## Spectral gaps and mode localization in Fibonacci chains of metal nanoparticles

Optics Express, Vol. 15, Issue 22, pp. 14396-14403 (2007)

http://dx.doi.org/10.1364/OE.15.014396

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### Abstract

In this paper we study the spectral, localization and dispersion properties of dipolar modes in quasi-periodically modulated nanoparticle chains based on the Fibonacci sequence. By developing a transfer matrix approach for the calculation of resonant frequencies, oscillation eigenvectors and integrated density of states (IDS) of spatially-modulated dipole chains, we demonstrate the presence of large spectral gaps and calculate the pseudo-dispersion diagram of Fibonacci plasmonic chains. The presence of plasmonic band-gaps and localized states in metal nanoparticle chains based on quasi-periodic order can have a large impact in the design and fabrication of novel nanophotonics devices.

© 2007 Optical Society of America

## 1. Introduction

1. M. Kohmoto, B. Sutherland, and C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model,” Phys. Rev. B **35**, 1020–1033 (1987). [CrossRef]

_{n}starts with an arbitrary seed element, F

_{0}=A for instance, and the inflation rule is repeatedly applied to obtain: F

_{1}=AB, F

_{2}=ABA, F

_{3}=ABAAB, etc, which displays the well known Fibonacci symmetry: F

_{n+1}={F

_{n}, F

_{n-1}} for j≥1.

2. D. Levine and P. J. Steinhardt, “Quasicrystals: definition and structure,” Phys. Rev. B **34**, 596–616 (1986). [CrossRef]

4. T. Fujiwara and T. Ogawa, *Quasicrystals* (Springer-Verlag, Berlin, 1990). [CrossRef]

5. R. B. Capaz, B. Koiller, and S. L. A. de Queiroz, “Gap states and localization properties of one-dimensional Fibonacci quasicrystals,” Phys. Rev. B **42**, 6402–6406 (1990). [CrossRef]

6. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization in Optics: Quasiperiodic media,” Phys. Rev. Lett. , **58**, 2436–2438 (1987). [CrossRef] [PubMed]

7. C. Benoit, G. Poussigue, and A. Azougarh, “Neutron scattering by phonons in quasi-crystals,” J. Phys.: Condens. Matter **2**, 2519–2536 (1990). [CrossRef]

8. E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. **376**, 225–337 (2003). [CrossRef]

8. E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. **376**, 225–337 (2003). [CrossRef]

9. A. Rudinger and F. Piechon, “On the multifractal spectrum of the Fibonacci chain,” J. Phys. A.: Math. Gen. **31**, 155–164 (1998). [CrossRef]

10. T. Fujiwara, M. Kohmoto, and T. Tokihiro, “Multifractal wavefunctions on a Fibonacci lattice,” Phys. Rev. B **40**, 7413–7416 (1989). [CrossRef]

11. F. Igloi, L. Turban, and H. Rieger, “Anomalous diffusion in aperiodic environments,” Phys. Rev. E. **59**, 1465–1474 (1999). [CrossRef]

6. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization in Optics: Quasiperiodic media,” Phys. Rev. Lett. , **58**, 2436–2438 (1987). [CrossRef] [PubMed]

6. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization in Optics: Quasiperiodic media,” Phys. Rev. Lett. , **58**, 2436–2438 (1987). [CrossRef] [PubMed]

12. W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. **72**, 633–636 (1994). [CrossRef] [PubMed]

13. T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B **50**, 4220–4223 (1994). [CrossRef]

14. L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, L. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. **90**, 055501 (2003). [CrossRef] [PubMed]

15. R. Zia, J. A. Schuller, and M. L. Brongersma, “Plasmonics: The Next Chip-Scale Technology,” Materials Today **9**, 20–27 (2006). [CrossRef]

16. S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, “Plasmonics -A Route to Nanoscale Optical Devices,” Adv. Mater. **13**, 1501 (2001). [CrossRef]

17. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. **2**, 229–232 (2003). [CrossRef] [PubMed]

15. R. Zia, J. A. Schuller, and M. L. Brongersma, “Plasmonics: The Next Chip-Scale Technology,” Materials Today **9**, 20–27 (2006). [CrossRef]

19. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B. **62**, 356–359 (2000). [CrossRef]

## 2. Computational method

17. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. **2**, 229–232 (2003). [CrossRef] [PubMed]

19. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B. **62**, 356–359 (2000). [CrossRef]

20. S. Y. Park and D. Stroud, “Surface-plasmon relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B. **69**, 125418 (2004). [CrossRef]

21. C. Girard and R. Quidant, “Near-field optical transmittance of metal particle chain waveguides,” Opt. Express , **12**, 6141 (2004). [CrossRef] [PubMed]

17. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. **2**, 229–232 (2003). [CrossRef] [PubMed]

15. R. Zia, J. A. Schuller, and M. L. Brongersma, “Plasmonics: The Next Chip-Scale Technology,” Materials Today **9**, 20–27 (2006). [CrossRef]

_{A}=A and d

_{B}=B in the symbolic Fibonacci sequence of any given order F

_{n}[see Fig. 1(a)].

_{11}), which is a good approximation of a “bulk” Fibonacci structure. In addition, we will choose d

_{A}=25 nm and d

_{B}=30 nm which, as explained above, ensures the validity of our dipole approach [17

**2**, 229–232 (2003). [CrossRef] [PubMed]

19. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B. **62**, 356–359 (2000). [CrossRef]

20. S. Y. Park and D. Stroud, “Surface-plasmon relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B. **69**, 125418 (2004). [CrossRef]

**62**, 356–359 (2000). [CrossRef]

*p*=

_{i,n}*qx*is the individual dipole,

_{i,n}*x*is the distance from equilibrium in the

_{i,n}*i*polarization direction at point n along the chain,

*ω*

_{0}is the metal plasma frequency, Γ

_{e}is the electronic relaxation frequency, Γ

_{R}is the relaxation frequency due to radiation into the far-field,

*γ*is a polarization-dependent term (

_{i}*γ*= 1 for transverse modes and

*γ*= - 2 for longitudinal) and α

^{2}

_{n}∝ 1/

*d*

_{n}^{3}is the near-field coupling term [19

**62**, 356–359 (2000). [CrossRef]

**62**, 356–359 (2000). [CrossRef]

*α*is not constant, because nearest neighbors of the

_{n}*n*th dipole do not repeat regularly, but they are now modulated according to the quasi-periodic Fibonacci sequence. Therefore, the positional disorder of the Fibonacci chain induces a nonuniform coupling along the chain. As a result, an analytical solution cannot be found for the eigenmodes of the Fibonacci dipole chain. However, simple transfer matrix approaches exist for the numerical solution of harmonic chains with spatially-modulated coupling [22

22. F. A. B. F. de Moura, L. P. Viana, and A. C. Frery, “Vibrational modes in aperiodic one-dimensional harmonic chains,” Phys. Rev. B. **73**, 212302 (2006). [CrossRef]

23. P. K. Datta and K. Kundu, “The absence of localization in one-dimensional disordered harmonic chains,” J. Phys: Condens. Matter **6**, 4465–4478 (1994). [CrossRef]

*u*is the Fourier transform of the oscillation amplitude of dipole

_{n}*n*. Equation (2) can now be reduced to a simpler transfer matrix form as:

*T*, relating the

_{n}*u*amplitudes to each other, as:

_{n}*α*, which depend purely on the geometrical arrangement of the particles, are known “in advance” by the application of the Fibonacci rule. Once all the

_{n}*α*have been found, the displacement of the

_{n}*N*th dipole with frequency

*ω*,

*u*(

_{N}*ω*), can be easily calculated by cascading the individual transfer matrices:

*u*

_{0}=

*u*

_{N+1}= 0 , which describes an N-particle chain, we see that the vibration frequencies of the plasmonic chain must satisfy the eigenvalue equation:

**62**, 356–359 (2000). [CrossRef]

*Q*

_{11}, restricting the root finding procedure to real numbers. This choice will allow us to directly focus on the plasmonic transport properties uniquely induced by the quasi-periodic geometry of the Fibonacci chain, regardless of the specific materials parameters. However, it remains clear that the described approach is also suited for arbitrary dispersion parameters, as long as complex-plane root finding is allowed. Once the oscillation properties of the plasmonic chain have been obtained, the integrated density of states (IDS) of the chain can be simply obtained as [3]:

*e*(

_{n}*ω*) is the eigenvector defining the polarization of the atomic displacement for the mode of frequency

_{k}*ω*on the atom at position

_{k}*n*. When the orthogonality relation is imposed on the eigenvectors:

*P*(

*ω*) ≈ 1, while for the most localized modes (where |

_{k}*e*(

_{n}*ω*)|

_{k}^{2}= 0 for all

*n*except one)

*P*(

*ω*) ≈ 1/

_{k}*N*, which is almost zero in the limit of long chains. The participation ratio of an arbitrary mode can therefore take a value between 1/

*N*and 1, depending on the degree of localization of the modes.

## 3. Results and discussion

*Q*

_{11}for a periodic and a Fibonacci quasi-periodic chain, respectively. As expected for a general N-particles chain, in both the cases the number of oscillation frequencies (the zeros of Eq. 6) equals the number of particles in the chain. However, in the case of the Fibonacci chain, the distribution of zeros shows non-periodic frequency gaps separating groups of allowed modes. In order to clearly relate this observation with the plasmon transport along the Fibonacci chain, we calculated the Integrated Density of States (IDS) of periodic and Fibonacci plasmon chains, as described in the previous section.

*ω*=

*f*(

*k*) becomes inappropriate in non-periodic chains. In the case of a quasi-periodic chain, the spatial disorder (Fibonacci quasi-periodicity) induces spatially localized modes which can be described by a superposition of numerous Fourier components with different relative intensities. This situation is reflected in the Fibonacci mode profiles shown in Fig. 3. However, by determining the spatial frequencies associated with Fibonacci modes of different angular frequencies, it is possible to define dispersion diagrams, known as pseudo-Brillouin zones, even in the case of non-periodic structures [3].

4. T. Fujiwara and T. Ogawa, *Quasicrystals* (Springer-Verlag, Berlin, 1990). [CrossRef]

7. C. Benoit, G. Poussigue, and A. Azougarh, “Neutron scattering by phonons in quasi-crystals,” J. Phys.: Condens. Matter **2**, 2519–2536 (1990). [CrossRef]

## 4. Summary

## Acknowledgments

## References and links

1. | M. Kohmoto, B. Sutherland, and C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model,” Phys. Rev. B |

2. | D. Levine and P. J. Steinhardt, “Quasicrystals: definition and structure,” Phys. Rev. B |

3. | C. Janot, |

4. | T. Fujiwara and T. Ogawa, |

5. | R. B. Capaz, B. Koiller, and S. L. A. de Queiroz, “Gap states and localization properties of one-dimensional Fibonacci quasicrystals,” Phys. Rev. B |

6. | M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization in Optics: Quasiperiodic media,” Phys. Rev. Lett. , |

7. | C. Benoit, G. Poussigue, and A. Azougarh, “Neutron scattering by phonons in quasi-crystals,” J. Phys.: Condens. Matter |

8. | E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. |

9. | A. Rudinger and F. Piechon, “On the multifractal spectrum of the Fibonacci chain,” J. Phys. A.: Math. Gen. |

10. | T. Fujiwara, M. Kohmoto, and T. Tokihiro, “Multifractal wavefunctions on a Fibonacci lattice,” Phys. Rev. B |

11. | F. Igloi, L. Turban, and H. Rieger, “Anomalous diffusion in aperiodic environments,” Phys. Rev. E. |

12. | W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. |

13. | T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B |

14. | L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, L. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. |

15. | R. Zia, J. A. Schuller, and M. L. Brongersma, “Plasmonics: The Next Chip-Scale Technology,” Materials Today |

16. | S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, “Plasmonics -A Route to Nanoscale Optical Devices,” Adv. Mater. |

17. | S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. |

18. | U. Kreibig and M. Vollmer, |

19. | M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B. |

20. | S. Y. Park and D. Stroud, “Surface-plasmon relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B. |

21. | C. Girard and R. Quidant, “Near-field optical transmittance of metal particle chain waveguides,” Opt. Express , |

22. | F. A. B. F. de Moura, L. P. Viana, and A. C. Frery, “Vibrational modes in aperiodic one-dimensional harmonic chains,” Phys. Rev. B. |

23. | P. K. Datta and K. Kundu, “The absence of localization in one-dimensional disordered harmonic chains,” J. Phys: Condens. Matter |

24. | M. Schroeder, |

25. | R. C. Hilborn, |

26. | U. Frisch, |

**OCIS Codes**

(230.3990) Optical devices : Micro-optical devices

(240.5420) Optics at surfaces : Polaritons

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: July 17, 2007

Revised Manuscript: September 26, 2007

Manuscript Accepted: October 1, 2007

Published: October 17, 2007

**Citation**

Luca Dal Negro and Ning-Ning Feng, "Spectral gaps and mode localization in Fibonacci chains of metal nanoparticles," Opt. Express **15**, 14396-14403 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-22-14396

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### References

- M. Kohmoto, B. Sutherland, and C. Tang, "Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model," Phys. Rev. B 35, 1020-1033 (1987). [CrossRef]
- D. Levine and P. J. Steinhardt, "Quasicrystals: definition and structure," Phys. Rev. B 34, 596-616 (1986). [CrossRef]
- C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).
- T. Fujiwara and T. Ogawa, Quasicrystals (Springer-Verlag, Berlin, 1990). [CrossRef]
- R. B. Capaz, B. Koiller, and S. L. A. de Queiroz, "Gap states and localization properties of one-dimensional Fibonacci quasicrystals," Phys. Rev. B 42, 6402-6406 (1990). [CrossRef]
- M. Kohmoto, B. Sutherland, and K. Iguchi, "Localization in Optics: Quasiperiodic media," Phys. Rev. Lett., 58, 2436-2438 (1987). [CrossRef] [PubMed]
- C. Benoit, G. Poussigue, and A. Azougarh, "Neutron scattering by phonons in quasi-crystals," J. Phys.: Condens. Matter 2, 2519-2536 (1990). [CrossRef]
- E. L. Albuquerque and M. G. Cottam, "Theory of elementary excitations in quasiperiodic structures," Phys. Rep. 376, 225-337 (2003). [CrossRef]
- A. Rudinger and F. Piechon, "On the multifractal spectrum of the Fibonacci chain," J. Phys. A.: Math. Gen. 31, 155-164 (1998).Q1 [CrossRef]
- T. Fujiwara, M. Kohmoto, and T. Tokihiro, "Multifractal wavefunctions on a Fibonacci lattice," Phys. Rev. B 40, 7413-7416 (1989). [CrossRef]
- F. Igloi, L. Turban, and H. Rieger, "Anomalous diffusion in aperiodic environments," Phys. Rev. E. 59, 1465-1474 (1999). [CrossRef]
- W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, "Localization of light waves in Fibonacci dielectric multilayers," Phys. Rev. Lett. 72, 633-636 (1994). [CrossRef] [PubMed]
- T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, "Photonic dispersion relation in a one-dimensional quasicrystal," Phys. Rev. B 50, 4220-4223 (1994). [CrossRef]
- L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, L. Colocci, and D. S. Wiersma, "Light transport through the band-edge states of Fibonacci quasicrystals," Phys. Rev. Lett. 90, 055501 (2003). [CrossRef] [PubMed]
- R. Zia, J. A. Schuller and M. L. Brongersma, "Plasmonics: The Next Chip-Scale Technology," Materials Today 9, 20-27 (2006). [CrossRef]
- S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A Route to Nanoscale Optical Devices," Adv. Mater. 13, 1501 (2001). [CrossRef]
- S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003). [CrossRef] [PubMed]
- U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).
- M. L. Brongersma, J. W. Hartman, and H. A. Atwater, "Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit," Phys. Rev. B. 62, 356-359 (2000). [CrossRef]
- S. Y. Park and D. Stroud, "Surface-plasmon relations in chains of metallic nanoparticles: an exact quasistatic calculation," Phys. Rev. B. 69, 125418 (2004). [CrossRef]
- C. Girard and R. Quidant, "Near-field optical transmittance of metal particle chain waveguides," Opt. Express, 12, 6141 (2004). [CrossRef] [PubMed]
- F. A. B. F. de Moura, L. P. Viana, A. C. Frery, "Vibrational modes in aperiodic one-dimensional harmonic chains," Phys. Rev. B. 73, 212302 (2006). [CrossRef]
- P. K. Datta and K. Kundu, "The absence of localization in one-dimensional disordered harmonic chains," J. Phys: Condens. Matter 6, 4465-4478 (1994). [CrossRef]
- M. Schroeder, Fractals, Chaos, Power Laws (Freeman, NY, 1991).
- R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford University Press, 2000).
- U. Frisch, Turbolence (Cambridge University Press, 2004).

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