## A semi-Dirac point and an electromagnetic topological transition in a dielectric photonic crystal |

Optics Express, Vol. 22, Issue 2, pp. 1906-1917 (2014)

http://dx.doi.org/10.1364/OE.22.001906

Acrobat PDF (3058 KB)

### Abstract

Accidental degeneracy in a photonic crystal consisting of a square array of elliptical dielectric cylinders leads to both a semi-Dirac point at the center of the Brillouin zone and an electromagnetic topological transition (ETT). A perturbation method is deduced to affirm the peculiar linear-parabolic dispersion near the semi-Dirac point. An effective medium theory is developed to explain the simultaneous semi-Dirac point and ETT and to show that the photonic crystal is either a zero-refractive-index material or an epsilon-near-zero material at the semi-Dirac point. Drastic changes in the wave manipulation properties at the semi-Dirac point, resulting from ETT, are described.

© 2014 Optical Society of America

## 1. Introduction

*Zitterbewegung*, and anti-localization, because the

1. P. R. Wallace, “The band theory of Graphite,” Phys. Rev. **71**(9), 622–634 (1947). [CrossRef]

3. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. **81**(1), 109–162 (2009). [CrossRef]

2. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. **6**(3), 183–191 (2007). [CrossRef] [PubMed]

22. D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B **87**(11), 115143 (2013). [CrossRef]

5. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A **78**(3), 033834 (2008). [CrossRef]

6. R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A **75**(6), 063813 (2007). [CrossRef]

7. S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. **104**(4), 043903 (2010). [CrossRef] [PubMed]

*Zitterbewegung*[8

8. X. Zhang, “Observing Zitterbewegung for Photons near the Dirac Point of a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. **100**(11), 113903 (2008). [CrossRef] [PubMed]

9. X. Zhang and Z. Liu, “Extremal Transmission and Beating Effect of Acoustic Waves in Two-Dimensional Sonic Crystals,” Phys. Rev. Lett. **101**(26), 264303 (2008). [CrossRef] [PubMed]

10. 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,” Nat. Mater. **10**(8), 582–586 (2011). [CrossRef] [PubMed]

23. V. Pardo and W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO_{2}/VO_{2} Nanostructures,” Phys. Rev. Lett. **102**(16), 166803 (2009). [CrossRef] [PubMed]

24. S. Banerjee, R. R. P. Singh, V. Pardo, and W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. **103**(1), 016402 (2009). [CrossRef] [PubMed]

23. V. Pardo and W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO_{2}/VO_{2} Nanostructures,” Phys. Rev. Lett. **102**(16), 166803 (2009). [CrossRef] [PubMed]

26. M. O. Goerbig, “Electronic properties of graphene in a strong magnetic field,” Rev. Mod. Phys. **83**(4), 1193–1243 (2011). [CrossRef]

26. M. O. Goerbig, “Electronic properties of graphene in a strong magnetic field,” Rev. Mod. Phys. **83**(4), 1193–1243 (2011). [CrossRef]

27. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science **336**(6078), 205–209 (2012). [CrossRef] [PubMed]

27. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science **336**(6078), 205–209 (2012). [CrossRef] [PubMed]

10. 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,” Nat. Mater. **10**(8), 582–586 (2011). [CrossRef] [PubMed]

28. J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. **96**(10), 101109 (2010). [CrossRef]

30. Y. Wu and J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. **102**(18), 183105 (2013). [CrossRef]

31. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. **97**(15), 157403 (2006). [CrossRef] [PubMed]

34. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. **100**(3), 033903 (2008). [CrossRef] [PubMed]

35. N. Engheta, “Materials Science. Pursuing Near-Zero Response,” Science **340**(6130), 286–287 (2013). [CrossRef] [PubMed]

31. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. **97**(15), 157403 (2006). [CrossRef] [PubMed]

10. 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,” Nat. Mater. **10**(8), 582–586 (2011). [CrossRef] [PubMed]

19. J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B **86**(3), 035141 (2012). [CrossRef]

20. Y. Li, Y. Wu, X. Chen, and J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express **21**(6), 7699–7711 (2013). [CrossRef] [PubMed]

## 2. The photonic crystal system and its dispersion relation

**10**(8), 582–586 (2011). [CrossRef] [PubMed]

17. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express **20**(4), 3898–3917 (2012). [CrossRef] [PubMed]

20. Y. Li, Y. Wu, X. Chen, and J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express **21**(6), 7699–7711 (2013). [CrossRef] [PubMed]

27. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science **336**(6078), 205–209 (2012). [CrossRef] [PubMed]

## 3. A perturbation method and the confirmation of the semi-Dirac dispersion

19. J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B **86**(3), 035141 (2012). [CrossRef]

20. Y. Li, Y. Wu, X. Chen, and J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express **21**(6), 7699–7711 (2013). [CrossRef] [PubMed]

19. J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B **86**(3), 035141 (2012). [CrossRef]

**21**(6), 7699–7711 (2013). [CrossRef] [PubMed]

36. B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter **12**(34), R435–R461 (2000). [CrossRef]

*not sufficient*if I consider only the doubly-degenerate states at Point A to compute the linear slopes. The reason is that the eigenstate marked as “B” in Fig. 1(a) is very close to the doubly-degenerate states and its contribution to the linear term of the dispersion relation near Point A is not negligible. Therefore, I need to solve the following

*all*directions

*except for*the

36. B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter **12**(34), R435–R461 (2000). [CrossRef]

**21**(6), 7699–7711 (2013). [CrossRef] [PubMed]

**10**(8), 582–586 (2011). [CrossRef] [PubMed]

37. Y. Wu, J. Li, Z.-Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B **74**(8), 085111 (2006). [CrossRef]

## 4. Anisotropic effective medium theory and the electromagnetic topological transition

37. Y. Wu, J. Li, Z.-Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B **74**(8), 085111 (2006). [CrossRef]

37. Y. Wu, J. Li, Z.-Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B **74**(8), 085111 (2006). [CrossRef]

38. Y. Lai, Y. Wu, P. Sheng, and Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. **10**(8), 620–624 (2011). [CrossRef] [PubMed]

39. H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. **14**(12), 123010 (2012). [CrossRef]

41. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. **108**(21), 213903 (2012). [CrossRef] [PubMed]

**10**(8), 582–586 (2011). [CrossRef] [PubMed]

35. N. Engheta, “Materials Science. Pursuing Near-Zero Response,” Science **340**(6130), 286–287 (2013). [CrossRef] [PubMed]

## 5. Conclusions

## Appendix

^{th}branch of the dispersion relations.

## Acknowledgments

## References and links

1. | P. R. Wallace, “The band theory of Graphite,” Phys. Rev. |

2. | A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. |

3. | A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. |

4. | F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. |

5. | S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A |

6. | R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A |

7. | S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. |

8. | X. Zhang, “Observing Zitterbewegung for Photons near the Dirac Point of a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. |

9. | X. Zhang and Z. Liu, “Extremal Transmission and Beating Effect of Acoustic Waves in Two-Dimensional Sonic Crystals,” Phys. Rev. Lett. |

10. | 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,” Nat. Mater. |

11. | T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B |

12. | V. Yannopapas, “Photonic analog of a spin-polarized system with Rashba spin-orbit coupling,” Phys. Rev. B |

13. | J. Bravo-Abad, J. D. Joannopoulos, and M. Soljačić, “Enabling single-mode behavior over large areas with photonic Dirac cones,” Proc. Natl. Acad. Sci. U.S.A. |

14. | A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. |

15. | M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature |

16. | Y. P. Bliokh, V. Freilikher, and F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: A comparative study,” Phys. Rev. B |

17. | K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express |

18. | K. Sakoda, “Proof of the universality of mode symmetries in creating photonic Dirac cones,” Opt. Express |

19. | J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B |

20. | Y. Li, Y. Wu, X. Chen, and J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express |

21. | D. Torrent and J. Sánchez-Dehesa, “Acoustic Analogue of Graphene: Observation of Dirac Cones in Acoustic Surface Waves,” Phys. Rev. Lett. |

22. | D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B |

23. | V. Pardo and W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO |

24. | S. Banerjee, R. R. P. Singh, V. Pardo, and W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. |

25. | G. Montambaux, F. Piéchon, J. N. Fuchs, and M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B |

26. | M. O. Goerbig, “Electronic properties of graphene in a strong magnetic field,” Rev. Mod. Phys. |

27. | H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science |

28. | J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. |

29. | V. C. Nguyen, L. Chen, and K. Halterman, “Total Transmission and Total Reflection by Zero Index Metamaterials with Defects,” Phys. Rev. Lett. |

30. | Y. Wu and J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. |

31. | M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. |

32. | A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B |

33. | A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B |

34. | B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. |

35. | N. Engheta, “Materials Science. Pursuing Near-Zero Response,” Science |

36. | B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter |

37. | Y. Wu, J. Li, Z.-Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B |

38. | Y. Lai, Y. Wu, P. Sheng, and Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. |

39. | H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. |

40. | J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. |

41. | Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. |

**OCIS Codes**

(260.2030) Physical optics : Dispersion

(160.3918) Materials : Metamaterials

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: November 26, 2013

Revised Manuscript: January 2, 2014

Manuscript Accepted: January 10, 2014

Published: January 21, 2014

**Citation**

Ying Wu, "A semi-Dirac point and an electromagnetic topological transition in a dielectric photonic crystal," Opt. Express **22**, 1906-1917 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-1906

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### References

- P. R. Wallace, “The band theory of Graphite,” Phys. Rev. 71(9), 622–634 (1947). [CrossRef]
- A. K. Geim, K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007). [CrossRef] [PubMed]
- A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009). [CrossRef]
- F. D. M. Haldane, S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008). [CrossRef] [PubMed]
- S. Raghu, F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008). [CrossRef]
- R. A. Sepkhanov, Y. B. Bazaliy, C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007). [CrossRef]
- S. R. Zandbergen, M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010). [CrossRef] [PubMed]
- X. Zhang, “Observing Zitterbewegung for Photons near the Dirac Point of a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 100(11), 113903 (2008). [CrossRef] [PubMed]
- X. Zhang, Z. Liu, “Extremal Transmission and Beating Effect of Acoustic Waves in Two-Dimensional Sonic Crystals,” Phys. Rev. Lett. 101(26), 264303 (2008). [CrossRef] [PubMed]
- X. Huang, Y. Lai, Z. H. Hang, H. Zheng, C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011). [CrossRef] [PubMed]
- T. Ochiai, M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009). [CrossRef]
- V. Yannopapas, “Photonic analog of a spin-polarized system with Rashba spin-orbit coupling,” Phys. Rev. B 83(11), 113101 (2011). [CrossRef]
- J. Bravo-Abad, J. D. Joannopoulos, M. Soljačić, “Enabling single-mode behavior over large areas with photonic Dirac cones,” Proc. Natl. Acad. Sci. U.S.A. 109(25), 9761–9765 (2012). [CrossRef] [PubMed]
- A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012). [CrossRef] [PubMed]
- M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013). [CrossRef] [PubMed]
- Y. P. Bliokh, V. Freilikher, F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: A comparative study,” Phys. Rev. B 87(24), 245134 (2013). [CrossRef]
- K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20(4), 3898–3917 (2012). [CrossRef] [PubMed]
- K. Sakoda, “Proof of the universality of mode symmetries in creating photonic Dirac cones,” Opt. Express 20(22), 25181–25194 (2012). [CrossRef] [PubMed]
- J. Mei, Y. Wu, C. T. Chan, Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012). [CrossRef]
- Y. Li, Y. Wu, X. Chen, J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013). [CrossRef] [PubMed]
- D. Torrent, J. Sánchez-Dehesa, “Acoustic Analogue of Graphene: Observation of Dirac Cones in Acoustic Surface Waves,” Phys. Rev. Lett. 108(17), 174301 (2012). [CrossRef] [PubMed]
- D. Torrent, D. Mayou, J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013). [CrossRef]
- V. Pardo, W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO2/VO2 Nanostructures,” Phys. Rev. Lett. 102(16), 166803 (2009). [CrossRef] [PubMed]
- S. Banerjee, R. R. P. Singh, V. Pardo, W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009). [CrossRef] [PubMed]
- G. Montambaux, F. Piéchon, J. N. Fuchs, M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (2009). [CrossRef]
- M. O. Goerbig, “Electronic properties of graphene in a strong magnetic field,” Rev. Mod. Phys. 83(4), 1193–1243 (2011). [CrossRef]
- H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012). [CrossRef] [PubMed]
- J. Hao, W. Yan, M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010). [CrossRef]
- V. C. Nguyen, L. Chen, K. Halterman, “Total Transmission and Total Reflection by Zero Index Metamaterials with Defects,” Phys. Rev. Lett. 105(23), 233908 (2010). [CrossRef] [PubMed]
- Y. Wu, J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. 102(18), 183105 (2013). [CrossRef]
- M. Silveirinha, N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006). [CrossRef] [PubMed]
- A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013). [CrossRef]
- A. Alu, M. G. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007). [CrossRef]
- B. Edwards, A. Alù, M. E. Young, M. Silveirinha, N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008). [CrossRef] [PubMed]
- N. Engheta, “Materials Science. Pursuing Near-Zero Response,” Science 340(6130), 286–287 (2013). [CrossRef] [PubMed]
- B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter 12(34), R435–R461 (2000). [CrossRef]
- Y. Wu, J. Li, Z.-Q. Zhang, C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006). [CrossRef]
- Y. Lai, Y. Wu, P. Sheng, Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011). [CrossRef] [PubMed]
- H. F. Ma, J. H. Shi, B. G. Cai, T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012). [CrossRef]
- J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012). [CrossRef]
- Q. Cheng, W. X. Jiang, T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012). [CrossRef] [PubMed]

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