Introduction to Dirac materials and topological insulators

Jérôme Cayssol

doi:10.1016/j.crhy.2013.09.012

Topological insulators/Isolants topologiques

Introduction to Dirac materials and topological insulators
[Introduction aux matériaux de Dirac et aux isolants topologiques]

Jérôme Cayssol ^1,²

¹ Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany
² LOMA (UMR 5798), CNRS and University Bordeaux-1, 33045 Talence, France

Comptes Rendus. Physique, Topological insulators / Isolants topologiques, Volume 14 (2013) no. 9-10, pp. 760-778.

Résumé
Abstract

Nous présentons dans cet article une courte introduction didactique à la physique des matériaux de Dirac, restreinte au graphène et à des isolants topologiques en deux dimensions. Nous commençons par un bref rappel des équations de Dirac et de Weyl dans le contexte de la physique des particules. Abordant les systèmes relatifs à la matière condensée, le graphène semi-métallique et divers isolants de Dirac sont présentés, parmi lesquels les isolants topologiques de Haldane et de Kane–Mele. Nous discutons aussi brièvement les réalisations expérimentales avec des matériaux à fort couplage spin–orbite.

We present a short pedagogical introduction to the physics of Dirac materials, restricted to graphene and two-dimensional topological insulators. We start with a brief reminder of the Dirac and Weyl equations in the particle physics context. Turning to condensed matter systems, semimetallic graphene and various Dirac insulators are introduced, including the Haldane and the Kane–Mele topological insulators. We also discuss briefly experimental realizations in materials with strong spin–orbit coupling.

Publié le : 2013-10-14

DOI : 10.1016/j.crhy.2013.09.012

Keywords: Dirac fermions, Graphene, Topological insulators, Edge modes
Mots-clés : Fermions de Dirac, Graphène, Isolants topologiques, États de bord

Affiliations des auteurs :

Jérôme Cayssol ^{1, 2}

¹ Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany
² LOMA (UMR 5798), CNRS and University Bordeaux-1, 33045 Talence, France

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Jérôme Cayssol. Introduction to Dirac materials and topological insulators. Comptes Rendus. Physique, Topological insulators / Isolants topologiques, Volume 14 (2013) no. 9-10, pp. 760-778. doi : 10.1016/j.crhy.2013.09.012. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2013.09.012/

Bibliographie
Cité par

[1] A. Zee Quantum Field Theory in a Nutshell, Princeton University Press, 2010

[2] S. Weinberg The Quantum Theory of Fields, Volume 1: Foundations, Cambridge University Press, 2005

[3] P.A.M. Dirac The quantum theory of the electron, P. Roy. Soc. Lond. Ser., Volume 117 (1928) no. 778, pp. 610-624

[4] P.A.M. Dirac A theory of electrons and protons, P. Roy. Soc. Lon. Ser.-A, Volume 126 (1930) no. 801, pp. 360-365

[5] O. Klein Z. Phys., 37 (1926), p. 895

[6] H. Weyl Electron and gravitation, Z. Phys., Volume 56 (1929), pp. 330-352

[7] E. Majorana Theory of the symmetry of electrons and positrons, Nuovo Cim., Volume 14 (1937), pp. 171-184

[8] P.B. Pal Dirac Majorana and Weyl fermions, 2010 | arXiv

[9] M.Z. Hasan; C.L. Kane Colloquium: Topological insulators, Rev. Mod. Phys., Volume 82 ( Nov. 2010 ), pp. 3045-3067

[10] Xiao-Liang Qi; Shou-Cheng Zhang Topological insulators and superconductors, Rev. Mod. Phys., Volume 83 ( Oct. 2011 ), pp. 1057-1110

[11] Markus König; Hartmut Buhmann; Laurens W. Molenkamp; Taylor Hughes; Chao-Xing Liu; Xiao-Liang Qi; Shou-Cheng Zhang The quantum spin Hall effect: theory and experiment, J. Phys. Soc. Jpn., Volume 77 ( March 2008 ) no. 3, p. 031007

[12] X.-L. Qi; S.-C. Zhang The quantum spin Hall effect and topological insulators, Phys. Today, Volume 63 (2010), p. 33

[13] B. Bernevig Topological Insulators and Topological Superconductors, Cambridge University Press, 2013

[14] C.L. Kane; E.J. Mele Quantum spin Hall effect in graphene, Phys. Rev. Lett., Volume 95 (2005), p. 226801

[15] C.L. Kane; E.J. Mele Topological order and the quantum spin Hall effect, Phys. Rev. Lett., Volume 95 (2005), p. 146802

[16] Frank T. Avignone; Steven R. Elliott; Jonathan Engel Double beta decay, Majorana neutrinos, and neutrino mass, Rev. Mod. Phys., Volume 80 ( Apr. 2008 ), pp. 481-516

[17] K.S. Novoselov; A.K. Geim; S.V. Morosov; D. Jiang; Y. Zhang; S.V. Dubonos; I.V. Grigorieva; A.A. Firsov Electric field effect in atomically thin carbon films, Science, Volume 306 (2004), p. 666

[18] K.S. Novoselov; A.K. Geim; S.V. Morosov; D. Jiang; M.I. Katsnelson; S.V. Grigorieva; I.V. Dubonos; A.A. Firsov Two-dimensional gas of massless Dirac fermions in graphene, Nature, Volume 438 (2005), p. 197

[19] Y. Zhang; Y.-W. Tan; H.L. Stormer; P. Kim Experimental observation of the quantum Hall effect and Berryʼs phase in graphene, Nature, Volume 438 (2005), p. 201

[20] P.R. Wallace The band theory of graphite, Phys. Rev., Volume 71 ( May 1947 ), pp. 622-634

[21] D.P. DiVincenzo; E.J. Mele Self-consistent effective-mass theory for intralayer screening in graphite intercalation compounds, Phys. Rev. B, Volume 29 ( Feb. 1984 ), pp. 1685-1694

[22] Gordon W. Semenoff Condensed-matter simulation of a three-dimensional anomaly, Phys. Rev. Lett., Volume 53 (1984), p. 2449

[23] F.D.M. Haldane Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett., Volume 61 (1988), p. 2015

[24] O. Klein Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac, Z. Phys., Volume 53 (1929), pp. 3-4

[25] Vadim V. Cheianov; Vladimir I. Falʼko Selective transmission of Dirac electrons and ballistic magnetoresistance of $n – p$ junctions in graphene, Phys. Rev. B, Volume 74 ( Jul. 2006 ), p. 041403

[26] M.I. Katsnelson; K.S. Novoselov; A.K. Geim Chiral tunnelling and the Klein paradox in graphene, Nat. Phys., Volume 2 (2006), p. 620 | DOI

[27] J. Cayssol; B. Huard; D. Goldhaber-Gordon Contact resistance and shot noise in graphene transistors, Phys. Rev. B, Volume 79 ( Feb. 2009 ), p. 075428

[28] Ai Yamakage; Ken-Ichiro Imura; Jérôme Cayssol; Yoshio Kuramoto Interfacial charge and spin transport in $Z_{2}$ topological insulators, Phys. Rev. B, Volume 83 (2011), p. 125401

[29] B. Huard; J.A. Sulpizio; N. Stander; K. Todd; B. Yang; D. Goldhaber-Gordon Transport measurements across a tunable potential barrier in graphene, Phys. Rev. Lett., Volume 98 ( Jun. 2007 ), p. 236803

[30] J.R. Williams; L. DiCarlo; C.M. Marcus Quantum Hall effect in a gate-controlled p–n junction of graphene, Science, Volume 317 (2007), p. 638

[31] Barbaros Özyilmaz; Pablo Jarillo-Herrero; Dmitri Efetov; Dmitry A. Abanin; Leonid S. Levitov; Philip Kim Electronic transport and quantum Hall effect in bipolar graphene $p – n – p$ junctions, Phys. Rev. Lett., Volume 99 ( Oct. 2007 ), p. 166804

[32] N. Stander; B. Huard; D. Goldhaber-Gordon Evidence for Klein tunneling in graphene $p – n$ junctions, Phys. Rev. Lett., Volume 102 ( Jan. 2009 ), p. 026807

[33] A.F. Young; P. Kim Quantum interference and carrier collimation in graphene heterojunctions, Nat. Phys., Volume 5 (2009), pp. 222-226

[34] T. Ando; T. Nakanishi; R. Saito Berryʼs phase and absence of back scattering in carbon nanotubes, J. Phys. Soc. Jpn., Volume 67 (1998), p. 2857

[35] P.E. Alain; J.N. Fuchs Klein tunneling in graphene: optics with massless electrons, Eur. Phys. J. B, Volume 83 (2011), pp. 301-317

[36] A.H. Castro Neto; F. Guinea; N.M.R. Peres; K.S. Novoselov; A.K. Geim The electronic properties of graphene, Rev. Mod. Phys., Volume 81 (2009), p. 109

[37] M.O. Goerbig Electronic properties of graphene in a strong magnetic field, Rev. Mod. Phys., Volume 83 (2011), p. 1193

[38] Valeri; N. Kotov; Bruno Uchoa; Vitor M. Pereira; F. Guinea; A.H. Castro Neto Electron–electron interactions in graphene: Current status and perspectives, Rev. Mod. Phys., Volume 84 ( Jul. 2012 ), pp. 1067-1125

[39] M.O. Goerbig; J.-N. Fuchs; G. Montambaux; F. Piéchon Phys. Rev. B, 78 ( Jul. 2008 ), p. 045415

[40] F. Guinea; M.I. Katsnelson; A.K. Geim Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering, Nat. Phys., Volume 6 (2010), p. 30

[41] Pouyan Ghaemi; Jérôme Cayssol; D.N. Sheng; Ashvin Vishwanath Fractional topological phases and broken time-reversal symmetry in strained graphene, Phys. Rev. Lett., Volume 108 ( Jun. 2012 ), p. 266801

[42] Shinsei Ryu; Christopher Mudry; Chang-Yu Hou; Claudio Chamon Masses in graphenelike two-dimensional electronic systems: Topological defects in order parameters and their fractional exchange statistics, Phys. Rev. B, Volume 80 (2009), p. 205319

[43] J. Zak Berryʼs phase for energy bands in solids, Phys. Rev. Lett., Volume 62 ( Jun. 1989 ), pp. 2747-2750

[44] L. Tarruell; D. Greif; T. Uehlinger; G. Jotzu; T. Esslinger Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice, Nature, Volume 483 (2012), p. 302

[45] E. Tang; J.-W. Mei; X.-G. Wen High-temperature fractional quantum Hall states, Phys. Rev. Lett., Volume 106 (2011), p. 236802

[46] K. Sun; Z. Gu; H. Katsura; S. Das Sarma Nearly flatbands with nontrivial topology, Phys. Rev. Lett., Volume 106 (2011), p. 236803

[47] T. Neupert; L. Santos; C. Chamon; C. Mudry Fractional quantum Hall states at zero magnetic field, Phys. Rev. Lett., Volume 106 (2011), p. 236804

[48] N. Regnault; B. Andrei Bernevig Fractional Chern insulator, Phys. Rev. X, Volume 1 (2011), p. 021014

[49] Y.-F. Wang; Z.-C. Gu; C.-D. Gong; D.N. Sheng Fractional quantum Hall effect of hard-core bosons in topological flat bands, Phys. Rev. Lett., Volume 107 (2011), p. 146803

[50] T. Neupert; L. Santos; S. Ryu; C. Chamon; C.ʼ Mudry Fractional topological liquids with time-reversal symmetry and their lattice realization, Phys. Rev. B, Volume 84 (2011), p. 165107

[51] C.-Z. Chang; J. Zhang; X. Feng; J. Shen; Z. Zhang; M. Guo; K. Li; Y. Ou; P. Wei; L.-L. Wang; Y. Feng; S. Ji; X. Chen; J. Jia; X. Dai; Z. Fang; S.-C. Zhang; K. He; Y. Wang; L. Lu; X.-C. Ma; Q.-K. Xue Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator, Science, Volume 340 (2013) no. 6129, pp. 167-170

[52] Shuichi Murakami; Naoto Nagaosa; Shou-Cheng Zhang Dissipationless quantum spin current at room temperature, Science, Volume 301 (2003) no. 5638, pp. 1348-1351

[53] Jairo Sinova; Dimitrie Culcer; Q. Niu; N.A. Sinitsyn; T. Jungwirth; A.H. MacDonald Universal intrinsic spin Hall effect, Phys. Rev. Lett., Volume 92 ( Mar. 2004 ), p. 126603

[54] Y.K. Kato; R.C. Myers; A.C. Gossard; D.D. Awschalom Observation of the spin Hall effect in semiconductors, Science, Volume 306 (2004) no. 5703, pp. 1910-1913

[55] J. Wunderlich; B. Kaestner; J. Sinova; T. Jungwirth Experimental observation of the spin-Hall effect in a two-dimensional spin–orbit coupled semiconductor system, Phys. Rev. Lett., Volume 94 ( Feb. 2005 ), p. 047204

[56] Daniel Huertas-Hernando; F. Guinea; Arne Brataas Spin–orbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps, Phys. Rev. B, Volume 74 ( Oct. 2006 ), p. 155426

[57] Hongki Min; J.E. Hill; N.A. Sinitsyn; B.R. Sahu; Leonard Kleinman; A.H. MacDonald Intrinsic and Rashba spin–orbit interactions in graphene sheets, Phys. Rev. B, Volume 74 ( Oct. 2006 ), p. 165310

[58] B.A. Bernevig; Taylor L. Hughes; Shou-Cheng Zhang Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science, Volume 314 (2006) no. 5806, p. 1757

[59] M. König; S. Wiedmann; C. Brüne; A. Roth; H. Buhmann; L.W. Molenkamp; X.L. Qi; S.C. Zhang Quantum spin Hall insulator state in HgTe quantum wells, Science, Volume 318 ( November 2007 ) no. 5851, pp. 766-770

[60] A. Roth; C. Brune; H. Buhmann; L.W. Molenkamp; J. Maciejko; X.L. Qi; S.C. Zhang Nonlocal transport in the quantum spin Hall state, Science, Volume 325 ( July 2009 ) no. 5938, pp. 294-297

[61] Chaoxing Liu; Taylor L. Hughes; Xiao-Liang Qi; Kang Wang; Shou-Cheng Zhang Quantum spin Hall effect in inverted type-ii semiconductors, Phys. Rev. Lett., Volume 100 ( Jun. 2008 ), p. 236601

[62] Ivan Knez; Rui-Rui Du; Gerard Sullivan Evidence for helical edge modes in inverted $InAs / GaSb$ quantum wells, Phys. Rev. Lett., Volume 107 ( Sep. 2011 ), p. 136603

[63] Chao-Xing Liu; Xiao-Liang Qi; Xi Dai; Zhong Fang; Shou-Cheng Zhang Quantum anomalous Hall effect in ${hg}_{1 - y} {mn}_{y} Te$ quantum wells, Phys. Rev. Lett., Volume 101 ( Oct. 2008 ), p. 146802

[64] Jing Wang; Biao Lian; Haijun Zhang; Shou-Cheng Zhang Anomalous edge transport in the quantum anomalous Hall state, Phys. Rev. Lett., Volume 111 ( Aug. 2013 ), p. 086803

[65] D.J. Thouless; M. Kohmoto; M.P. Nightingale; M. den Nijs Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett., Volume 49 ( Aug. 1982 ), pp. 405-408

[66] Xiao-Liang Qi; Yong-Shi Wu; Shou-Cheng Zhang Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors, Phys. Rev. B, Volume 74 ( Aug. 2006 ), p. 085308

[67] Doru Sticlet; Frederic Piéchon; Jean-Noël Fuchs; Pavel Kalugin; Pascal Simon Geometrical engineering of a two-band Chern insulator in two dimensions with arbitrary topological index, Phys. Rev. B, Volume 85 ( Apr. 2012 ), p. 165456

[68] R. Jackiw; C. Rebbi Solitons with fermion number 1/2, Phys. Rev. D, Volume 13 ( Jun. 1976 ), pp. 3398-3409

[69] W.P. Su; J.R. Schrieffer; A.J. Heeger Solitons in polyacetylene, Phys. Rev. Lett., Volume 42 ( Jun. 1979 ), pp. 1698-1701

[70] W.P. Su; J.R. Schrieffer; A.J. Heeger Soliton excitations in polyacetylene, Phys. Rev. B, Volume 22 ( Aug. 1980 ), pp. 2099-2111

[71] Takashi Oka; Hideo Aoki Photovoltaic Hall effect in graphene, Phys. Rev. B, Volume 79 (2009), p. 081406

[72] Takuya Kitagawa; Takashi Oka; Arne Brataas; Liang Fu; Eugene Demler Transport properties of nonequilibrium systems under the application of light: Photoinduced quantum Hall insulators without Landau levels, Phys. Rev. B, Volume 84 ( Dec. 2011 ), p. 235108

[73] Zhenghao Gu; H.A. Fertig; Daniel P. Arovas; Assa Auerbach Floquet spectrum and transport through an irradiated graphene ribbon, Phys. Rev. Lett., Volume 107 ( Nov. 2011 ), p. 216601

[74] Manuel Torres; Alejandro Kunold Kubo formula for Floquet states and photoconductivity oscillations in a two-dimensional electron gas, Phys. Rev. B, Volume 71 ( Mar. 2005 ), p. 115313

[75] Netanel H. Lindner; Gil Refael; Victor Galitski Floquet topological insulator in semiconductor quantum wells, Nat. Phys., Volume 7 (2011), pp. 490-495

[76] J. Cayssol; B. Dora; F. Simon; R. Moessner Floquet topological insulators, Phys. Status Solidi, Volume 7 (2013) no. 1–2, pp. 101-108

[77] M. Polini; F. Guinea; M. Lewenstein; H.C. Manoharan; V. Pellegrini Artificial graphene as a tunable Dirac material, 2013 | arXiv

[78] K.K. Gomes; W. Mar; W. Ko; W. Guinea; H.C. Manoharan Designer Dirac fermions and topological phases in molecular graphene, Nature, Volume 483 (2012), p. 306

[79] A. Singha et al. Two-dimensional Mott–Hubbard electrons in an artificial honeycomb lattice, Science, Volume 332 (2011) no. 6034, pp. 1176-1179

[80] C.H. Park; S. Louie Making massless Dirac fermions from patterned two-dimensional electron gases, Nano Lett., Volume 9 (2009), pp. 1793-1797

[81] G.E. Volovik An analog of the quantum Hall effect in a superfluid ³He film, JETP, Volume 67 (1988), pp. 1804-1811

[82] G.E. Volovik, The Universe in a Helium Droplet, The International Series of Monographs on Physics, vol. 117, Oxford.

[83] A.B. Khanikaev; S.H. Mousavi; W.-K. Tse; M. Kargarian; A.H. MacDonald; G. Shvets Photonic topological insulators, Nature Materials, Volume 12 (2013), pp. 233-239

[84] M.C. Rechtsman; J.M. Zeuner; Y. Plotnik; Y. Lumer; D. Podolsky; F. Dreisow; S. Nolte; M. Segev; A. Szameit Photonic Floquet topological insulators, Nature, Volume 496 (2013), pp. 196-200

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