow assessing the usefulness of a selected system for experimental purposes can only be obtained from ab initio calculations. Employing density functional theory (DFT), thin Bi and Sb films offer a nice opportunity to study both the evolution of a 2D TI to a trivial 3D semimetal and the transformation of a three-dimensional (3D) topological semimetal to a trivial 2D insulator. For the structurally more complicated compounds, such as Bi2Te3, GeBi2Te4, or Pb2Bi2Te5, the chapter also focuses on thin films and their topological properties as a function of film thickness. In particular, it describes the identification of stable phases with sufficiently large bandgaps, which allows the observation of topological protection in transport experiments.
References
- Haldane, F.D.M. (1988) Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett, 61, 2015.
- Kane, C.L. and Mele, E.J. (2005) Quantum spin Hall effect in graphene. Phys. Rev. Lett, 95, 226801.
- Fu, L. and Kane, C.L. (2007) Topological insulators with inversion symmetry. Phys. Rev. B, 76, 045302.
- Hsieh, D., Xia, Y., Wray, L., Qian, D., Pal, A., Dil, J.H., Osterwalder, J., Meier, F., Bihlmayer, G., Kane, C.L., Hor, Y.S., Cava, R.J., and Hasan, M.Z. (2009) Observation of unconventional quantum spin textures in topological insulators. Science, 323, 919.
- König, M., Wiedmann, S., Brüne, C., Roth, A., Buhmann, H., Molenkamp, L.W., Qi, X.-L., and Zhang, S.-C. (2007) Quantum spin Hall insulator state in HgTe quantum wells. Science, 318, 766.
- Murakami, S. (2006) Quantum spin Hall effect and enhanced magnetic response by spin-orbit coupling. Phys. Rev. Lett, 97, 236805.
- Li, J., Martin, I., Büttiker, M., and Morpurgo, A.F. (2012) Marginal topological properties of graphene: a comparison with topological insulators. Phys. Scr., T146, 014021.
- DiVincenzo, D.P. and Mele, E.J. (1984) Self-consistent effective-mass theory for intralayer screening in graphite intercalation compounds. Phys. Rev. B, 29, 1685.
- Gmitra, M., Konschuh, S., Ertler, C., Ambrosch-Draxl, C., and Fabian, J. (2009) Band-structure topologies of graphene: spin-orbit coupling effects from first principles. Phys. Rev. B, 80, 235431.
- Wimmer, E., Krakauer, H., Weinert, M., and Freeman, A. (1981) Full-potential self-consistent linearized-augmented-plane-wave method for calculating the electronic structure of molecules and surfaces: O2 molecule. Phys. Rev. B, 24, 864.
- Fleur, for a program description, see http://www.flapw.de (accessed 21 October 2014).
- Konschuh, S., Gmitra, M., and Fabian, J. (2010) Tight-binding theory of the spin-orbit coupling in graphene. Phys. Rev. B, 82, 245412.
- Qi, X.-L. and Zhang, S.S. (2010) The quantum spin Hall effect and topological insulators. Phys. Today, 63, 33.
- Fujita, M., Wakabayashi, K., Nakada, K., and Kusakabe, K. (1996) Peculiar localized state at zigzag graphite edge. J. Phys. Soc. Jpn., 65, 1920.
- Sau, J.D., Lutchyn, R.M., Tewari, S., and Sarma, S.D. (2010) Generic new platform for topological quantum computation using semiconductor heterostructures. Phys. Rev. Lett., 104, 040502.
- Varykhalov, A., Sánchez-Barriga, J., Shikin, A.M., Biswas, C., Vescovo, E., Rybkin, A., Marchenko, D., and Rader, O. (2008) Electronic and magnetic properties of quasifreestanding graphene on Ni. Phys. Rev. Lett., 101, 157601.
- Marchenko, D., Varykhalov, A., Scholz, M.R., Bihlmayer, G., Rashba, E.I., Rybkin, A., Shikin, A.M., and Rader, O. (2012) Giant Rashba splitting in graphene due to hybridization with gold. Nat. Commun., 3, 1232.
- Marchenko, D., Varykhalov, A., Scholz, M.R., Sánchez-Barriga, J., Rader, O., Rybkina, A., Shikin, A.M., Seyller, T., and Bihlmayer, G. (2013) Spin-resolved photoemission and ab initio theory of graphene/SiC. Phys. Rev. B, 88, 075422.
- Rashba, E.I. (2009) Graphene with structure-induced spin-orbit coupling: spin-polarized states, spin zero modes, and quantum Hall effect. Phys. Rev. B, 79, 161409(R).
- Weeks, C., Hu, J., Alicea, J., Franz, M., and Wu, R. (2011) Engineering a robust quantum spin Hall state in graphene via adatom deposition. Phys. Rev. X, 1, 021001.
- Bernevig, B.A., Hughes, T.L., and Zhang, S.-C. (2006) Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science, 314, 1757.
- Kane, E.O. (1957) Band structure of indium antimonide. J. Phys. Chem. Solids, 1, 249.
- Luttinger, J.M. and Kohn, W. (1955) Motion of electrons and holes in perturbed periodic fields. Phys. Rev., 97, 869.
- Christensen, N.E., Gorczyca, I., Christensen, O.B., Schmid, U., and Cardona, M. (1990) Band structure and heterojunctions of II-VI materials. J. Cryst. Growth, 101, 318.
- Barfuss, A., Dudy, L., Scholz, M.R., Roth, H., Höpfner, P., Blumenstein, C., Landolt, G., Dil, J.H., Plumb, N.C., Radovic, M., Bostwick, A., Rotenberg, E., Fleszar, A., Bihlmayer, G., Wortmann, D., Hanke, W., Claessen, R., and Schäfer, J. (2013) Elemental topological insulator with a tunable Fermi level: strained α-Sn on InSb(001). Phys. Rev. Lett., 111, 157205.
- Cade, N.A. and Lee, P.M. (1985) Self consistent energy band structures for HgTe and CdTe. Solid State Commun., 56, 637.
- Sakuma, R., Friedrich, C., Miyake, T., Blügel, S., and Aryasetiawan, F. (2011) GW calculations including spin-orbit coupling: application to Hg chalcogenides. Phys. Rev. B, 84, 085144.
- Zhang, H., Freimuth, F., Bihlmayer, G., Blügel, S., and Mokrousov, Y. (2012) Topological phases of Bi(111) bilayer in an external exchange field. Phys. Rev. B, 86, 035104.
- Avron, J.E., Osadchy, D., and Seiler, R. (2003) A topological look at the quantum Hall effect. Phys. Today, 56 (8), 38–42.
- Volovik, G.E. (2003) The Universe in a Helium Droplet, The International Series of Monographs on Physics, vol. 117 (eds J. Birman and S.F. Edwards), Clarendon Press, Oxford.
- Brüne, C., Liu, C.X., Novik, E.G., Hankiewicz, E.M., Buhmann, H., Chen, Y.L., Qi, X.L., Shen, Z.X., Zhang, S.C., and Molenkamp, L.W. (2011) Quantum Hall effect from the topological surface states of strained bulk HgTe. Phys. Rev. Lett., 106, 126803.
- Mönig, H., Sun, J., Koroteev, Y.M., Bihlmayer, G., Chulkov, E.V., Pohl, K., and Hofmann, P. (2005) The structure of the (111) surface of bismuth. Phys. Rev. B, 72, 085410.
- Zhang, H., Freimuth, F., Bihlmayer, G., Lezaic, M., Blügel, S., and Mokrousov, Y. (2013) Engineering quantum anomalous Hall phases with orbital and spin degrees of freedom. Phys. Rev. B, 87, 205132.
- Eremeev, S.V., Bihlmayer, G., Vergniory, M., Koroteev, Y.M., Menshikova, T.V., Henk, J., Ernst, A., and Chulkov, E.V. (2011) Ab initio electronic structure of thallium-based topological insulators. Phys. Rev. B, 83, 205129.
- Wada, M., Murakami, S., Freimuth, F., and Bihlmayer, G. (2011) Localized edge states in two-dimensional topological insulators: ultrathin Bi films. Phys. Rev. B, 83, 121310(R).
- Yang, F., Miao, L., Wang, Z.F., Yao, M.-Y., Zhu, F., Song, Y.R., Wang, M.-X., Xu, J.-P., Fedorov, A.V., Sun, Z., Zhang, G.B., Liu, C., Liu, F., Qian, D., Gao, C.L., and Jia, J.-F. (2012) Spatial and energy distribution of topological edge states in single Bi(111) bilayer. Phys. Rev. Lett., 109, 016801.
- Hirahara, T., Miyamoto, K., Kimura, A., Niinuma, Y., Bihlmayer, G., Chulkov, E.V., Nagao, T., Matsuda, I., Qiao, S., Shimada, K., Namatame, H., Taniguchi, M., and Hasegawa, S. (2008) Origin of the surface-state band-splitting in ultrathin Bi films: from a Rashba effect to a parity effect. New J. Phys., 10, 083038.
- Nagao, T., Sadowski, J.T., Saito, M., Yaginuma, S., Fujikawa, Y., Kogure, T., Ohno, T., Hasegawa, Y., Hasegawa, S., and Sakurai, T. (2004) Nanofilm allotrope and phase transformation of ultrathin Bi film on Si(111)7 × 7. Phys. Rev. Lett., 93, 105501.
- Hofmann, P. (2006) The surfaces of bismuth: structural and electronic properties. Prog. Surf. Sci., 81, 191–245.
- Liu, Z., Liu, C.-X., Wu, Y.-S., Duan, W.-H., Liu, F., and Wu, J. (2011) Stable nontrivial ℤ2 topology in ultrathin Bi (111) films: a first-principles study. Phys. Rev. Lett., 107, 136805.
- Koroteev, Y.M., Bihlmayer, G., Chulkov, E.V., and Blügel, S. (2008) First-principles investigation of structural and electronic properties of ultrathin Bi films. Phys. Rev. B, 77, 045428.
- Zhang, P., Liu, Z., Duan, W., Liu, F., and Wu, J. (2012) Topological and electronic transitions in a Sb(111) nanofilm: the interplay between quantum confinement and surface effect. Phys. Rev. B, 85, 201410.
- Tichovolski, E.J. and Mavroides, J.G. (1969) Magnetoreflectance studies on the bandstructure of Bismuth-Antimony alloys. Solid State Commun., 7, 927.
- Ast, C.R. and Höchst, H. (2001) Fermi surface of Bi(111) measured by photoemission spectroscopy. Phys. Rev. Lett., 87, 177602.
- Höchst, H. and Ast, C.R. (2004) The Fermi surfaces of thin Sb(111) films. J. Electron. Spectrosc. Relat. Phenom., 137-140, 441–444.
- Sugawara, K., Sato, T., Souma, S., Takahashi, T., Arai, M., and Sasaki, T. (2006) Fermi surface and anisotropic spin-orbit coupling of Sb(111) studied by angle-resolved photoemission spectroscopy. Phys. Rev. Lett., 96, 046411.
- Koroteev, Y.M., Bihlmayer, G., Gayone, J.E., Chulkov, E.V., Blügel, S., Echenique, P.M., and Hofmann, P. (2004) Strong spin-orbit splitting on Bi surfaces. Phys. Rev. Lett., 93, 046403.
- Hsieh, D., Qian, D., Wray, L., Xia, Y., Hor, Y.S., Cava, R.J., and Hasan, M.Z. (2008) A topological Dirac insulator in a quantum spin Hall phase. Nature, 452, 970.
- Guo, H., Sugawara, K., Takayama, A., Souma, S., Sato, T., Satoh, N., Ohnishi, A., Kitaura, M., Sasaki, M., Xue, Q.-K., and Takahashi, T. (2011) Evolution of surface states in Bi1−xSbx alloys across the topological phase transition. Phys. Rev. B, 83, 201104.
- Bihlmayer, G., Koroteev, Y.M., Chulkov, E.V., and Blügel, S. (2010) Surface and edge-states in ultrathin Bi-Sb films. New J. Phys., 12, 065006.
- Wang, D., Chen, L., Liu, H., and Wang, X. (2013) Electronic structures and topological properties of Bi(111) ultrathin films. J. Phys. Soc. Jpn., 82, 094712.
- Edelman, V.S. (2008) Tunneling spectra at terrace boundaries on the bismuth surface. J. Exp. Theor. Phys., 107, 251.
- Yaginuma, S., Nagaoka, K., Nagao, T., Bihlmayer, G., Koroteev, Y.M., Chulkov, E.V., and Nakayama, T. (2008) Electronic structure of ultrathin bismuth films with A7 and black-phosphorus-like structures. J. Phys. Soc. Jpn., 77, 014701.
-
Kotaka, H., Ishii, F., Saito, M., Nagao, T., and Yaginuma, S. (2012) Edge states of Bi nanoribbons on Bi substrates: first-principles density functional study. Jpn. J. Appl. Phys., 51, 025201.
10.7567/JJAP.51.025201 Google Scholar
- Wells, J.W., Dil, J.H., Meier, F., Lobo-Checa, J., Petrov, V.N., Osterwalder, J., Ugeda, M.M., Fernandez-Torrente, I., Pascual, J.I., Rienks, E.D.L., Jensen, M.F., and Hofmann, P. (2009) Nondegenerate metallic states on Bi(114): a one-dimensional topological metal. Phys. Rev. Lett., 102, 096802.
- Sabater, C., Gosálbez-Martínez, D., Fernández-Rossier, J., Rodrigo, J.G., Untiedt, C., and Palacios, J.J. (2013) Topologically protected quantum transport in locally exfoliated bismuth at room temperature. Phys. Rev. Lett., 110, 176802.
- Hirahara, T., Bihlmayer, G., Sakamoto, Y., Yamada, M., Miyazaki, H., Kimura, S., Blügel, S., and Hasegawa, S. (2011) Interfacing 2D and 3D topological insulators: Bi(111) bilayer on Bi2Te3. Phys. Rev. Lett., 107, 166801.
- Miao, L., Wang, Z.F., Ming, W., Yao, M.-Y., Wang, M., Yang, F., Song, Y.R., Zhu, F., Fedorov, A.V., Sun, Z., Gao, C.L., Liu, C., Xue, Q.-K., Liu, C.-X., Liu, F., Qian, D., and Jia, J.-F. (2013) Quasiparticle dynamics in reshaped helical Dirac cone of topological insulators. Proc. Natl. Acad. Sci. U.S.A., 110 (8), 2758–2762.
- Wang, L.-L. and Johnson, D.D. (2011) Ternary tetradymite compounds as topological insulators. Phys. Rev. B, 83, 241309.
- Zhang, H., Liu, C.-X., Qi, X.-L., Dai, X., Fang, Z., and Zhang, S.-C. (2009) Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nat. Phys., 5, 438.
- Pauly, C., Bihlmayer, G., Liebmann, M., Grob, M., Georgi, A., Subramaniam, D., Scholz, M.R., Sànchez-Barriga, J., Varykhalov, A., Blügel, S., Rader, O., and Morgenstern, M. (2012) Probing two topological surface bands of Sb2Te3 by spin-polarized photoemission spectroscopy. Phys. Rev. B, 86, 235106.
- Park, K., Heremans, J.J., Scarola, V.W., and Minic, D. (2010) Robustness of topologically protected surface states in layering of Bi2Te3 thin films. Phys. Rev. Lett., 105, 186801.
- Kuroda, K., Miyahara, H., Ye, M., Eremeev, S.V., Koroteev, Y.M., Krasovskii, E.E., Chulkov, E.V., Hiramoto, S., Moriyoshi, C., Kuroiwa, Y., Miyamoto, K., Okuda, T., Arita, M., Shimada, K., Namatame, H., Taniguchi, M., Ueda, Y., and Kimura, A. (2012) Experimental verification of PbBi2Te4 as a 3D topological insulator. Phys. Rev. Lett., 108, 206803.
- Menshchikova, T., Eremeev, S., Koroteev, Y., Kuznetsov, V., and Chulkov, E. (2011) Ternary compounds based on binary topological insulators as an efficient way for modifying the Dirac cone. JETP Lett., 93, 15–20.
- Eremeev, S.V., Landolt, G., Menshchikova, T.V., Slomski, B., Koroteev, Y.M., Aliev, Z.S., Babanly, M.B., Henk, J., Ernst, A., Patthey, L., Eich, A., Khajetoorians, A.A., Hagemeister, J., Pietzsch, O., Wiebe, J., Wiesendanger, R., Echenique, P.M., Tsirkin, S.S., Amiraslanov, I.R., Dil, J.H., and Chulkov, E.V. (2012) Atom-specific spin mapping and buried topological states in a homologous series of topological insulators. Nat. Commun., 3, 635.
- Silkin, I.V., Koroteev, Y.M., Eremeev, S.V., Bihlmayer, G., and Chulkov, E.V. (2011) Three- and two-dimensional topogical insulators in Pb2Sb2Te5, Pb2Bi2Te5, and Pb2Bi2Se5 layered compounds. JETP Lett., 94, 217.
