## Plasmon modes and negative refraction in metal nanowire composites

Optics Express, Vol. 11, Issue 7, pp. 735-745 (2003)

http://dx.doi.org/10.1364/OE.11.000735

Acrobat PDF (3348 KB)

### Abstract

Optical properties of metal nanowires and nanowire composite materials are studied. An incident electromagnetic wave can effectively couple to the propagating surface plasmon polariton (SPP) modes in metal nanowires resulting in very large local fields. The excited SPP modes depend on the structure of nanowires and their orientation with respect to incident radiation. A nanowire percolation composite is shown to have a broadband spectrum of localized plasmon modes. We also show that a composite of nanowires arranged into parallel pairs can act as a left-handed material with the effective magnetic permeability and dielectric permittivity both negative in the visible and near-infrared spectral ranges.

© 2003 Optical Society of America

## 1. Introduction

1. S. D. M. Brown, P. Corio, A. Marucci, M. A. Pimenta, M. S. Dresselhaus, and G. Dresselhaus, “Second-order resonant Raman spectra of single-walled carbon nanotubes,” Phys. Rev. B **61**, 7734–7742 (2000); [CrossRef]

2. K. B. Shelimov and M. Moskovits, “Composite Nanostructures Based on Template-Grown Boron Nitride Nanotubules,” Chemistry of Materials , **12**, 250 (2000); [CrossRef]

3. J. Li, C. Papadopoulos, J.M. Xu, and M. Moskovits, “Highly-ordered carbon nanotube arrays for electronics applications,” Applied Physics Letters **75**, 367 (1999); [CrossRef]

4. J.B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. **85**, 3966 (2000); [CrossRef] [PubMed]

5. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ∊ and μ,” Soviet Physics Uspekhi **10**, 509 (1968). [CrossRef]

*et al.*[6

6. D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Shultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. **84**, 4184 (2000); [CrossRef] [PubMed]

7. G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys.Rev.B67, 035109 (2003) [CrossRef]

8. V.A. Podolskiy, A.K. Sarychev, and V.M. Shalaev, “Plasmon modes in metal nanowires and left-handed materials,” Journal of Nonlinear Optical Physics and Materials **11**, 65 (2002) [CrossRef]

*et al.*[9

9. L.V. Panina, A.N. Grigorenko, and D.P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,” Phys.Rev.B **66**, 155411 (2002) [CrossRef]

8. V.A. Podolskiy, A.K. Sarychev, and V.M. Shalaev, “Plasmon modes in metal nanowires and left-handed materials,” Journal of Nonlinear Optical Physics and Materials **11**, 65 (2002) [CrossRef]

^{3}. We also describe how nanowire composites can be used for developing LHMs

*in the near-IR and visible*parts of the spectrum.

## 2. Coupled Dipole Equations (CDEs)

8. V.A. Podolskiy, A.K. Sarychev, and V.M. Shalaev, “Plasmon modes in metal nanowires and left-handed materials,” Journal of Nonlinear Optical Physics and Materials **11**, 65 (2002) [CrossRef]

*b*

_{2}, we assume to be much smaller than the wavelength of the incident light, λ; the radius, however, can be comparable with the optical skin-depth in the metal. The length of the nanowire, 2

*b*

_{1}, can be on the order of light wavelength. Such relations between parameters of the system makes it very hard to solve the Maxwell equations exactly; they can be, however, analyzed numerically.

10. E.M. Purcell and C.R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophysical Journal **186**, 705 (1973) [CrossRef]

*N*) of interacting polarizable spheres placed on a cubic lattice (see Fig. 1). The lattice period

*a*(and thus the radius of individual spheres,

*R*) is much smaller than the wavelength of the incident light. If the relation

*a*≪ λ is fulfilled, then each dipole can be treated in the quasi-static approximation with the field given by the sum of the incident field and the fields due to all other dipoles, as described by the following coupled-dipole equations (CDEs)

*E*represents the incident field at the location of the

_{inc}*i*-th dipole,

**r**

_{i},

*Ĝ*(

**r**

_{i}-

**r**

_{j})

**d**

_{j}represents the EM field scattered by dipole

*j*at this point, with

*Ĝ*being a regular part of the free-space dyadic Green function. The latter is defined as

*Ĝ*

**d**=

*G*

_{αβ}

*d*

_{β}. The Greek indices represent the Cartesian components of vectors and the summation over the repeated indices is implied.

_{0}, usually given by the Clausius-Mossotti relation (see, for example [11]) with the radiative correction introduced by Draine [12

12. B.T. Draine, “Discrete dipole approximation and its application to interstellar graphite grains,” Astrophys.J. **333**, 848 (1988) [CrossRef]

_{LL}is the Lorentz-Lorenz polarizability without the radiation correction.

*R*to the lattice size

*a*. In their original work, Purcell and Pennypacker [10

10. E.M. Purcell and C.R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophysical Journal **186**, 705 (1973) [CrossRef]

*a*/

*R*< 2; such intersection, phenomenologically, takes into account the multipolar corrections in the depolarization factor, remaining within the dipolar approximation. As shown later by Markel [13, 14

14. V.A. Markel, “Scattering of light from two interacting spherical particles,” J.Mod.Opt **39**853 (1992) [CrossRef]

*a*/

*R*≈ 1.688. According to Draine’s calculations [15] the value given by Eq. (5) leads to small (about 1%) error in the absorption coefficient in the quasi-static case for a spherical system. Our simulations suggest that results of simulations strongly depend on the intersection ratio

*a*/

*R*. In our simulations below, we choose the value of this parameter from the condition that the solution to the CDEs give the correct depolarization factor for a single nanowire.

## 3. SPP resonance in single nanowires and percolation nanowire composites

*m*long illuminated by a plane wave with the vacuum wavelength of 540 nm. We model the nanowire by four parallel arrays of polarizable spheres (see Fig. 1). This 2×2 in cross-section system allows us to account for the skin-effect [8

**11**, 65 (2002) [CrossRef]

**11**, 65 (2002) [CrossRef]

^{3}. The spatial area, where the field is concentrated is highly localized around the nanowire, and can be as small as 100 nm (Fig. 3). Note also that the SPP resonance in a single nanowire is narrow, with the width of about 50

*nm*.

19. S. Ducourtieux, et al, “Near-field optical studies of semicontinuous metal films,” Phys.Rev.B64 165403 (2001) [CrossRef]

20. V. M. Shalaev (editor) *Optical Properties of Nanostructured Random Media*, *Topics in Applied Physics*, v. 82, (Springer Verlag, Berlin, 2002) [CrossRef]

*b*

_{2}/

*b*

_{1}and can be made arbitrary small [21

21. A.N. Lagarkov and A.K. Sarychev, “Electromagnetic properties of composites containing elongated conducting inclusions,” Phys.Rev.B **53**, 6318 (1996) [CrossRef]

## 4. Effective dipole and magnetic response of the nanowire pairs

*R*away from the nanowire pair with dimensions 2

*b*

_{1}×

*d*×2

*b*

_{2}(see Fig. 5(a)) are derived from the vector potential

**A**that for the distances

*R*≫ λ,

*b*

_{1},

*b*

_{2},

*d*takes the standard form

**A**=(

*e*/

^{ikr}*cR*) ∫

*e*

^{-ik(n·r)}

**j**(

**r**)

*d*

**r**, where

**j**(

**r**) is the current density inside the nanowires and vector

**n**is the unit vector in the direction of observation. We introduce vector

**d**directed from one nanowire to another and assume that the coordinate system has its origin in the center of the system so that centers of each wires have coordinates

**d**/2 and -

**d**/2, respectively. The direction of propagation of electromagnetic wave is such that the wavevector

**k**‖

**d**(see Fig. 5).

**A**can be written as

**j**

_{1}and

**j**

_{2}are the currents in the wires, and ρ is the coordinate along the wires (ρ⊥

**d**). It is well known that the dipole component is dominating in the scattering of a thin antenna, even for the antenna size comparable with a wavelength (see, e.g., [11]). Therefore we can replace the term

*e*

^{-ikn·ρ}in Eq. (6) by unity. Note that for the forward and back scattering, which is responsible for the effective properties of the medium, this term exactly equals one.

*d*between the wires is much smaller than the wavelength and expand Eq. (6) in series of

*d*,

**P**for the system of two nanowires and its contribution to the scattering can be written as

**A**

_{d}=-

*ik*(

*e*

^{ikR}/

*R*)

**P**, where

**p**being the local polarization; the integration in Eq. (8) is over the volume of both wires. The second term in Eq. (7) gives the magnetic dipole and quadrupole contributions to the vector potential:

**M**is the magnetic moment of the two wires,

**P**and magnetic

**M**moments given by Eqs. (8) and (10), respectively, since they provide the main contributions to the forward and back scattering. The second term in Eq. (9) describes a quadrupole contribution; the magnetic field associated with this term is proportional to

*d*ρ in the far zone. This term equals zero in the direction

**k**of the incident wave (and the opposite direction) and it achieves the maximum in the direction of the wires

**n**‖ρ. This “oblique” scattering is dumped when the coupled wires are arranged in a regular array. Yet, the oblique scattering can result in the excitation of a surface wave in a layer of the coupled wires; we do not consider this effect here.

## 5. Negative refraction in nanowire composites

*dielectric*and

*magnetic*optical responses of such a system have strong resonances.We suggest that this specific nanowire configuration can be employed as a structure unit for a left-handed material in the near-IF and visible frequency ranges. We illustrate this idea by considering magnetic and dielectric polarization properties of a layer made of such nanowire pairs subjected to a plane electromagnetic wave incident normally onto the layer.

*b*

_{2}is much smaller than the wavelength of the incident light, while the length of nanowires 2

*b*

_{1}can be comparable with the wavelength. The distance between the wires,

*d*, is much smaller than the wire length.

*E*is parallel to the axis of nanowires in the pair, whereas the magnetic component

*H*is perpendicular to the nanowire pair (the nanowires within each pair are placed on top of each other and the pairs form a layer as shown in Fig. 5).

*effective dipole moments*.

*the magnetic moments*in the nanowire pairs associated with the closed-loop currents.

*single*nanowires using the DDA approach described above and compare these moments with those excited in a two-wire system (Fig. 6). For the dipole moment response in a single wire, we see the strong “antenna” resonance when the wavelength of the incident light is close to 4

*b*

_{1}. In the two wire system this resonance becomes somewhat weaker and broader. The induced magnetic moment in a single nanowire is extremely small, as expected; we still can identify the resonance corresponding to the excitation of a circular current on the nanowire surface. This resonance is greatly enhanced in the two-wire system as seen in Fig. 6.

**11**, 65 (2002) [CrossRef]

21. A.N. Lagarkov and A.K. Sarychev, “Electromagnetic properties of composites containing elongated conducting inclusions,” Phys.Rev.B **53**, 6318 (1996) [CrossRef]

*b*

_{1}/

*b*

_{2}(see Fig. 6):

_{m}is the bulk metal conductivity. Although the numerical results shown in Fig. 6 and the theoretical calculations presented in Fig. 7 are qualitatively similar there are also some differences.

*b*

_{1}≫

*d*≫

*b*

_{2}, which is not exactly the case in our numerical simulations. Also, when considering the response to the electric field in derivation of Eqs. (11), we assumed that the two wires in the system interact with the electric field independently, which is indeed only a rough approximation. In accordance with this, when comparing the lines in Fig. 6, we see that the dipole response in Eqs. (11) represents better the dipole moment of an isolated nanowire rather than that for the pair. Note that the dipole response of the pair is weaker then the response of a single nanowire due to radiation losses. Furthermore, in derivation of the magnetic response in Eqs. (11), we didn’t take into account the radiation decay in the magnetic response, which can significantly increase the width of the magnetic resonance, as can be seen in Fig. 6. Despite the differences, we can conclude that theoretical formulas (11) do predict qualitatively similar behavior for the electric and magnetic responses when compared with the numerical simulations and thus they can be useful in estimates of the electric and magnetic responses in metal nanowires.

*b*

_{2}. However, we should note that this parameter should be kept comparable to the skin depth in the nanowire material [8

**11**, 65 (2002) [CrossRef]

*d*), the interaction between the wires becomes smaller and the radiative losses in the system increase. As a result, both the dielectric and magnetic resonances become weaker, resulting in a decreased region for the “negative responses”; the magnetic resonance practically vanishes at larger distances, when the interaction between the wires is negligible (Fig. 8). The dielectric moment, however, starts to grow again after some critical distance between the wires have been achieved, regaining its value for the isolated nanowire (not shown). We didn’t vary the length of nanowires since as mentioned above, the nanowire length should be closed to the half of the resonant wavelength, λ

_{res}≈ 4

*b*

_{1}, for the efficient excitation of SPP in nanowires. We also note that the radiative losses could be significant in the coupled nanowire systems. Our simulations suggest that losses increase as we increase the distance between the wires

*d*or the thickness of the wire

*b*

_{2}(Fig. 9). With a decrease of the wire thickness, the effect becomes stronger and losses smaller; we expect that the effect is particularly strong at the wire thickness close to the skin-depth (20 nm).

## 6. Conclusions

## Acknowledgements

## References and links

1. | S. D. M. Brown, P. Corio, A. Marucci, M. A. Pimenta, M. S. Dresselhaus, and G. Dresselhaus, “Second-order resonant Raman spectra of single-walled carbon nanotubes,” Phys. Rev. B |

2. | K. B. Shelimov and M. Moskovits, “Composite Nanostructures Based on Template-Grown Boron Nitride Nanotubules,” Chemistry of Materials , |

3. | J. Li, C. Papadopoulos, J.M. Xu, and M. Moskovits, “Highly-ordered carbon nanotube arrays for electronics applications,” Applied Physics Letters |

4. | J.B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. |

5. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ∊ and μ,” Soviet Physics Uspekhi |

6. | D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Shultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. |

7. | G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys.Rev.B67, 035109 (2003) [CrossRef] |

8. | V.A. Podolskiy, A.K. Sarychev, and V.M. Shalaev, “Plasmon modes in metal nanowires and left-handed materials,” Journal of Nonlinear Optical Physics and Materials |

9. | L.V. Panina, A.N. Grigorenko, and D.P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,” Phys.Rev.B |

10. | E.M. Purcell and C.R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophysical Journal |

11. | J.D. Jackson |

12. | B.T. Draine, “Discrete dipole approximation and its application to interstellar graphite grains,” Astrophys.J. |

13. | V.A. Markel, “Antisymmetrical optical states,” J.Opt.Soc.Am. |

14. | V.A. Markel, “Scattering of light from two interacting spherical particles,” J.Mod.Opt |

15. | B.T. Draine “The discrete dipole approximation for light scattering by irregular targets” in |

16. | M. Moskovits, private communication |

17. | N. Yamamoto, K. Araya, M. Nakano, and F.J.Garsia de Abajo, “Direct imaging of plasmons in nanostructures”, annual OSA meeting 2002, Orlando, Florida |

18. | D. Stauffer and A. Aharony |

19. | S. Ducourtieux, et al, “Near-field optical studies of semicontinuous metal films,” Phys.Rev.B64 165403 (2001) [CrossRef] |

20. | V. M. Shalaev (editor) |

21. | A.N. Lagarkov and A.K. Sarychev, “Electromagnetic properties of composites containing elongated conducting inclusions,” Phys.Rev.B |

**OCIS Codes**

(160.4670) Materials : Optical materials

(260.5740) Physical optics : Resonance

(310.6860) Thin films : Thin films, optical properties

**ToC Category:**

Focus Issue: Negative refraction and metamaterials

**History**

Original Manuscript: February 10, 2003

Revised Manuscript: March 31, 2003

Published: April 7, 2003

**Citation**

Viktor Podolskiy, Andrey Sarychev, and Vladimir Shalaev, "Plasmon modes and negative refraction in metal nanowire composites," Opt. Express **11**, 735-745 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-7-735

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### References

- S. D. M. Brown, P. Corio, A. Marucci, M. A. Pimenta, M. S. Dresselhaus, and G. Dresselhaus, �??Second-order resonant Raman spectra of single-walled carbon nanotubes,�?? Phys. Rev. B 61, 7734�??7742 (2000) [CrossRef]
- K. B. Shelimov, and M. Moskovits, �??Composite Nanostructures Based on Template-Grown Boron Nitride Nanotubules,�?? Chem. Materials 12, 250 (2000) [CrossRef]
- J. Li, C. Papadopoulos, J.M. Xu, and M. Moskovits, �??Highly-ordered carbon nanotube arrays for electronics applications,�?? Appl. Phys. Lett. 75, 367 (1999) [CrossRef]
- J.B. Pendry, �??Negative Refraction Makes a Perfect Lens,�?? Phys. Rev. Lett. 85, 3966 (2000) [CrossRef] [PubMed]
- V. G. Veselago, �??The electrodynamics of substances with simultaneously negative values of ε and µ,�?? Soviet Phys. Usp. 10, 509 (1968). [CrossRef]
- D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Shultz, �??Composite Medium with Simultaneously Negative Permeability and Permittivity,�?? Phys. Rev. Lett. 84, 4184 (2000) [CrossRef] [PubMed]
- G.Shvets, �??Photonic approach to making a material with a negative index of refraction,�?? Phys. Rev. B 67, 035109 (2003) [CrossRef]
- V.A.Podolskiy, A.K. Sarychev, and V.M. Shalaev, �??Plasmon modes in metal nanowires and lefthanded materials,�?? J. Nonlinear Opt. Phys. Materials 11, 65 (2002) [CrossRef]
- L.V. Panina, A.N. Grigorenko, D.P. Makhnovskiy, �??Optomagnetic composite medium with conducting nanoelements,�?? Phys. Rev. B 66, 155411 (2002) [CrossRef]
- E.M. Purcell and C.R. Pennypacker, �??Scattering and absorption of light by nonspherical dielectric grains,�?? Astrophys. J. 186, 705 (1973) [CrossRef]
- J.D. Jackson, Classical Electrodynamics, (J. Wiley & Sons, Inc, 1999)
- B.T.Draine, �??Discrete dipole approximation and its application to interstellar graphite grains,�?? Astrophys. J. 333, 848 (1988) [CrossRef]
- V.A. Markel, �??Antisymmetrical optical states,�?? J. Opt. Soc. Am. B 12 1783 (1995)
- V.A.Markel, �??Scattering of light from two interacting spherical particles,�?? J. Mod. Opt. 39 853 (1992) [CrossRef]
- B.T.Draine �??The discrete dipole approximation for light scattering by irregular targets�?? in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, Acad.Press (2000)
- M. Moskovits, private communication
- N. , K.Araya, M. Nakano, F.J. Garsia de Abajo, �??Direct imaging of plasmons in nanostructures,�?? OSA Annual Meeting (Optical Society of America, Washington, D.C., 2002).
- D. Stauffer and A. Aharony, Introduction to percolation theory, (Taylor and Fransis, 1994)
- S. Ducourtieux, et al, �??Near-field optical studies of semicontinuous metal films,�?? Phys. Rev. B 64 165403 (2001) [CrossRef]
- V. M. Shalaev (editor) Optical Properties of Nanostructured Random Media , Topics in Applied Physics, v. 82, (Springer Verlag, Berlin, 2002) [CrossRef]
- A.N. Lagarkov and A.K. Sarychev, �??Electromagnetic properties of composites containing elongated conducting inclusions,�?? Phys. Rev. B 53, 6318 (1996) [CrossRef]

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