{{Short description|Quantum effect in some non-metals}} {{use dmy dates|date=December 2020}} [[File:Graphene Brillouin Zone & Linear Dispersion.PNG|thumb|upright=1.75|Electronic band structure of monolayer graphene, with a zoomed inset showing the Dirac cones. There are 6&nbsp;cones corresponding to the 6&nbsp;vertices of the hexagonal first Brillouin zone.|alt=Brillouin zone in graphene ]] In physics, '''Dirac cones''' are features that occur in some electronic band structures that describe unusual electron transport properties of materials like graphene and topological insulators.<ref name=Novoselov-Geim-2007> {{cite journal |last1=Novoselov |first1=K.S. |last2=Geim |first2=A.K. |date=2007 |title=The rise of graphene |journal=Nature Materials |volume=6 |issue=3 |pages=183–191 |doi=10.1038/nmat1849 |pmid=17330084 |bibcode=2007NatMa...6..183G |s2cid=14647602 }} </ref><ref name=Hasan-Kane-2010> {{cite journal |last1=Hasan |first1=M.Z. |last2=Kane |first2=C.L. |date=2010 |title=Topological Insulators |journal=Rev. Mod. Phys. |volume=82 |issue=4 |page=3045 |doi=10.1103/revmodphys.82.3045 |arxiv=1002.3895 |bibcode=2010RvMP...82.3045H |s2cid=16066223 |url=https://repository.upenn.edu/cgi/viewcontent.cgi?article=1043&context=physics_papers }} </ref><ref name=AMIR-TohokuU-2016-08-29> {{cite web |title=Superconductors: Dirac cones come in pairs |date=29 Aug 2011 |series=Research Highlights |publisher=Tohoku University |website=wpi-aimr.tohoku.ac.jp |department=Advanced Institute for Materials Research |url=http://www.wpi-aimr.tohoku.ac.jp/en/aimresearch/highlight/2011/20110829_000812.html |access-date=2 Mar 2018 |language=en }}</ref> In these materials, at energies near the Fermi level, the valence band and conduction band take the shape of the upper and lower halves of a conical surface, meeting at what are called '''Dirac points'''.

Typical examples include graphene, topological insulators, bismuth antimony thin films and some other novel nanomaterials,<ref name=Novoselov-Geim-2007 /><ref>[https://physicsworld.com/a/dirac-cones-could-exist-in-bismuth-antimony-films/ Dirac cones could exist in bismuth–antimony films]. Physics World, Institute of Physics, 17 April 2012.</ref><ref>{{cite journal|last1=Hsieh|first1=David|date=2008|title=A topological Dirac insulator in a quantum spin Hall phase|journal=Nature|volume=452|issue=7190|pages=970–974|doi=10.1038/nature06843|pmid=18432240|arxiv=0902.1356 |bibcode=2008Natur.452..970H|url=https://authors.library.caltech.edu/49766/|access-date=18 August 2023|archive-date=22 August 2023|archive-url=https://web.archive.org/web/20230822123717/https://authors.library.caltech.edu/49766/}}</ref> in which the electronic energy and momentum have a linear dispersion relation such that the electronic band structure near the Fermi level takes the shape of an upper conical surface for the electrons and a lower conical surface for the holes. The two conical surfaces touch each other and form a zero-band gap semimetal.

The name of Dirac cone comes from the Dirac equation that can describe relativistic particles in quantum mechanics, proposed by Paul Dirac. Isotropic Dirac cones in graphene were first predicted by P. R. Wallace in 1947<ref name="graph">{{cite journal|author=Wallace, P. R.|year=1947|title=The Band Theory of Graphite|journal=Physical Review|volume=71|issue=9|pages=622–634|doi=10.1103/PhysRev.71.622|bibcode=1947PhRv...71..622W}}</ref> and experimentally observed by the Nobel Prize laureates Andre Geim and Konstantin Novoselov in 2005.<ref>[https://www.nobelprize.org/nobel_prizes/physics/laureates/2010/press.html The Nobel Prize in Physics 2010 Press Release]. Nobelprize.org, 5 October 2010. Retrieved 2011-12-31.</ref>

==Description== thumb|Tilted Dirac cones in momentum space. From left to right, the tilt increases, from no tilt in the first cone to overtilt in the last. The three first are Type-I Weyl semimetals, the last one is a Type-II Weyl semimetal. In quantum mechanics, Dirac cones are a kind of crossing-point which electrons avoid,<ref name=Fuchs-etal-2012>{{cite journal |first1 = Jean-Noël |last1 = Fuchs |first2 = Lih-King |last2 = Lim |first3 = Gilles |last3 = Montambaux |year = 2012 |title = Interband tunneling near the merging transition of Dirac cones |journal = Physical Review&nbsp;A |volume = 86 |issue = 6 |article-number = 063613 |doi = 10.1103/PhysRevA.86.063613 |arxiv = 1210.3703 |bibcode = 2012PhRvA..86f3613F |s2cid = 67850936 |url = https://www.equipes.lps.u-psud.fr/Montambaux/reprints/178-interband-tunneling.pdf |access-date = 29 August 2018 |archive-date = 21 January 2023 |archive-url = https://web.archive.org/web/20230121154439/https://www.equipes.lps.u-psud.fr/Montambaux/reprints/178-interband-tunneling.pdf }}</ref> where the energy of the valence and conduction bands are not equal anywhere in two dimensional lattice {{mvar|k}}-space, except at the zero dimensional Dirac points. As a result of the cones, electrical conduction can be described by the movement of charge carriers which are massless fermions, a situation which is handled theoretically by the relativistic Dirac equation.<ref name=Novoselov-Morozov-etal-2005> {{cite journal |first1=K.S. |last1=Novoselov |first2=A.K. |last2=Geim |first3=S.V. |last3=Morozov |first4=D. |last4=Jiang |first5=M.I. |last5=Katsnelson |first6=I.V. |last6=Grigorieva |first7=S.V. |last7=Dubonos |first8=A.A. |last8=Firsov |display-authors=6 |date=10 Nov 2005 |title=Two-dimensional gas of massless Dirac fermions in graphene |journal=Nature |volume=438 |issue=7065 |pages=197–200 |doi=10.1038/nature04233 |pmid=16281030 |arxiv=cond-mat/0509330 |bibcode=2005Natur.438..197N |s2cid=3470761 |url=https://www.nature.com/articles/nature04233 |access-date=2 Mar 2018 }} </ref> The massless fermions lead to various quantum Hall effects, magnetoelectric effects in topological materials, and ultra high carrier mobility.<ref name="MyUser_Phys.org_May_25_2016c"> {{cite news |title=Two-dimensional Dirac materials: Structure, properties, and rarity |website=Phys.org |url=http://phys.org/news/2015-04-two-dimensional-dirac-materials-properties-rarity.html |access-date=25 May 2016 }} </ref><ref name=Hasan-Moore-2011> {{cite journal |last1=Hasan |first1=M.Z. |last2=Moore |first2=J.E. |year=2011 |title=Three-dimensional topological insulators |journal=Annual Review of Condensed Matter Physics |volume=2 |pages=55–78 |doi=10.1146/annurev-conmatphys-062910-140432 |arxiv=1011.5462 |bibcode=2011ARCMP...2...55H |s2cid=11516573 |language=En }} </ref> Dirac cones were observed in 2008-2009, using angle-resolved photoemission spectroscopy (ARPES) on the potassium-graphite intercalation compound KC<sub>8</sub><ref name=Grüneis-etal-2009> {{cite journal |first1=A. |last1=Grüneis |first2=C. |last2=Attaccalite |first3=A. |last3=Rubio |first4=D.V. |last4=Vyalikh |first5=S.L. |last5=Molodtsov |first6=J. |last6=Fink |first7=R. |last7=Follath |first8=W. |last8=Eberhardt |first9=B. |last9=Büchner |first10=T. |last10=Pichler |display-authors=6 |year=2009 |title=Angle-resolved photoemission study of the graphite intercalation compound KC{{sub|8}}: A key to graphene |journal=Physical Review&nbsp;B |volume=80 |issue=7 |article-number=075431 |doi=10.1103/PhysRevB.80.075431 |bibcode=2009PhRvB..80g5431G |hdl=10261/95912 |hdl-access=free }} </ref> and on several bismuth-based alloys.<ref name=Hsieh-Qian-etal-2008> {{Cite journal |last1=Hsieh |first1=D. |last2=Qian |first2=D. |last3=Wray |first3=L. |last4=Xia |first4=Y. |last5=Hor |first5=Y.S. |last6=Cava |first6=R.J. |last7=Hasan |first7=M.Z. |year=2008 |title=A topological Dirac insulator in a quantum spin Hall phase |journal=Nature |volume=452 |issue=7190 |pages=970–974 |arxiv=0902.1356 |doi=10.1038/nature06843 |issn=0028-0836 |pmid=18432240 |bibcode=2008Natur.452..970H |s2cid=4402113 |language=En }} </ref><ref name=Hsieh-Xia-etal-2009> {{Cite journal |last1=Hsieh |first1=D. |last2=Xia |first2=Y. |last3=Qian |first3=D. |last4=Wray |first4=L. |last5=Dil |first5=J.H. |last6=Meier |first6=F. |last7=Osterwalder |first7=J. |last8=Patthey |first8=L. |last9=Checkelsky |first9=J.G. |last10=Ong |first10=N.P. |last11=Fedorov |first11=A.V. |last12=Lin |first12=H. |last13=Bansil |first13=A. |last14=Grauer |first14=D. |last15=Hor |first15=Y.S. |last16=Cava |first16=R.J. |last17=Hasan |first17=M.Z. |display-authors=6 |year=2009 |title=A tunable, topological insulator in the spin helical Dirac transport regime |journal=Nature |volume=460 |issue=7259 |pages=1101–1105 |arxiv=1001.1590 |bibcode=2009Natur.460.1101H |doi=10.1038/nature08234 |pmid=19620959 |s2cid=4369601 }} </ref><ref name=Hasan-Moore-2011/>

As an object with three dimensions, Dirac cones are a feature of two-dimensional materials or surface states, based on a linear dispersion relation between energy and the two components of the crystal momentum {{mvar|k}}<sub>x</sub> and {{mvar|k}}<sub>y</sub>. However, this concept can be extended to three dimensions, where '''Dirac semimetals''' are defined by a linear dispersion relation between energy and {{mvar|k}}<sub>x</sub>, {{mvar|k}}<sub>y</sub>, and {{mvar|k}}<sub>z</sub>. In {{mvar|k}}-space, this shows up as a hypercone, which have doubly degenerate bands which also meet at Dirac points.<ref name=Hasan-Moore-2011/> Dirac semimetals contain both time reversal and spatial inversion symmetry; when one of these is broken, the Dirac points are split into two constituent Weyl points, and the material becomes a Weyl semimetal.<ref name=Wehling-etal-2014> {{cite journal |first1=T.O. |last1=Wehling |first2=A.M. |last2=Black-Schaffer |first3=A.V. |last3=Balatsky |year=2014 |title=Dirac materials |journal=Advances in Physics |volume=63 |issue=1 |page=1 |arxiv=1405.5774 |doi=10.1080/00018732.2014.927109 |bibcode=2014AdPhy..63....1W |s2cid=118557449 }}</ref><ref> {{cite journal |last1=Singh |first1=Bahadur |last2=Sharma |first2=Ashutosh |last3=Lin |first3=H. |last4=Hasan |first4=M.Z. |last5=Prasad |first5=R. |last6=Bansil |first6=A. |date=2012-09-18 |title=Topological electronic structure and Weyl semimetal in the TlBiSe2 class |journal=Physical Review&nbsp;B |volume=86 |issue=11 |article-number=115208 |doi=10.1103/PhysRevB.86.115208 |arxiv=1209.5896 |s2cid=119109505 }} </ref><ref name=Huang-Xu-etal-2015> {{cite journal |first1=S.-M. |last1=Huang |first2=S.-Y. |last2=Xu |first3=I. |last3=Belopolski |first4=C.-C. |last4=Lee |first5=G. |last5=Chang |first6=B.K. |last6=Wang |first7=N. |last7=Alidoust |first8=G. |last8=Bian |first9=M. |last9=Neupane |first10=C. |last10=Zhang |first11=S. |last11=Jia |first12=A. |last12=Bansil |first13=H. |last13=Lin |first14=M.Z. |last14=Hasan |display-authors=6 |date=2015 |title=A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class |journal=Nature Communications |volume=6 |page=7373 |doi=10.1038/ncomms8373 |pmid=26067579 |pmc=4490374 |bibcode=2015NatCo...6.7373H }} </ref><ref name=Weng-etal-2015> {{cite journal |last1=Weng |first1=Hongming |last2=Fang |first2=Chen |last3=Fang |first3=Zhong |last4=Bernevig |first4=B. 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== Analog systems == Dirac points have been realized in many physical areas such as plasmonics, phononics, or nanophotonics (microcavities,<ref>{{Cite journal |last1=Terças |first1=H. |last2=Flayac |first2=H. |last3=Solnyshkov |first3=D. D. |last4=Malpuech |first4=G. |date=2014-02-11 |title=Non-Abelian Gauge Fields in Photonic Cavities and Photonic Superfluids |url=https://link.aps.org/doi/10.1103/PhysRevLett.112.066402 |journal=Physical Review Letters |volume=112 |issue=6 |article-number=066402 |doi=10.1103/PhysRevLett.112.066402|pmid=24580697 |arxiv=1303.4286 |bibcode=2014PhRvL.112f6402T |s2cid=10674352 }}</ref> photonic crystals<ref>{{Cite journal |last1=He |first1=Wen-Yu |last2=Chan |first2=C. T. |date=2015-02-02 |title=The Emergence of Dirac points in Photonic Crystals with Mirror Symmetry |journal=Scientific Reports |language=en |volume=5 |issue=1 |page=8186 |doi=10.1038/srep08186 |issn=2045-2322 |pmc=4650825 |pmid=25640993|arxiv=1409.3939 |bibcode=2015NatSR...5E8186H }}</ref>).

== See also == * Dirac matter

==References== {{reflist|25em}}

==Further reading== * {{cite journal |last1=Wehling |first1=T.O. |last2=Black-Schaffer |first2=A.M. |last3=Balatsky |first3=A.V. |year=2014 |title=Dirac materials |journal=Advances in Physics |volume=63 |issue=1 |page=1 |doi=10.1080/00018732.2014.927109 |arxiv=1405.5774 |bibcode=2014AdPhy..63....1W |s2cid=118557449 }}

* {{cite news |last=Johnston |first=Hamish |date=23 July 2015 |title=Weyl fermions are spotted at long last |website=Physics World |url=http://physicsworld.com/cws/article/news/2015/jul/23/weyl-fermions-are-spotted-at-long-last |access-date=22 November 2018 }}

* {{cite journal |last=Ciudad |first=David |date=20 August 2015 |title=Massless, yet real |journal=Nature Materials |volume=14 |issue=9 |page=863 |doi=10.1038/nmat4411 |pmid=26288972 |issn=1476-1122|doi-access=free}}

* {{cite journal |last=Vishwanath |first=Ashvin |date=8 September 2015 |title=Where the Weyl things are |journal=Physics |volume=8 |page=84 |doi=10.1103/Physics.8.84 |bibcode=2015PhyOJ...8...84V |url=http://physics.aps.org/articles/v8/84 |access-date=22 November 2018 |doi-access=free }}

* {{cite journal |last1=Jia |first1=Shuang |last2=Xu |first2=Su-Yang |last3=Hasan |first3=M. Zahid |date=25 October 2016 |title=Weyl semimetals, Fermi arcs, and chiral anomaly |journal=Nature Materials |volume=15 |issue=11 |pages=1140–1144 |doi=10.1038/nmat4787 |pmid=27777402 |arxiv=1612.00416 |bibcode=2016NatMa..15.1140J |s2cid=1115349 |url=https://www.nature.com/articles/nmat4787 }}

* {{cite book |last1 = Hasan |first1 = M. Z. |last2 = Xu |first2 = S.-Y. |last3 = Neupane |first3 = M. |year = 2015 |chapter = Chapter&nbsp;4: Topological insulators, topological Dirac semimetals, topological crystalline insulators, and topological Kondo insulators |pages = 55–100 |editor1-last=Ortmann |editor1-first=Frank |editor2-last=Roche |editor2-first=Stephan |editor3-last=Valenzuela |editor3-first=Sergio O. |title = Topological Insulators: Fundamentals and Perspectives |publisher = Wiley |isbn = 978-3-527-33702-6 |arxiv=1406.1040 |bibcode=2014arXiv1406.1040Z }}

Category:Electronic band structures Category:Semimetals