## Dirac dynamics in one-dimensional graphene-like plasmonic crystals: pseudo-spin, chirality, and diffraction anomaly |

Optics Express, Vol. 18, Issue 24, pp. 25329-25338 (2010)

http://dx.doi.org/10.1364/OE.18.025329

Acrobat PDF (1240 KB)

### Abstract

We introduce a new class of plasmonic crystals possessing graphene-like internal symmetries and Dirac-type spectrum in *k*-space. We study dynamics of surface plasmon polaritons supported in the plasmonic crystals by employing the formalism of Dirac dynamics for relativistic quantum particles. Through an analogy with graphene, we introduce a concept of pseudo-spin and chirality to indicate built-in symmetry of the plasmonic crystals near Dirac point. The surface plasmon polaritons with different pseudo-spin states are shown to split in the crystals into two beams, analogous to spin Hall effect.

© 2010 OSA

## 1. Introduction

1. A. K. Geim, “Graphene: status and prospects,” Science **324**(5934), 1530–1534 (2009). [CrossRef] [PubMed]

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

3. 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]

7. O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. **104**(6), 063901 (2010). [CrossRef] [PubMed]

8. S. H. Nam, A. J. Taylor, and A. Efimov, “Diabolical point and conical-like diffraction in periodic plasmonic nanostructures,” Opt. Express **18**(10), 10120–10126 (2010). [CrossRef] [PubMed]

## 2. Lattice symmetry, pseudo-spin, and chirality

8. S. H. Nam, A. J. Taylor, and A. Efimov, “Diabolical point and conical-like diffraction in periodic plasmonic nanostructures,” Opt. Express **18**(10), 10120–10126 (2010). [CrossRef] [PubMed]

8. S. H. Nam, A. J. Taylor, and A. Efimov, “Diabolical point and conical-like diffraction in periodic plasmonic nanostructures,” Opt. Express **18**(10), 10120–10126 (2010). [CrossRef] [PubMed]

10. A. A. Sukhorukov and Y. S. Kivshar, “Discrete gap solitons in modulated waveguide arrays,” Opt. Lett. **27**(23), 2112–2114 (2002). [CrossRef]

*β*and

*A*and

*B*are the amplitudes of the Floquet-Bloch eigenmodes in A and B sublattice, respectively.

*K*exhibits linear Dirac-type spectrum near Dirac point (or diabolical point) as shown in Fig. 1(b). Along the each linear dispersion curves (blue and red lines), the eigenvectors take approximately the same sublattice symmetry, which can be easily confirmed from Eq. (2b) since

11. A. H. Castro Neto, 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]

*K*and positive momentum

*κ*belongs to the same branch of the spectrum (blue) for the SPP mode with the opposite

*K*and

*κ*.

*C*as Eq. (3), which is a projection of pseudo-spin on the direction of momentum

*κ*. where is a 2x2 matrix and y-component of the Pauli vector, and

*γ*is the eigenvalue of

*C.*

*γ*of the chirality operator

*C*takes + 1 and −1 for positive

*K*and negative

*K*SPPs, respectively. For

*K*), the direction of pseudo-spin (θ = 0 for blue line and θ = π for red line) is parallel to the momentum

*κ*and antiparallel for

*K*). This property indicates the SPP states near Dirac point have well defined chirality. The chirality is used to refer to the additional built-in symmetry between the SPP modes with positive and negative

*K*analogous to graphene [11

11. A. H. Castro Neto, 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]

12. M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nat. Phys. **2**(9), 620–625 (2006). [CrossRef]

13. T. Ando, T. Nakanishi, and R. Saito, “Berry's phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. **67**(8), 2857–2862 (1998). [CrossRef]

14. Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, “Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening,” ACS Nano **2**(11), 2301–2305 (2008). [CrossRef]

15. F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. **6**(1), 30–33 (2010). [CrossRef]

## 3. Dirac dynamics of SPP

**18**(10), 10120–10126 (2010). [CrossRef] [PubMed]

*κ*acts on the pseudo-spin

*κ*accounts for SPP coupling between A and B sublattices [16

16. A. K. Geim and A. H. MacDonald, “Graphene: Exploring carbon flatland,” Phys. Today **60**(8), 35–41 (2007). [CrossRef]

17. J. K. Furdyna, “Split light,” Physics **3**, 56 (2010). [CrossRef]

19. O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science **319**(5864), 787–790 (2008). [CrossRef] [PubMed]

*κ*as in Eq. (4): These positive and negative

*K*states are analogous to positive and negative energy states of relativistic particles in QED. If a Gaussian beam is normally incident on the plasmonic crystal and the beam width

*Λ*, the two sublattices are approximately equally excited. Thus the initial spinor state in position space may be expressed as Eq. (5a). By performing Fourier transform, we also obtain an expression in momentum space as Eq. (5b): where

*A*is a normalization factor of the initial state given as

*z*is finally given as Eq. (8) in the momentum and position space, respectively. Note that the above expression is derived only from the Hamiltonian in Eq. (1) and no specific physical parameters of plasmonic crystal are assumed.

*K*states in the momentum space,

*K*state than the negative one (

*K*state are symmetric and antisymmetric, respectively. Thus the input Gaussian wave packet preferentially couples to the positive state at

**18**(10), 10120–10126 (2010). [CrossRef] [PubMed]

*K*state is significantly reduced as seen in Fig. 3(a) and 3(b). This is due to the band deformation near

*K*dispersion curves are not symmetric due to strong coupling in the real structure. Nevertheless, they are in good agreement with the overall pattern.

*K*state [Fig. 4(a) and (b) ]. The wave packet evolution shows conventional diffraction broadening in Fig. 4(c) and no anomaly occurs, which agrees with the numerical simulation for the structure with increased metal thickness to 15nm in the lattice.

## 4. Conclusion

## Appendix

*K*in a similar manner the original Dirac Hamiltonian Eq. (A-2) was derived by linearizing the energy-momentum relation (A-1):

*κ*as in Eq. (A-5), both Hamiltonians have the same eigenvalues

## Acknowledgements

## References and links

1. | A. K. Geim, “Graphene: status and prospects,” Science |

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

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

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

5. | X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. |

6. | O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. |

7. | O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. |

8. | S. H. Nam, A. J. Taylor, and A. Efimov, “Diabolical point and conical-like diffraction in periodic plasmonic nanostructures,” Opt. Express |

9. | S. Hyun Nam, E. Ulin-Avila, G. Bartal, and X. Zhang, “Deep subwavelength surface modes in metal-dielectric metamaterials,” Opt. Lett. |

10. | A. A. Sukhorukov and Y. S. Kivshar, “Discrete gap solitons in modulated waveguide arrays,” Opt. Lett. |

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

12. | M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nat. Phys. |

13. | T. Ando, T. Nakanishi, and R. Saito, “Berry's phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. |

14. | Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, “Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening,” ACS Nano |

15. | F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. |

16. | A. K. Geim and A. H. MacDonald, “Graphene: Exploring carbon flatland,” Phys. Today |

17. | J. K. Furdyna, “Split light,” Physics |

18. | M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. |

19. | O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science |

20. | B. Thaller, |

**OCIS Codes**

(020.5580) Atomic and molecular physics : Quantum electrodynamics

(240.6680) Optics at surfaces : Surface plasmons

(260.1960) Physical optics : Diffraction theory

(160.3918) Materials : Metamaterials

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: September 7, 2010

Revised Manuscript: November 1, 2010

Manuscript Accepted: November 5, 2010

Published: November 19, 2010

**Citation**

Sung Hyun Nam, Jiangfeng Zhou, Antoinette J. Taylor, and Anatoly Efimov, "Dirac dynamics in one-dimensional graphene-like plasmonic crystals: pseudo-spin, chirality, and diffraction anomaly," Opt. Express **18**, 25329-25338 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-25329

Sort: Year | Journal | Reset

### References

- A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009). [CrossRef] [PubMed]
- A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007). [CrossRef] [PubMed]
- 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]
- T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009). [CrossRef]
- 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]
- O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007). [CrossRef] [PubMed]
- O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010). [CrossRef] [PubMed]
- S. H. Nam, A. J. Taylor, and A. Efimov, “Diabolical point and conical-like diffraction in periodic plasmonic nanostructures,” Opt. Express 18(10), 10120–10126 (2010). [CrossRef] [PubMed]
- S. Hyun Nam, E. Ulin-Avila, G. Bartal, and X. Zhang, “Deep subwavelength surface modes in metal-dielectric metamaterials,” Opt. Lett. 35(11), 1847–1849 (2010). [CrossRef] [PubMed]
- A. A. Sukhorukov and Y. S. Kivshar, “Discrete gap solitons in modulated waveguide arrays,” Opt. Lett. 27(23), 2112–2114 (2002). [CrossRef]
- A. H. Castro Neto, 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]
- M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nat. Phys. 2(9), 620–625 (2006). [CrossRef]
- T. Ando, T. Nakanishi, and R. Saito, “Berry's phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. 67(8), 2857–2862 (1998). [CrossRef]
- Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, “Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening,” ACS Nano 2(11), 2301–2305 (2008). [CrossRef]
- F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. 6(1), 30–33 (2010). [CrossRef]
- A. K. Geim and A. H. MacDonald, “Graphene: Exploring carbon flatland,” Phys. Today 60(8), 35–41 (2007). [CrossRef]
- J. K. Furdyna, “Split light,” Physics 3, 56 (2010). [CrossRef]
- M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004). [CrossRef] [PubMed]
- O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008). [CrossRef] [PubMed]
- B. Thaller, Advanced visual quantum mechanics, (Springer, 2005).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.