Abstract

Localized surface plasmon resonances of metal nanoparticles of arbitrary shape are analyzed in the near-static limit with retardation included to the second order. Starting from the electrostatic approximation, the second-order correction to the resonant dielectric constant is expressed by means of a triple surface integral. For arbitrary nano particles with cylindrical symmetry we show how the triple surface integral can be significantly simplified, resulting in a computationally efficient scheme for evaluation of nanoparticle plasmon eigenresonances in the near-static limit. The approach allows for calculation of both dipolar and higher-order resonances.

© 2011 Optical Society of America

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    [CrossRef]
  2. M. Moskovits, Rev. Mod. Phys. 57, 783 (1985).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. S. A. Maier and H. A. Atwater, J. Appl. Phys. 98, 011101 (2005).
    [CrossRef]
  6. J. W. Strutt (Lord Rayleigh), Philos. Mag. 41, 447 (1871).
  7. L. Lorenz, K. Dan. Vidensk. Selsk. Skr. 6, 1 (1890).
  8. G. Mie, Ann. Phys. 330, 377 (1908).
    [CrossRef]
  9. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).
  10. A. M. Kern and O. J. F. Martin, J. Opt. Soc. Am. A 26, 732 (2009).
    [CrossRef]
  11. T. Søndergaard, Phys. Status Solidi B 244, 3448 (2007).
    [CrossRef]
  12. H. Kawata, H. Tamarup, K. Esumi, and K. Miyano, Appl. Phys. Lett. 83, 4625 (2003).
    [CrossRef]
  13. I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, Phys. Rev. B 72, 155412 (2005).
    [CrossRef]
  14. J. Jung, T. G. Pedersen, T. Søndergaard, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B 81, 125413 (2010).
    [CrossRef]
  15. J. Jung, T. Søndergaard, T. G. Pedersen, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B (to be published).

2010 (2)

H. Atwater and A. Polman, Nat. Mater. 9, 205 (2010).
[CrossRef] [PubMed]

J. Jung, T. G. Pedersen, T. Søndergaard, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B 81, 125413 (2010).
[CrossRef]

2009 (1)

2007 (2)

S. Lal, S. Link, and N. J. Halas, Nat. Photon. 1, 641 (2007).
[CrossRef]

T. Søndergaard, Phys. Status Solidi B 244, 3448 (2007).
[CrossRef]

2005 (2)

S. A. Maier and H. A. Atwater, J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, Phys. Rev. B 72, 155412 (2005).
[CrossRef]

2003 (1)

H. Kawata, H. Tamarup, K. Esumi, and K. Miyano, Appl. Phys. Lett. 83, 4625 (2003).
[CrossRef]

2002 (1)

Y. C. Cao, R. Jin, and C. A. Mirkin, Science 297, 1536 (2002).
[CrossRef] [PubMed]

1985 (1)

M. Moskovits, Rev. Mod. Phys. 57, 783 (1985).
[CrossRef]

1908 (1)

G. Mie, Ann. Phys. 330, 377 (1908).
[CrossRef]

1890 (1)

L. Lorenz, K. Dan. Vidensk. Selsk. Skr. 6, 1 (1890).

1871 (1)

J. W. Strutt (Lord Rayleigh), Philos. Mag. 41, 447 (1871).

Atwater, H.

H. Atwater and A. Polman, Nat. Mater. 9, 205 (2010).
[CrossRef] [PubMed]

Atwater, H. A.

S. A. Maier and H. A. Atwater, J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

Cao, Y. C.

Y. C. Cao, R. Jin, and C. A. Mirkin, Science 297, 1536 (2002).
[CrossRef] [PubMed]

Esumi, K.

H. Kawata, H. Tamarup, K. Esumi, and K. Miyano, Appl. Phys. Lett. 83, 4625 (2003).
[CrossRef]

Fredkin, D. R.

I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, Phys. Rev. B 72, 155412 (2005).
[CrossRef]

Halas, N. J.

S. Lal, S. Link, and N. J. Halas, Nat. Photon. 1, 641 (2007).
[CrossRef]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

Jin, R.

Y. C. Cao, R. Jin, and C. A. Mirkin, Science 297, 1536 (2002).
[CrossRef] [PubMed]

Jung, J.

J. Jung, T. Søndergaard, T. G. Pedersen, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B (to be published).

J. Jung, T. G. Pedersen, T. Søndergaard, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B 81, 125413 (2010).
[CrossRef]

Kawata, H.

H. Kawata, H. Tamarup, K. Esumi, and K. Miyano, Appl. Phys. Lett. 83, 4625 (2003).
[CrossRef]

Kern, A. M.

Lal, S.

S. Lal, S. Link, and N. J. Halas, Nat. Photon. 1, 641 (2007).
[CrossRef]

Larsen, A. L.

J. Jung, T. Søndergaard, T. G. Pedersen, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B (to be published).

J. Jung, T. G. Pedersen, T. Søndergaard, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B 81, 125413 (2010).
[CrossRef]

Link, S.

S. Lal, S. Link, and N. J. Halas, Nat. Photon. 1, 641 (2007).
[CrossRef]

Lorenz, L.

L. Lorenz, K. Dan. Vidensk. Selsk. Skr. 6, 1 (1890).

Maier, S. A.

S. A. Maier and H. A. Atwater, J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

Martin, O. J. F.

Mayergoyz, I. D.

I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, Phys. Rev. B 72, 155412 (2005).
[CrossRef]

Mie, G.

G. Mie, Ann. Phys. 330, 377 (1908).
[CrossRef]

Mirkin, C. A.

Y. C. Cao, R. Jin, and C. A. Mirkin, Science 297, 1536 (2002).
[CrossRef] [PubMed]

Miyano, K.

H. Kawata, H. Tamarup, K. Esumi, and K. Miyano, Appl. Phys. Lett. 83, 4625 (2003).
[CrossRef]

Moskovits, M.

M. Moskovits, Rev. Mod. Phys. 57, 783 (1985).
[CrossRef]

Nielsen, B. B.

J. Jung, T. Søndergaard, T. G. Pedersen, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B (to be published).

J. Jung, T. G. Pedersen, T. Søndergaard, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B 81, 125413 (2010).
[CrossRef]

Pedersen, K.

J. Jung, T. Søndergaard, T. G. Pedersen, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B (to be published).

J. Jung, T. G. Pedersen, T. Søndergaard, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B 81, 125413 (2010).
[CrossRef]

Pedersen, T. G.

J. Jung, T. Søndergaard, T. G. Pedersen, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B (to be published).

J. Jung, T. G. Pedersen, T. Søndergaard, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B 81, 125413 (2010).
[CrossRef]

Polman, A.

H. Atwater and A. Polman, Nat. Mater. 9, 205 (2010).
[CrossRef] [PubMed]

Rayleigh,

J. W. Strutt (Lord Rayleigh), Philos. Mag. 41, 447 (1871).

Søndergaard, T.

J. Jung, T. Søndergaard, T. G. Pedersen, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B (to be published).

J. Jung, T. G. Pedersen, T. Søndergaard, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B 81, 125413 (2010).
[CrossRef]

T. Søndergaard, Phys. Status Solidi B 244, 3448 (2007).
[CrossRef]

Strutt, J. W.

J. W. Strutt (Lord Rayleigh), Philos. Mag. 41, 447 (1871).

Tamarup, H.

H. Kawata, H. Tamarup, K. Esumi, and K. Miyano, Appl. Phys. Lett. 83, 4625 (2003).
[CrossRef]

Zhang, Z.

I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, Phys. Rev. B 72, 155412 (2005).
[CrossRef]

Ann. Phys. (1)

G. Mie, Ann. Phys. 330, 377 (1908).
[CrossRef]

Appl. Phys. Lett. (1)

H. Kawata, H. Tamarup, K. Esumi, and K. Miyano, Appl. Phys. Lett. 83, 4625 (2003).
[CrossRef]

J. Appl. Phys. (1)

S. A. Maier and H. A. Atwater, J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

J. Opt. Soc. Am. A (1)

K. Dan. Vidensk. Selsk. Skr. (1)

L. Lorenz, K. Dan. Vidensk. Selsk. Skr. 6, 1 (1890).

Nat. Mater. (1)

H. Atwater and A. Polman, Nat. Mater. 9, 205 (2010).
[CrossRef] [PubMed]

Nat. Photon. (1)

S. Lal, S. Link, and N. J. Halas, Nat. Photon. 1, 641 (2007).
[CrossRef]

Philos. Mag. (1)

J. W. Strutt (Lord Rayleigh), Philos. Mag. 41, 447 (1871).

Phys. Rev. B (3)

I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, Phys. Rev. B 72, 155412 (2005).
[CrossRef]

J. Jung, T. G. Pedersen, T. Søndergaard, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B 81, 125413 (2010).
[CrossRef]

J. Jung, T. Søndergaard, T. G. Pedersen, K. Pedersen, A. L. Larsen, and B. B. Nielsen, Phys. Rev. B (to be published).

Phys. Status Solidi B (1)

T. Søndergaard, Phys. Status Solidi B 244, 3448 (2007).
[CrossRef]

Rev. Mod. Phys. (1)

M. Moskovits, Rev. Mod. Phys. 57, 783 (1985).
[CrossRef]

Science (1)

Y. C. Cao, R. Jin, and C. A. Mirkin, Science 297, 1536 (2002).
[CrossRef] [PubMed]

Other (1)

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

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Figures (3)

Fig. 1
Fig. 1

Three-dimensional nanoparticle with dielectric constant ε 1 ( ω ) bounded by the surface S with outward normal vector n ^ . The dielectric constant outside is ε 2 .

Fig. 2
Fig. 2

Second-order correction to the resonant dielectric constant versus the ratio b / a . The small (blue) circles follow a second-order polynomial in b / a . (a) Oblate spheroids, (b) disks with hemispherical edges.

Fig. 3
Fig. 3

Resonance wavelength of the horizontally polarized dipole mode versus the thickness of silver particles embedded in rutile TiO 2 . (a) Oblate spheroids, (b) disks with hemispherical edges.

Equations (14)

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ε 1 ( ω res ) = ε 1 ( 0 ) + ( a k 0 ) ε 1 ( 1 ) + ( a k 0 ) 2 ε 1 ( 2 ) + .
σ ( r ) = λ ( 0 ) 2 π S H ( r , r ) σ ( r ) d S ,
ε 1 ( 2 ) = ( ε 1 ( 0 ) ε 2 ) 2 8 π S τ ( r ) n ^ · e 1 ( 0 ) ( r ) d S S τ ( r ) n ^ · S [ n ^ × e 1 ( 0 ) ( r ) ] × r r | r r | d S d S ,
n ^ · e 1 ( 0 ) ( r ) = σ ( r ) 2 ε 2 + 1 4 π ε 2 S H ( r , r ) σ ( r ) d S = σ ( r ) ε 1 ( 0 ) ε 2 ,
n ^ × e 1 ( 0 ) ( r ) = 1 4 π ε 2 S n ^ × ( r r ) | r r | 3 σ ( r ) d S ,
τ ( r ) = λ ( 0 ) 2 π S τ ( r ) H ( r , r ) d S .
ε 1 ( 2 ) = ( λ ( 0 ) 1 λ ( 0 ) ) 3 ε 2 2 4 π 2 S τ ( r ) n ^ · S [ n ^ × S r r | r r | 3 σ ( r ) d S ] × r r | r r | d S d S .
0 2 π 0 2 π 0 2 π n ^ · [ n ^ × ( r r ) ] × ( r r ) | r r | | r r | 3 d φ d φ d φ
J v ( θ , θ , θ ) = A F 0 , 0 ( θ , θ ) F 0 , 1 ( θ , θ ) + B F 1 , 0 ( θ , θ ) × F 0 , 1 ( θ , θ ) + ( C + E ) F 1 , 0 ( θ , θ ) F 1 , 1 ( θ , θ ) + D F 0 , 0 ( θ , θ ) F 1 , 1 ( θ , θ ) ,
J h ( θ , θ , θ ) = A F 1 , 0 ( θ , θ ) F 1 , 1 ( θ , θ ) + B F 2 , 0 ( θ , θ ) × F 1 , 1 ( θ , θ ) + C [ ( 2 F 2 , 0 ( θ , θ ) F 0 , 0 ( θ , θ ) ) × F 2 , 1 ( θ , θ ) ( F 2 , 0 ( θ , θ ) F 0 , 0 ( θ , θ ) ) × F 0 , 1 ( θ , θ ) ] + D F 1 , 0 ( θ , θ ) F 2 , 1 ( θ , θ ) + E F 2 , 0 ( θ , θ ) F 2 , 1 ( θ , θ ) ,
F m , n ( θ , θ ) = 0 2 π cos m φ d φ [ ζ ( θ , θ ) + ξ ( θ , θ ) cos φ ] ( 2 n + 1 ) / 2 ,
ε 1 ( 2 ) = ( λ ( 0 ) 1 λ ( 0 ) ) 3 ε 2 2 4 π 2 0 π 0 π 0 π τ ( θ ) J v , h ( θ , θ , θ ) × σ ( θ ) d S ( θ ) d S ( θ ) d S ( θ ) ,
ε 1 ( 0 ) = l + 1 l ε 2 , ε 1 ( 2 ) = 2 ( l + 1 ) ( 2 l + 1 ) l 2 [ 4 l ( l + 1 ) 3 ] ε 2 2 .
ω res ( 2 ) = ω p ε + ε 2 λ ( 0 ) + 1 λ ( 0 ) 1 + α ε 2 2 ( a / c ) 2 ω p 3 2 [ ε + ε 2 λ ( 0 ) + 1 λ ( 0 ) 1 ] 5 / 2 .

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