Dirac cone

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Brillouin zone in graphene
Electronic band structure of monolayer graphene, with a zoomed inset showing the Dirac cones. There are 6 cones corresponding to the 6 vertices of the hexagonal first Brillouin zone.
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.

Dirac cones, named after Paul Dirac, are features that occur in some electronic band structures that describe unusual electron transport properties of materials like graphene and topological insulators.[1][2][3] 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.


In quantum mechanics, Dirac cones are a kind of crossing-point which electrons avoid,[4] where the energy of the valence and conduction bands are not equal anywhere in two dimensional lattice 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.[5] The massless fermions lead to various quantum Hall effects, magnetoelectric effects in topological materials, and ultra high carrier mobility.[6][7] Dirac cones were observed in 2008-2009, using angle-resolved photoemission spectroscopy (ARPES) on the potassium-graphite intercalation compound KC8.[8] and on several bismuth-based alloys.[9][10][7]

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 kx and ky. However, this concept can be extended to three dimensions, where Dirac semimetals are defined by a linear dispersion relation between energy and kx, ky, and kz. In k-space, this shows up as a hypercone, which have doubly degenerate bands which also meet at Dirac points.[7] 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.[11][12][13][14][15][16][17][18][19][20][21] In 2014, direct observation of the Dirac semimetal band structure using ARPES was conducted on the Dirac semimetal cadmium arsenide.[22][23][24]

Analog systems[edit]

Dirac points have been realized in many physical areas such as plasmonics, phononics, or nanophotonics (microcavities,[25] photonic crystals[26]).

See also[edit]


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  2. ^ Hasan, M.Z.; Kane, C.L. (2010). "Topological Insulators". Rev. Mod. Phys. 82 (4): 3045. arXiv:1002.3895. Bibcode:2010RvMP...82.3045H. doi:10.1103/revmodphys.82.3045. S2CID 16066223.
  3. ^ "Superconductors: Dirac cones come in pairs". Advanced Institute for Materials Research. wpi-aimr.tohoku.ac.jp. Research Highlights. Tohoku University. 29 August 2011. Retrieved 2 March 2018.
  4. ^ Fuchs, Jean-Noël; Lim, Lih-King; Montambaux, Gilles (2012). "Interband tunneling near the merging transition of Dirac cones" (PDF). Physical Review A. 86 (6): 063613. arXiv:1210.3703. Bibcode:2012PhRvA..86f3613F. doi:10.1103/PhysRevA.86.063613. S2CID 67850936.
  5. ^ Novoselov, K.S.; Geim, A.K.; Morozov, S.V.; Jiang, D.; Katsnelson, M.I.; Grigorieva, I.V.; et al. (10 November 2005). "Two-dimensional gas of massless Dirac fermions in graphene". Nature. 438 (7065): 197–200. arXiv:cond-mat/0509330. Bibcode:2005Natur.438..197N. doi:10.1038/nature04233. PMID 16281030. S2CID 3470761. Retrieved 2 March 2018.
  6. ^ "Two-dimensional Dirac materials: Structure, properties, and rarity". Phys.org. Retrieved 25 May 2016.
  7. ^ a b c Hasan, M.Z.; Moore, J.E. (2011). "Three-dimensional topological insulators". Annual Review of Condensed Matter Physics. 2: 55–78. arXiv:1011.5462. Bibcode:2011ARCMP...2...55H. doi:10.1146/annurev-conmatphys-062910-140432. S2CID 11516573.
  8. ^ Grüneis, A.; Attaccalite, C.; Rubio, A.; Vyalikh, D.V.; Molodtsov, S.L.; Fink, J.; et al. (2009). "Angle-resolved photoemission study of the graphite intercalation compound KC8: A key to graphene". Physical Review B. 80 (7): 075431. Bibcode:2009PhRvB..80g5431G. doi:10.1103/PhysRevB.80.075431. hdl:10261/95912.
  9. ^ Hsieh, D.; Qian, D.; Wray, L.; Xia, Y.; Hor, Y.S.; Cava, R.J.; Hasan, M.Z. (2008). "A topological Dirac insulator in a quantum spin Hall phase". Nature. 452 (7190): 970–974. arXiv:0902.1356. Bibcode:2008Natur.452..970H. doi:10.1038/nature06843. ISSN 0028-0836. PMID 18432240. S2CID 4402113.
  10. ^ Hsieh, D.; Xia, Y.; Qian, D.; Wray, L.; Dil, J.H.; Meier, F.; et al. (2009). "A tunable, topological insulator in the spin helical Dirac transport regime". Nature. 460 (7259): 1101–1105. arXiv:1001.1590. Bibcode:2009Natur.460.1101H. doi:10.1038/nature08234. PMID 19620959. S2CID 4369601.
  11. ^ Wehling, T.O.; Black-Schaffer, A.M.; Balatsky, A.V. (2014). "Dirac materials". Advances in Physics. 63 (1): 1. arXiv:1405.5774. Bibcode:2014AdPhy..63....1W. doi:10.1080/00018732.2014.927109. S2CID 118557449.
  12. ^ Singh, Bahadur; Sharma, Ashutosh; Lin, H.; Hasan, M.Z.; Prasad, R.; Bansil, A. (18 September 2012). "Topological electronic structure and Weyl semimetal in the TlBiSe2 class". Physical Review B. 86 (11): 115208. arXiv:1209.5896. doi:10.1103/PhysRevB.86.115208. S2CID 119109505.
  13. ^ Huang, S.-M.; Xu, S.-Y.; Belopolski, I.; Lee, C.-C.; Chang, G.; Wang, B.K.; et al. (2015). "A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class". Nature Communications. 6: 7373. Bibcode:2015NatCo...6.7373H. doi:10.1038/ncomms8373. PMC 4490374. PMID 26067579.
  14. ^ Weng, Hongming; Fang, Chen; Fang, Zhong; Bernevig, B. Andrei; Dai, Xi (2015). "Weyl semimetal phase in non-centrosymmetric transition-metal monophosphides". Physical Review X. 5 (1): 011029. arXiv:1501.00060. Bibcode:2015PhRvX...5a1029W. doi:10.1103/PhysRevX.5.011029. S2CID 15298985.
  15. ^ Xu, S.-Y.; Belopolski, I.; Alidoust, N.; Neupane, M.; Bian, G.; Zhang, C.; et al. (2015). "Discovery of a Weyl Fermion semimetal and topological Fermi arcs". Science. 349 (6248): 613–617. arXiv:1502.03807. Bibcode:2015Sci...349..613X. doi:10.1126/science.aaa9297. PMID 26184916. S2CID 206636457.
  16. ^ Xu, Su-Yang; Alidoust, Nasser; Belopolski, Ilya; Yuan, Zhujun; Bian, Guang; Chang, Tay-Rong; et al. (2015). "Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide". Nature Physics. 11 (9): 748–754. arXiv:1504.01350. doi:10.1038/nphys3437. ISSN 1745-2481. S2CID 119118252.
  17. ^ Huang, Xiaochun; Zhao, Lingxiao; Long, Yujia; Wang, Peipei; Chen, Dong; Yang, Zhanhai; et al. (2015). "Observation of the chiral-anomaly-induced negative magnetoresistance in 3‑D Weyl semimetal TaAs". Physical Review X. 5 (3): 031023. arXiv:1503.01304. Bibcode:2015PhRvX...5c1023H. doi:10.1103/PhysRevX.5.031023. S2CID 55929760.
  18. ^ Zhang, Cheng-Long; Xu, Su-Yang; Belopolski, Ilya; Yuan, Zhujun; Lin, Ziquan; Tong, Bingbing; et al. (25 February 2016). "Signatures of the Adler–Bell–Jackiw chiral anomaly in a Weyl fermion semimetal". Nature Communications. 7 (1): 10735. doi:10.1038/ncomms10735. ISSN 2041-1723. PMC 4773426. PMID 26911701.
  19. ^ Schoop, Leslie M.; Ali, Mazhar N.; Straßer, Carola; Topp, Andreas; Varykhalov, Andrei; Marchenko, Dmitry; et al. (2016). "Dirac cone protected by non-symmorphic symmetry and three-dimensional Dirac line node in ZrSiS". Nature Communications. 7 (1): 11696. arXiv:1509.00861. Bibcode:2016NatCo...711696S. doi:10.1038/ncomms11696. ISSN 2041-1723. PMC 4895020. PMID 27241624.
  20. ^ Neupane, M.; Belopolski, I.; Hosen, Md.M.; Sanchez, D.S.; Sankar, R.; Szlawska, M.; et al. (2016). "Observation of topological nodal fermion semimetal phase in ZrSiS". Physical Review B. 93 (20): 201104(R). arXiv:1604.00720. doi:10.1103/PhysRevB.93.201104. ISSN 2469-9969. S2CID 118446447.
  21. ^ Lu, Ling; Fu, Liang; Joannopoulos, John D.; Soljačic, Marin (17 March 2013). "Weyl points and line nodes in gyroid photonic crystals" (PDF). Nature Photonics. 7 (4): 294–299. arXiv:1207.0478. Bibcode:2013NaPho...7..294L. doi:10.1038/nphoton.2013.42. S2CID 5144108. Retrieved 2 March 2018.
  22. ^ Neupane, Madhab; Xu, Su-Yang; Sankar, Raman; Nasser, Alidoust; Bian, Guang; Liu, Chang; et al. (2014). "Observation of a three-dimensional topological Dirac semimetal phase in high-mobility Cd3As2". Nature Communications. 5: 3786. arXiv:1309.7892. Bibcode:2014NatCo...5.3786N. doi:10.1038/ncomms4786. PMID 24807399.
  23. ^ Sankar, R.; Neupane, M.; Xu, S.-Y.; Butler, C.J.; Zeljkovic, I.; Panneer Muthuselvam, I.; et al. (2015). "Large single crystal growth, transport property, and spectroscopic characterizations of three-dimensional Dirac semimetal Cd3As2". Scientific Reports. 5: 12966. Bibcode:2015NatSR...512966S. doi:10.1038/srep12966. PMC 4642520. PMID 26272041.
  24. ^ Borisenko, Sergey; Gibson, Quinn; Evtushinsky, Danil; Zabolotnyy, Volodymyr; Büchner, Bernd; Cava, Robert J. (2014). "Experimental realization of a three-dimensional Dirac semimetal". Physical Review Letters. 113 (2): 027603. arXiv:1309.7978. Bibcode:2014PhRvL.113b7603B. doi:10.1103/PhysRevLett.113.027603. ISSN 0031-9007. PMID 25062235. S2CID 19882802.
  25. ^ Terças, H.; Flayac, H.; Solnyshkov, D. D.; Malpuech, G. (11 February 2014). "Non-Abelian Gauge Fields in Photonic Cavities and Photonic Superfluids". Physical Review Letters. 112 (6): 066402. arXiv:1303.4286. Bibcode:2014PhRvL.112f6402T. doi:10.1103/PhysRevLett.112.066402. PMID 24580697. S2CID 10674352.
  26. ^ He, Wen-Yu; Chan, C. T. (2 February 2015). "The Emergence of Dirac points in Photonic Crystals with Mirror Symmetry". Scientific Reports. 5 (1): 8186. doi:10.1038/srep08186. ISSN 2045-2322. PMC 4650825. PMID 25640993.

Further reading[edit]

  • Hasan, M. Z.; Xu, S.-Y.; Neupane, M. (2015). "Chapter 4: Topological insulators, topological Dirac semimetals, topological crystalline insulators, and topological Kondo insulators". In Ortmann, Frank; Roche, Stephan; Valenzuela, Sergio O. (eds.). Topological Insulators: Fundamentals and Perspectives. Wiley. pp. 55–100. arXiv:1406.1040. Bibcode:2014arXiv1406.1040Z. ISBN 978-3-527-33702-6.