We show how one may obtain conical (Dirac) dispersions in photonic crystals, and in some cases, such conical dispersions can be used to create a metamaterial with an effective zero refractive index. We show specifically that in two-dimensional photonic crystals with symmetry, we can adjust the system parameters to obtain accidental triple degeneracy at Γ point, whose band dispersion comprises two linear bands that generate conical dispersion surfaces and an additional flat band crossing the Dirac-like point. If this triply degenerate state is formed by monopole and dipole excitations, the system can be mapped to an effective medium with permittivity and permeability equal to zero simultaneously, and this system can transport wave as if the refractive index is effectively zero. However, not all the triply degenerate states can be described by monopole and dipole excitations and in those cases, the conical dispersion may not be related to an effective zero refractive index. Using multiple scattering theory, we calculate the Berry phase of the eigenmodes in the Dirac-like cone to be equal to zero for modes in the Dirac-like cone at the zone center, in contrast with the Berry phase of for Dirac cones at the zone boundary. 1. Introduction The Dirac equation is the wave equation formulated to describe relativistic spin 1/2 particles [1]. In the special case where the effective mass of the spin 1/2 particle is zero, and the solution to Dirac equation has a linear dispersion in the sense that the energy is linearly proportional to the wave vector . The electric band structure of graphene near the Fermi level can be described by the massless Dirac equation and hence exhibit the Dirac dispersion [2–15]. The electronic band dispersion is linear near the six corners of the two-dimensional (2D) hexagonal Brillouin zone at the and points, and the dispersion close to the Fermi energy at each of these corner -points can be visualized as two cones meeting at the Fermi level at one point called the Dirac point, and the conical dispersion near the Dirac point is usually referred to as Dirac cones. This rather singular electronic band structure of graphene near the Fermi level gives rise to many unusual transport properties [2–15], including quantum hall effect [4–6], Zitterbewegung [7–11], and Klein paradox [12]. Dirac cone dispersions are not limited to graphene but can also be found in classical wave periodic systems such as photonic crystals [16–24]. In fact, linear dispersions at the Brillouin zone boundary for 2D triangular photonic crystals appeared in the photonic
References
[1]
P. A. M. Dirac, “The quantum theory of the electron,” Proceedings of the Royal Society A, vol. 117, p. 610, 1928.
[2]
K. S. Novoselov, A. K. Geim, S. V. Morozov et al., “Electric field in atomically thin carbon films,” Science, vol. 306, no. 5696, pp. 666–669, 2004.
[3]
K. S. Novoselov, A. K. Geim, S. V. Morozov et al., “Two-dimensional gas of massless Dirac fermions in graphene,” Nature, vol. 438, no. 7065, pp. 197–200, 2005.
[4]
Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry's phase in graphene,” Nature, vol. 438, no. 7065, pp. 201–204, 2005.
[5]
V. P. Gusynin and S. G. Sharapov, “Unconventional integer quantum hall effect in graphene,” Physical Review Letters, vol. 95, no. 14, Article ID 146801, 2005.
[6]
K. S. Novoselov, Z. Jiang, Y. Zhang et al., “Room-temperature quantum hall effect in graphene,” Science, vol. 315, no. 5817, p. 1379, 2007.
[7]
M. I. Katsnelsona, “Zitterbewegung, chirality, and minimal conductivity in graphene,” The European Physical Journal B, vol. 51, no. 2, pp. 157–160, 2006.
[8]
J. Cserti and G. David, “Unified description of Zitterbewegung for spintronic, graphene, and superconducting systems,” Physical Review B, vol. 74, no. 17, Article ID 172305, 2006.
[9]
T. M. Rusin and W. Zawadzki, “Transient Zitterbewegung of charge carriers in mono and bilayer graphene, and carbon nanotubes,” Physical Review B, vol. 76, Article ID 195439, 2007.
[10]
G. David and J. Cserti, “General theory of the Zitterbewegung,” Physical Review B, vol. 81, Article ID 121417, 2010.
[11]
A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Reviews of Modern Physics, vol. 81, no. 1, pp. 109–162, 2009.
[12]
M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nature Physics, vol. 2, no. 9, pp. 620–625, 2006.
[13]
S. V. Morozov, K. S. Novoselov, M. I. Katsnelson et al., “Strong suppression of weak localization in graphene,” Physical Review Letters, vol. 97, no. 1, Article ID 016801, 2006.
[14]
A. K. Geim and A. H. MacDonald, “Graphene: exploring carbon flatland,” Physics Today, vol. 60, no. 8, pp. 35–41, 2007.
[15]
A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Materials, vol. 6, no. 3, pp. 183–191, 2007.
[16]
M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Physical Review B, vol. 44, no. 16, pp. 8565–8571, 1991.
[17]
R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Physical Review A, vol. 75, no. 6, Article ID 063813, 2007.
[18]
M. Diema, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B, vol. 405, no. 14, pp. 2990–2995, 2010.
[19]
R. A. Sepkhanov, J. Nilsson, and C. W. J. Beenakker, “How to detect the pseudospin-1/2 Berry phase in a photonic crystal with a Dirac spectrum,” Physical Review B, vol. 78, Article ID 045122, 2008.
[20]
X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Physical Review Letters, vol. 100, no. 11, p. 113903, 2008.
[21]
S. Raghu and F. D. M. Haldane, “Analogs of quantum Hall effect edge states in photonic crystals,” Physical Review A, vol. 78, Article ID 033834, 2008.
[22]
F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Physical Review Letters, vol. 100, Article ID 013904, 2008.
[23]
T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Physical Review B, vol. 80, no. 15, p. 155103, 2009.
[24]
T. Ochiai, “Topological properties of bulk and edge states in honeycomb lattice photonic crystals: the case of TE polarization,” Journal of Physics, vol. 22, no. 22, p. 225502, 2010.
[25]
E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Physical Review Letters, vol. 58, no. 20, pp. 2059–2062, 1987.
[26]
S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Physical Review Letters, vol. 58, no. 23, pp. 2486–2489, 1987.
[27]
K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Physical Review Letters, vol. 65, no. 25, pp. 3152–3155, 1990.
[28]
Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photoniccrystal,” Physical Review Letters, vol. 100, no. 1, p. 013905, 2008.
[29]
Z. Yu, G. s Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Physical Review Letters, vol. 100, no. 2, p. 023902, 2008.
[30]
Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Solja？i？, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature, vol. 461, no. 7265, pp. 772–775, 2009.
[31]
X. Ao, Z. Lin, and C. T. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Physical Review B, vol. 80, no. 3, p. 033105, 2009.
[32]
Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Physical Review Letters, vol. 106, no. 9, p. 093903, 2011.
[33]
X. Zhang and Z. Liu, “Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals,” Physical Review Letters, vol. 101, no. 26, p. 264303, 2008.
[34]
V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Soviet Physics Uspekhi, vol. 10, no. 4, p. 509, 1968.
[35]
R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, no. 5514, pp. 77–79, 2001.
[36]
D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Physical Review Letters, vol. 84, no. 18, pp. 4184–4187, 2000.
[37]
S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Physical Review Letters, vol. 95, no. 13, p. 137404, 2005.
[38]
V. M. Shalaev, W. Cai, U. K. Chettiar et al., “Negative index of refraction in optical metamaterials,” Optics Letters, vol. 30, no. 24, pp. 3356–3358, 2005.
[39]
G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Optics Letters, vol. 32, no. 1, pp. 53–55, 2007.
[40]
J. B. Pendry, “Negative refraction makes a perfect lens,” Physical Review Letters, vol. 85, no. 18, pp. 3966–3969, 2000.
[41]
N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science, vol. 308, no. 5721, pp. 534–537, 2005.
[42]
J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science, vol. 312, no. 5781, pp. 1780–1782, 2006.
[43]
U. Leonhardt, “Optical conformal mapping,” Science, vol. 312, no. 5781, pp. 1777–1780, 2006.
[44]
D. Schurig, J. J. Mock, B. J. Justice et al., “Metamaterial electromagnetic cloak at microwave frequencies,” Science, vol. 314, no. 5801, pp. 977–980, 2006.
[45]
M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photonics and Nanostructures, vol. 6, no. 1, pp. 87–95, 2008.
[46]
T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Optics Express, vol. 16, no. 22, pp. 18545–18550, 2008.
[47]
H. Y. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Applied Physics Letters, vol. 90, no. 24, p. 241105, 2007.
[48]
Y. Lai, J. Ng, H. Y. Chen et al., “Illusion optics: the optical transformation of an object into another object,” Physical Review Letters, vol. 102, no. 25, p. 253902, 2009.
[49]
M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Physical Review Letters, vol. 97, no. 15, p. 157403, 2006.
[50]
M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in ε-near-zero media,” Physical Review B, vol. 75, no. 7, p. 075119, 2007.
[51]
M. G. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials,” Physical Review B, vol. 76, no. 24, p. 245109, 2007.
[52]
A. Alu and N. Engheta, “Dielectric sensing in ε-near-zero narrow waveguide channels,” Physical Review B, vol. 78, p. 045102, 2008.
[53]
A. Alu, M. G. Silveirinha, and N. Engheta, “Transmission-line analysis of ε-near-zero–filled narrow channels,” Physical Review E, vol. 78, no. 1, p. 016604, 2008.
[54]
B. Edwards, A. Alu, M. G. Silveirinha, and N. Engheta, “Reflectionless sharp bends and corners in waveguides using ε-near-zero effects,” Journal of Applied Physics, vol. 105, no. 4, p. 044905, 2009.
[55]
R. Liu, Q. Cheng, T. Hand et al., “Experimental demonstration of electromagnetic tunneling through an ε-near-zero metamaterial at microwave frequencies,” Physical Review Letters, vol. 100, no. 2, p. 023903, 2008.
[56]
B. Edwards, A. Alu, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of ε-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Physical Review Letters, vol. 100, no. 3, p. 033903, 2008.
[57]
K. Halterman and S. Feng, “Resonant transmission of electromagnetic fields through subwavelength zero- ？ slits,” Physical Review A, vol. 78, no. 2, p. 021805, 2008.
[58]
R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Physical Review E, vol. 70, no. 4, p. 046608, 2004.
[59]
S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Physical Review Letters, vol. 89, no. 21, p. 213902, 2002.
[60]
A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “ε-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Physical Review B, vol. 75, no. 15, p. 155410, 2007.
[61]
J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Applied Physics Letters, vol. 96, no. 10, p. 101109, 2010.
[62]
Y. Jin and S. He, “Enhancing and suppressing radiation with some permeability-near-zero structures,” Optics Express, vol. 18, no. 16, pp. 16587–16593, 2010.
[63]
V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Physical Review Letters, vol. 105, no. 23, p. 233908, 2010.
[64]
Y. Xu and H. Chen, “Total reflection and transmission by ε-near-zero metamaterials with defects,” Applied Physics Letters, vol. 98, no. 11, p. 113501, 2011.
[65]
L. G. Wang, Z. G. Wang, J. X. Zhang, and S. Y. Zhu, “Realization of Dirac point with double cones in optics,” Optics Letters, vol. 34, no. 10, p. 1510, 2009.
[66]
X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Materials, vol. 10, no. 8, pp. 582–586, 2011.
[67]
M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society of London A, vol. 392, no. 1802, pp. 45–57, 1984.
[68]
Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Physical Review B, vol. 74, no. 8, p. 085111, 2006.
[69]
L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nature Photonics, vol. 3, no. 8, pp. 461–463, 2009.
[70]
K. Sakoda, Optical Properties of Photonic Crystals, Springer-Verlag, Berlin, Germany, 2nd edition, 2004.
[71]
Y. D. Chong, X. G. Wen, and M. Soljacic, “Effective theory of quadratic degeneracies,” Physical Review B, vol. 77, no. 23, p. 235125, 2008.
[72]
K. Sakoda and H. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Optics Express, vol. 18, no. 26, pp. 27371–27386, 2010.
[73]
K. Sakoda and H. Zhou, “Analytical study of two-dimensional degenerate metamaterial antennas,” Optics Express, vol. 19, no. 15, pp. 13899–13921, 2011.
[74]
K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Optics Express, vol. 20, no. 4, pp. 3898–3917, 2012.
[75]
K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Optics Express, vol. 20, no. 9, pp. 9925–9939, 2012.
[76]
T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics, Springer, Berlin, Germany, 1990.
[77]
J. Mei, Y. Wu, C. T. Chan, and Z. Q. Zhang, “Do linear dispersions of classical waves mean Dirac cones?” Physical Review B, vol. 86, p. 035141, 2012.
[78]
Z. Lan, N. Goldman, A. Bermudez, W. Lu, and P. Ohberg, “Dirac-Weyl fermions with arbitrary spin in two-dimensional optical superlattices,” Physical Review B, vol. 84, no. 16, p. 165115, 2011.
[79]
B. Dora, J. Kailasvuori, and R. Moessner, “Lattice generalization of the Dirac equation to general spin and the role of the flat band,” Physical Review B, vol. 84, no. 19, p. 195422, 2011.
[80]
E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Physical Review Letters, vol. 56, no. 14, pp. 1471–1474, 1986.
[81]
J. T. Costa and M. G. Silveirinha, “Mimicking the Veselago-Pendry lens with broadband matched double-negative metamaterials,” Physical Review B, vol. 84, no. 15, p. 155131, 2011.
[82]
C. T. Chan, X. Huang, F. Liu, and Z. H. Hang, “Dirac dispersion and zero-index in two dimensional and three dimensional photonic and phononic systems,” Progress In Electromagnetics Research B, vol. 44, pp. 163–190, 2012.
