Dirac-like cones at k=0

Huang Xue-Qin; Chan Che-Ting

doi:10.7498/aps.64.184208

Abstract

Dirac cones and Dirac points are found at the K (K') points in the Brillouin zones of electronic and classical waves systems with hexagonal or triangular lattices. Accompanying the conical dispersions, there are many intriguing phenomena including quantum Hall effect, Zitterbewegung and Klein tunneling. Such Dirac cones at the Brillouin zone boundary are the consequences of the lattice symmetry and time reversal symmetry. Conical dispersions are difficult to form in the zone center because of time reversal symmetry, which generally requires the band dispersions to be quadratic at k=0. However, the conical dispersions with a triply degenerate state at k=0 can be realized in two dimensional (2D) photonic crystal (PC) using accidental degeneracy. The triply degenerate state consists of two linear bands that generate Dirac cones and an additional flat band intersecting at the Dirac point. If the triply degenerate state is derived from the monopolar and dipolar excitations, effective medium theory can relate this 2D PC to a double zero-refractive-index material with effective permittivity and permeability equal to zero simultaneously. There is hence a subtle relationship between two seemingly unrelated concepts: Dirac-like cone and zero-refractive index. The all-dielectric double zero-refractive-index material has advantage over metallic zero-index metamaterials which are usually poorly impedance matched to the background and are lossy in high frequencies. The Dirac-like cone zero-index materials have impedances that can tune to match the background material and the loss is small as the system has an all-dielectric construction, enabling the possibility of realizing zero refractive index in optical frequencies. The realization of Dirac-like cones at k=0 can be extended from the electromagnetic wave system to acoustic and elastic wave systems and effective medium theory can also be applied to relate these systems to zero-index materials. The concept of Dirac/Dirac-like cone is intrinsically 2D. However, using accidental degeneracy and special symmetries, the concept of Dirac-like point can be extended from two to three dimensions in electromagnetic and acoustic waves. Effective medium theory is also applicable to these systems, and these systems can be related to isotropic media with effectively zero refractive indices. One interesting implication of Dirac-like cones in 2D PC is the existence of robust interface states. The existence of interface states is not a trivial problem and there is usually no assurance that localized state can be found at the boundary of photonic or phononic crystal. In order to create an interface state, one usually needs to decorate the interface with strong perturbations. Recently, it is found that interface state can always be found at the boundary separating two semi-infinite PCs which have their system parameters slightly perturbed from the Dirac-like cone formation condition. The assured existence of interface states in such a system can be explained by the sign of the surface impedance of the PCs on either side of the boundary which can be derived using a layer-by-layer multiple scattering theory. In a deeper level, the existence of the interface state can be accounted for by the geometric properties of the bulk band. It turns out that the geometric phases of the bulk band determine the surface impedance within the frequency range of the band gap. The geometric property of the momentum space can hence be used to explain the existence of interface states in real space through a bulk-interface correspondence.

Keywords:

Authors and contacts

1.
Department of Physics and Institute for Advanced Study, Hong Kong University of Science and Technology, Hong Kong, China

Corresponding author: Chan Che-Ting, phchan@ust.hk

Funds: Project supported by Hong Kong Research Grant Council (Grant No. AoE/P-02/12).

References

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[80]	Wang Z, Chong Y D, Joannopoulos J D, Soljacic M 2008 Phys. Rev. Lett. 100 013905
[81]	Yu Z, Veronis G, Wang Z, Fan S 2008 Phys. Rev. Lett. 100 023902
[82]	Kraus Y E, Lahini Y, Ringel Z, Verbin M, Zilberberg O 2012 Phys. Rev. Lett. 109 106402
[83]	Poo Y, Wu R X, Lin Z, Yang Y, Chan C T 2011 Phys. Rev. Lett. 106 093903
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[85]	Rechtsman M C, Plotnik Y, Zeuner J M, Song D, Chen Z, Szameit A, Segev M 2013 Phys. Rev. Lett. 111 103901
[86]	Zak J 1989 Phys. Rev. Lett. 62 2747

Cited By

[1]	Dirac P A M 1928 Proc. R. Soc. A: Math. Phys. Eng. Sci. 117 610
[2]	Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Gregorieva I V, Firsov A A 2004 Science 306 666
[3]	Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, Firsov A A 2005 Nature 438 197
[4]	Zhang Y, Tan Y W, Stormer H L, Kim P 2005 Nature 438 201
[5]	Gusynin V P, Sharapov S G 2005 Phys. Rev. Lett. 95 146801
[6]	Novoselov K S, Jiang Z, Zhang Y, Morozov S V, Stormer H L, Zeitler U, Maan J C, Boebinger G S, Kim P, Geim A K 2007 Science 315 1379
[7]	Katsnelsona M I 2006 Eur. Phys. J. B 51 157
[8]	Cserti J, David G 2006 Phys. Rev. B 74 172305
[9]	Rusin T M, Zawadzki W 2007 Phys. Rev. B 76 195439
[10]	David G, Cserti J 2010 Phys. Rev. B 81 121417
[11]	Neto A H C, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109
[12]	Katsnelson M I, Novoselov K S, Geim A K 2006 Nat. Phys. 2 620
[13]	Morozov S V, Novoselov K S, Katsnelson M I, Schedin F, Ponomarenko L A, Jiang D, Geim A K 2006 Phys. Rev. Lett. 97 016801
[14]	Geim A K, MacDonald A H 2007 Phys. Today 60 35
[15]	Geim A K, Novoselov K S 2007 Nat. Mater. 6 183
[16]	Plihal M, Maradudin A A 1991 Phys. Rev. B 44 8565
[17]	Sepkhanov R A, Bazaliy Ya B, Beenakker C W J 2007 Phys. Rev. A 75 063813
[18]	Diem M, Koschny T, Soukoulis C M 2010 Physica B 405 2990
[19]	Sepkhanov R A, Nilsson J, Beenakker C W J 2008 Phys. Rev. B 78 045122
[20]	Zhang X 2008 Phys. Rev. Lett. 100 113903
[21]	Zhang X, Liu Z 2008 Phys. Rev. Lett. 101 264303
[22]	Raghu S, Haldane F D M 2008 Phys. Rev. A 78 033834
[23]	Haldane F D M, Raghu S 2008 Phys. Rev. Lett. 100 013904
[24]	Ochiai T, Onoda M 2009 Phys. Rev. B 80 155103
[25]	Ochiai T 2010 J. Phys.: Condens. Matter 22 225502
[26]	Veselago V G 1968 Sov. Phys. Usp. 10 509
[27]	Shelby R A, Smith D R, Schultz S 2001 Science 292 77
[28]	Smith D R, Padilla W J, Vier D C, Nemat-Nasser S C, Schultz S 2000 Phys. Rev. Lett. 84 4184
[29]	Zhang S, Fan W, Panoiu N C, Malloy K J, Osgood R M, Brueck S R J 2005 Phys. Rev. Lett. 95 137404
[30]	Shalaev V M, Cai W, Chettiar U K, Yuan H K, Sarychev A K, Drachev V P, Kildishev A V 2005 Opt. Lett. 30 3356
[31]	Dolling G, Wegener M, Soukoulis C M, Linden S 2007 Opt. Lett. 32 53
[32]	Pendry J B 2000 Phys. Rev. Lett. 85 3966
[33]	Fang N, Lee H, Sun C, Zhang X 2005 Science 308 534
[34]	Pendry J B, Schurig D, Smith D R 2006 Science 312 1780
[35]	Leonhardt U 2006 Science 312 1777
[36]	Schurig D, Mock J J, Justice B J, Cummer S A, Pendry J B, Starr A F, Smith D R 2006 Science 314 977
[37]	Rahm M, Schurig D, Roberts D A, Cummer S A, Smith D R, Pendry J B 2008 Photon. Nanostruct. Fundam. Appl. 6 87
[38]	Yang T, Chen H Y, Luo X D, Ma H R 2008 Opt. Express 16 18545
[39]	Chen H Y, Chan C T 2007 Appl. Phys. Lett. 90 241105
[40]	Lai Y, Ng J, Chen H Y, Han D, Xiao J, Zhang Z Q, Chan C T 2009 Phys. Rev. Lett. 102 253902
[41]	Silveirinha M, Engheta N 2006 Phys. Rev. Lett. 97 157403
[42]	Silveirinha M, Engheta N 2007 Phys. Rev. B 75 075119
[43]	Silveirinha M G, Engheta N 2007 Phys. Rev. B 76 245109
[44]	Alu A, Engheta N 2008 Phys. Rev. B 78 045102
[45]	Alu A, Silveirinha M G, Engheta N 2008 Phys. Rev. E 78 016604
[46]	Edwards B, Alu A, Silveirinha M G, Engheta N 2009 J. Appl. Phys. 105 044905
[47]	Liu R, Cheng Q, Hand T, Mock J J, Cui T J, Cummer S A, Smith D R 2008 Phys. Rev. Lett. 100 023903
[48]	Edwards B, Alu A, Young M E, Silveirinha M, Engheta N 2008 Phys. Rev. Lett. 100 033903
[49]	Halterman K, Feng S 2008 Phys. Rev. A 78 021805
[50]	Ziolkowski R W 2004 Phys. Rev. E 70 046608
[51]	Enoch S, Tayeb G, Sabouroux P, Guerin N, Vincent P 2002 Phys. Rev. Lett. 89 213902
[52]	Alu A, Silveirinha M G, Salandrino A, Engheta N 2007 Phys. Rev. B 75 155410
[53]	Hao J, Yan W, Qiu M 2010 Appl. Phys. Lett. 96 101109
[54]	Jin Y, He S 2010 Opt. Express 18 16587
[55]	Nguyen V C, Chen L, Halterman K 2010 Phys. Rev. Lett. 105 233908
[56]	Xu Y, Chen H 2011 Appl. Phys. Lett. 98 113501
[57]	Huang X, Lai Y, Hang Z H, Zheng H, Chan C T 2011 Nat. Mater. 10 582
[58]	Chan C T, Hang Z H, Huang X 2012 Advances in OptoElectronics 2012 313984
[59]	Chan C T, Huang X, Liu F, Hang Z H 2012 PIER B 44 163
[60]	Wang L G, Wang Z G, Zhang J X, Zhu S Y 2009 Opt. Lett. 34 1510
[61]	Mei J, Wu Y, Chan C T, Zhang Z Q 2012 Phys. Rev. B 86 035141
[62]	Liu F, Huang X, Chan C T 2012 Appl. Phys. Lett. 100 071911
[63]	Liu F, Lai Y, Huang X, Chan C T 2011 Phys. Rev. B 84 224113
[64]	Sakoda K 2012 Opt. Express 20 3898
[65]	Huang X, Xiao M, Zhang Z Q, Chan C T 2014 Phys. Rev. B 90 075423
[66]	Sakoda K, Zhou H 2010 Opt. Express 18 27371
[67]	Sakoda K, Zhou H 2011 Opt. Express 19 13899
[68]	Wu Y, Li J, Zhang Z Q, Chan C T 2006 Phys. Rev. B 74 085111
[69]	Moitra P, Yang Y, Anderson Z, Kravchenko I I, Briggs D P, Valentine J 2013 Nat. Photon. 7 791
[70]	Joannopoulos J D, Johnson S G, Winn J N, Meade R D 2008 Photonic Crystals: Molding the Flow of Light (2nd Ed.) (Princeton: Princeton University Press)
[71]	Lawrence F J, Botten L C, Dossou K B, McPhedran R C, de Sterke C M 2010 Phys. Rev. A 82 053840
[72]	Lawrence F J, Botten L C, Dossou K B, de Sterke C M, McPhedran R C 2009 Phys. Rev. A 80 023826
[73]	Xiao M, Zhang Z Q, Chan C T 2014 Phys. Rev. X 4 021017
[74]	Wang Z, Chong Y, Joannopoulos J D, Soljacic M 2009 Nature 461 772
[75]	Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A 2013 Nature 496 196
[76]	Khanikaev A B, Mousavi S H, Tse W K, Kargarian M, MacDonald A H, Shvets G 2013 Nat. Mater. 12 233
[77]	Lu L, Fu L, Joannopoulos J D, Soljacic M 2013 Nat Photon. 7 294
[78]	Fang K, Yu Z, Fan S 2012 Nat Photon. 6 782
[79]	Hafezi M, Demler E A, Lukin M D, Taylor J M 2011 Nat. Phys. 7 907
[80]	Wang Z, Chong Y D, Joannopoulos J D, Soljacic M 2008 Phys. Rev. Lett. 100 013905
[81]	Yu Z, Veronis G, Wang Z, Fan S 2008 Phys. Rev. Lett. 100 023902
[82]	Kraus Y E, Lahini Y, Ringel Z, Verbin M, Zilberberg O 2012 Phys. Rev. Lett. 109 106402
[83]	Poo Y, Wu R X, Lin Z, Yang Y, Chan C T 2011 Phys. Rev. Lett. 106 093903
[84]	Plotnik Y, Rechtsman M C, Song D, Heinrich M, Zeuner J M, Nolte S, Lumer Y, Malkova N, Xu J, Szameit A, Chen Z, Segev M 2014 Nat. Mater. 13 57
[85]	Rechtsman M C, Plotnik Y, Zeuner J M, Song D, Chen Z, Szameit A, Segev M 2013 Phys. Rev. Lett. 111 103901
[86]	Zak J 1989 Phys. Rev. Lett. 62 2747

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