Dirac Matter PDF
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Progress in Mathematical Physics 71 Bertrand Duplantier Vincent Rivasseau Jean-Nöel Fuchs Editors Dirac Matter Progress in Mathematical Physics Volume 71 Editors-in-chief AnneBoutetdeMonvel,UniversitédeParisVIIUFRdeMathematiques, ParisCX05,France GeraldKaiser,CenterforSignalsandWaves,Portland,Oregon,USA EditorialBoard SirM.Berry,UniversityofBristol,UK P.Blanchard,UniversityofBielefeld,Germany M.Eastwood,UniversityofAdelaide,Australia A.S.Fokas,UniversityofCambridge,UK F.W.Hehl,UniversityofCologne,Germany andUniversityofMissouri-Columbia,USA D.Sternheimer,UniversitédeBourgogne,Dijon,France C.Tracy,UniversityofCalifornia,Davis,USA Moreinformationaboutthisseriesathttp://www.springer.com/series/4813 Bertrand Duplantier • Vincent Rivasseau Jean-Nöel Fuchs Editors Dirac Matter Editors Bertrand Duplantier Vincent Rivasseau Service de Physique Théorique Laboratoire de Physique Théorique CEA Saclay Université Pa ris-Sud Gif Sur Yvette Cedex, France Orsay, France Jean-Nöel Fuchs Laboratoire de Physique Université Pierre-et-Marie Curie Paris, France ISSN 1544-9998 ISSN2197-1846 (electronic) Progress inMathematicalPh ysic s ISBN 978-3-319-32535-4 ISBN 978-3-319-32536-1 (eBook) DOI 10.1007/978-3-319-32536-1 Library of Congress Control Number: 2016963686 Mathematics Subject Classification (2010): 82-01, 82-02, 81V70, 35Q41 © Springer International Publishing Switzerland 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This book is published under the trade name Birkhäuser, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Contents Foreword ................................................................. ix Philip Kim Graphene and Relativistic Quantum Physics 1 Introduction ........................................................... 1 2 Early experiment ...................................................... 2 3 Pseudospin chirality in graphene ....................................... 4 4 Berry phase in magneto-oscillations .................................... 7 5 Pseudospin and Klein tunneling in graphene ........................... 14 6 Conclusions ........................................................... 19 References ............................................................. 21 Mark Goerbig and Gilles Montambaux Dirac Fermions in Condensed Matter and Beyond 1 Introduction ........................................................... 25 2 Emergence of Dirac fermions in a generic two-band model ............. 26 3 Dirac fermions in tight-binding models and fermion doubling ........... 28 4 Dirac fermions in a magnetic field ..................................... 32 5 Motion and merging of time-reversal-symmetric Dirac points ........... 37 6 Manipulation of Dirac points in artificial graphenes .................... 40 7 More Dirac points ..................................................... 46 8 Conclusions ........................................................... 49 References ............................................................. 51 vi Contents Chuan Li, Sophie Gu´eron and H´el`ene Bouchiat Quantum Transport in Graphene: Impurity Scattering as a Probe of the Dirac Spectrum 1 Introduction ........................................................... 55 2 Impurity scattering in graphene: determination of the transport and elastic scattering times ............................................ 57 3 Quantum transport: proximity induced superconductivity and specular Andreev reflection. ....................................... 62 4 Perspectives: inducing new functionalities in graphene by creating scattering centers .......................................... 67 5 Conclusion ............................................................ 71 References ............................................................. 72 Laurent L´evy Experimental Signatures of Topological Insulators 1 Introduction ........................................................... 75 2 Bulked gap in strained mercury telluride ............................... 79 3 ARPES spectra and surface mercury telluride .......................... 80 4 Topological signatures in transport experiments ........................ 87 5 Conclusions ........................................................... 92 References ............................................................. 93 David Carpentier Topology of Bands in Solids: From Insulators to Dirac Matter 1 Introduction ........................................................... 95 2 Bloch theory .......................................................... 97 3 Geometrical phase and parallel transport .............................. 99 4 Topological properties of a valence band in an insulator ................ 107 5 From insulators to semi-metals ........................................ 120 6 Conclusion and perspectives ........................................... 125 Appendix: Two useful trivializations of the Bloch bundle ............... 125 References ............................................................. 127 Contributors Philip Kim Department of Physics, Harvard University, Cambridge, Massachussets, USA Mark Goerbig and Gilles Montambaux Laboratoire de Physique des Solides, Universit´e Paris Sud, Orsay, France Chuan Li, Sophie Gu´eron and H´el`ene Bouchiat Laboratoire de Physique des Solides, Universit´e Paris Sud, Orsay, France Laurent L´evy Institut N´eel, Universit´e Grenoble Alpes and CNRS, Grenoble, France David Carpentier Laboratoire de Physique, Ecole Normale Sup´erieure de Lyon, Lyon, France Foreword This book is the fifteenth in a series of Proceedings for the S´eminaire Poincar´e, which is directed towards a broad audience of physicists, mathematicians, and philosophers of science. The goal of this Seminar is to provide up-to-date information about general topics of great interest in physics. Both the theoretical and experimental aspects of the topic are covered, generally with some historical background. Inspired by the Nicolas Bourbaki Seminar in mathematics, hence nicknamed “Bourbaphy”, the Poincar´eSeminar is held twice a year at the Institut Henri Poincar´ein Paris, with written contributions prepared in advance. Particular care is devoted to the pedagogicalnature of the presentations,so that they may be accessible to a large audience of scientists. ThisnewvolumeofthePoincar´eSeminarSeries,DiracMatter,correspondsto the eighteenthsuchseminar,heldonJune28,2014.Its aimistoprovideageneral introductiontothephysicsofmaterials,theconductionpropertiesofwhicharedue to electrons described by the Dirac equation. The latter was introduced by Paul Dirac in 1928 in order to wed quantum mechanics and special relativity so as to describe a fast moving electron beyond the non-relativistic Schr¨odinger equation. Thisoutstandingcontributionwasrecognizedbytheawardofthe1933NobelPrize inPhysicstoDirac.Althoughinhigh-energyphysicsDirac’sequationwaslongago supersededby quantumfieldtheoryasthe descriptionoffundamental particles,it was recently resurrected as the correct low-energy description of many condensed matter systems, such as graphene and topological insulators. The present volume explains why Dirac still matters. Thefirstarticle,entitled“GrapheneandRelativisticQuantumPhysics”,and written by the experimental pioneer, Philip Kim, is devoted to graphene, a two- dimensional form of crystalline carbon organized in a honeycomb lattice. The author starts by recounting the path that lead to the discovery in 2004-2005 by Kostya Novoselov and Andre Geim of this new type of crystals, whose thickness is that of a single atom. The discovery of graphene and its use as an electronic devicewassoonafterrecognizedbythe awardtothe pairofthe 2010NobelPrize. Kim then goes on to explain the surprising behavior of electrons in this material by showing how the massless Dirac equation emerges as their low-energyeffective description.Indeed,thebandstructureofgrapheneissuchthattheconductionand valence bands touch at two inequivalent contact points which form a degenerate Fermisurface.Inthevicinityofthesecontactpoints,electronsarenotdescribedby
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