## Plasmon nanoparticle superlattices as optical-frequency magnetic metamaterials |

Optics Express, Vol. 20, Issue 14, pp. 15781-15796 (2012)

http://dx.doi.org/10.1364/OE.20.015781

Acrobat PDF (2819 KB)

### Abstract

Nanocrystal superlattices have emerged as a new platform for bottom-up metamaterial design, but their optical properties are largely unknown. Here, we investigate their emergent optical properties using a generalized semi-analytic, full-field solver based on rigorous coupled wave analysis. Attention is given to superlattices composed of noble metal and dielectric nanoparticles in unary and binary arrays. By varying the nanoparticle size, shape, separation, and lattice geometry, we demonstrate the broad tunability of superlattice optical properties. Superlattices composed of spherical or octahedral nanoparticles in cubic and AB_{2} arrays exhibit magnetic permeabilities tunable between 0.2 and 1.7, despite having non-magnetic constituents. The retrieved optical parameters are nearly polarization and angle-independent over a broad range of incident angles. Accordingly, nanocrystal superlattices behave as isotropic bulk metamaterials. Their tunable permittivities, permeabilities, and emergent magnetism may enable new, bottom-up metamaterials and negative index materials at visible frequencies.

© 2012 OSA

## 1. Introduction

1. R. M. Erb, H. S. Son, B. Samanta, V. M. Rotello, and B. B. Yellen, “Magnetic assembly of colloidal superstructures with multipole symmetry,” Nature (London) **457**, 999–1002 (2009). [CrossRef]

11. W. H. Evers, H. Friedrich, L. Filion, M. Dijkstra, and D. Vanmaekelbergh, “Observation of a ternary nanocrystal superlattice and its structural characterization by electron tomography,” Angew. Chem. Int. Ed. **48**, 9655–9657 (2009). [CrossRef]

8. E. V. Shevchenko, D. V. Talapin, N. A. Kotov, S. O’Brien, and C. B. Murray, “Structural diversity in binary nanoparticle superlattices,” Nature (London) **439**, 55–59 (2006). [CrossRef]

11. W. H. Evers, H. Friedrich, L. Filion, M. Dijkstra, and D. Vanmaekelbergh, “Observation of a ternary nanocrystal superlattice and its structural characterization by electron tomography,” Angew. Chem. Int. Ed. **48**, 9655–9657 (2009). [CrossRef]

8. E. V. Shevchenko, D. V. Talapin, N. A. Kotov, S. O’Brien, and C. B. Murray, “Structural diversity in binary nanoparticle superlattices,” Nature (London) **439**, 55–59 (2006). [CrossRef]

10. A. Dong, J. Chen, P. M. Vora, J. M. Kikkawa, and C. B. Murray, “Binary nanocrystal superlattice membranes self-assembled at the liquid-air interface,” Nature (London) **446**, 474–477 (2010). [CrossRef]

_{2}Te nanoparticles in AB

_{2}lattices have been shown to exhibit strongly enhanced conductance, with over a 100-fold increase in conductivity compared to unary PbTe or Ag

_{2}Te superlattices [9

9. J. J. Urban, D. V. Talapin, E. V. Shevchenko, C. R. Kagan, and C. B. Murray, “Synergism in binary nanocrystal superlattices leads to enhanced p-type conductivity in self-assembled PbTe/Ag_{2}Te thin films,” Nat. Mater. **6**, 115–121 (2007). [CrossRef] [PubMed]

12. Y. G. Zhao, A. Wu, H. L. Lu, S. Chang, W. K. Lu, S. T. Ho, M. E. van der Boom, and T. J. Marks, “Traveling wave electro-optic phase modulators based on intrinsically polar self-assembled chromophoric superlattices,” Appl. Phys. Lett. **79**, 587–589 (2001). [CrossRef]

13. D. G. Suna, Z. Liub, J. Mab, and S. T. Hob, “Design and fabrication of electro-optic waveguides with self-assembled superlattice films,” Opt. Laser Technol. **39**, 285–289 (2007). [CrossRef]

14. A. Alu and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express **17**, 5723–5730 (2009). [CrossRef] [PubMed]

15. A. Alu and N. Engheta, “Plasmonic and metamaterial cloaking: physical mechanisms and potentials,” J. Opt. A: Pure Appl. Opt. **10**, 093002 (2008). [CrossRef]

17. B. Auguie and W. L. Barnes, “Collective resonances in Gold nanoparticle arrays,” Phys. Rev. Lett. **101**, 143902 (2008). [CrossRef] [PubMed]

27. I. Romero and F. J. Garcia de Abajo, “Anisotropy and particle-size effects in nanostructured plasmonic metamaterials,” Opt. Express **17**, 22012–22022 (2009). [CrossRef] [PubMed]

28. C. L. Holloway, E. F. Kuster, J. B. Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. **51**, 2596–2603 (2003). [CrossRef]

29. J. Liu and N. Bowler, “Analysis of double-negative (DNG) bandwidth for a metamaterial composed of magnetodielectric spherical particleseEmbedded in a matrix,” IEEE Antennas Wirel. Propag. Lett. **10**, 399–402 (2011). [CrossRef]

18. A. L. Fructos, S. Campione, F. Capolino, and F. Mesa, “Characterization of complex plasmonic modes in two-dimensional periodic arrays of metal nanospheres,” J. Opt. Soc. Am. B **28**, 1446–1458 (2011). [CrossRef]

21. A. Alu, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express **14**, 1557–1567 (2006). [CrossRef] [PubMed]

27. I. Romero and F. J. Garcia de Abajo, “Anisotropy and particle-size effects in nanostructured plasmonic metamaterials,” Opt. Express **17**, 22012–22022 (2009). [CrossRef] [PubMed]

30. N. Liu, S. Mukherjee, K. Bao, L. Brown, J. Dorfmuller, P. Nordlander, and N. J. Halas, “Magnetic plasmon formation and propagation in artificial aromatic molecules,” Nano Lett. **12**, 364–369 (2012). [CrossRef]

33. J. B. Lassiter, H. Sobhani, J. A. Fan, J. Kundu, F. Capasso, P. Nordlander, and N. J. Halas, “Fano resonances in plasmonic nanoclusters: geometrical and chemical tunability,” Nano Lett. **10**, 3184–3189 (2010). [CrossRef] [PubMed]

30. N. Liu, S. Mukherjee, K. Bao, L. Brown, J. Dorfmuller, P. Nordlander, and N. J. Halas, “Magnetic plasmon formation and propagation in artificial aromatic molecules,” Nano Lett. **12**, 364–369 (2012). [CrossRef]

34. M. G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A **73**, 1105–1112 (1983). [CrossRef]

35. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B **65**, 195104 (2002). [CrossRef]

37. C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B **77**, 195328 (2008). [CrossRef]

## 2. Theoretical framework

38. Q. Cao, P. Lalanne, and J. P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A **19**, 335–338 (2002). [CrossRef]

41. M. Shahabadi, S. Atakaramians, and N. Hojjat, “Transmission line formulation for the full-wave analysis of two-dimensional dielectric photonic crystals,” IEE Proc.-Sci. Meas. Technol. **151**, 327–334 (2004). [CrossRef]

### 2.1. Two-dimensional superlattices

*xy*-plane. The materials are assumed to have a linear electromagnetic response. According to Bloch’s theorem [42

42. R. E. Collin, *Foundations for Microwave Engineering* (IEEE, 2001). [CrossRef]

**. Assuming a complex exponential time dependence of the form exp(−**

*k**iωt*), the pseudo-periodic electric and magnetic fields can be written as: where

**and**

*E*(*r*)**are vector fields with the same spatial periodicity as the lattice. Using a Fourier expansion,**

*H*(*r*)**and**

*E*(*r*)**can be represented as:**

*H*(*r*)*L*,

_{x}*L*) and (

_{y}*K*

_{x0},

*K*

_{y0}) are the lattice parameters and the Bloch wavevectors in the

*x*and

*y*directions, respectively. Due to the periodicity of the lattice, the dielectric function is also periodic in the

*xy*-plane, and can be expanded as a double Fourier series:

*ε*(

_{mn}*z*) is the

*mn*Fourier coefficient of the dielectric constant at any position

^{th}*z*orthogonal to the xy-plane. For numerical implementation, the series must be truncated. Including up to the

*N*harmonic in each direction, Maxwell’s equations can be written in terms of the unknown expansion coefficients of the electric and magnetic fields:

^{th}*K*] and [

_{x}*K*] are wavevector-type matrices of the form: and [

_{y}*N*] is a permittivity-type matrix: In this representation,

*e*denotes an

_{k}*N*x

*N*Toeplitz matrix of the permittivity Fourier expansion coefficients, starting from

*ε*

_{k0}up to

*ε*.

_{kN}*μ*

_{0}[L] and

*ε*

_{0}[C] have dimensions of inductance and capacitance per unit length, respectively, and are given by:

*E*

_{x,y}) and (

*H*

_{x,y}) in Eq. (10) can be determined by expanding the corresponding components of the incident fields in a Fourier series. Consequently, the unknown coefficients for any two-dimensional periodic array can be determined via field matching at the boundaries.

### 2.2. Extension to three-dimensional lattices of arbitrarily-shaped constituents

*z*direction. For example, single layers of spherical and polyhedral nanoparticles reported in this work are modelled as stacks of 25 and 50 layers of 2D arrays of circular disks and square patches, respectively. Cascaded S-matrices of these layers can then be used to determine the optical properties of the full three-dimensional lattice.

*T*of the lattice while

*R*. These quantities can be directly used to extract the superlattice optical parameters, as described below.

### 2.3. Permittivity, permeability, and refractive index retrieval

*n*″, can be uniquely determined from the first equation. The real component of the refractive index,

*n*′, has infinite solutions separated by

*λ*/

*d*in the complex plane. The appropriate choice of

*n*′, corresponding to a physical, causal material, can be determined by imposing Kramers-Kronig criteria on

*n*″. Among two possible answers for the normalized impedance, the solution

*Z*′ > 0 must be chosen to ensure a passive medium. Subsequently, the effective permittivity and permeability can be determined directly: For off-normal illumination at an angle

*θ*

_{1},

*n*must be replaced with

*n*cos

*θ*

_{2}while

*Z*must be substituted with either

*Z*cos

*θ*

_{2}or

*Z*/cos

*θ*

_{2}for in-plane and out-of-plane polarizations, respectively. Here,

*θ*

_{2}is the refracted angle in the second medium and satisfies Snell’s law:

*n*

_{1}sin

*θ*

_{1}=

*n*

_{2}sin

*θ*

_{2}.

## 3. Results

### 3.1. Unary nanoparticle superlattices

*R*of 5 nm, 10 nm, and 30 nm. Particles are assumed to be separated by 2 nm, approximating the interparticle separation of self-assembled superlattices due to ligands. The refractive index between particles is set to 1.5, typical of organic solvents and ligands; the regions above and below the lattice are taken to have an index

*n*=1 (air). The empirical Johnson and Christy dataset is used to describe permittivity of the gold nanoparticles [43

43. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

*P*) of Au nanoparticles superlattices composed of 1, 2, 3, and 4 layers. The transmitted power

_{T}*P*is determined by integrating the Poynting vector of the transmitted fields across the unit cell area. The incident light is assumed to be polarized in the

_{T}*x*-direction.

*z*-layers of the lattice, which become out-of-phase as the wave propagates through the lattice. As particle size increases to R=10 nm, the dominant peak broadens, increases, and red-shifts compared to the R=5 nm lattice. The red-shift, from

*λ*= 528 nm for R = 5 nm particles to

*λ*= 542 nm for this R=10 nm particle lattice, arises from the lower-energy resonance of individual particles. Further, the broadening arises from increased ohmic and radiation losses. As the particle size increases, the superlattice spectra begins to deviate significantly from a single resonant peak. For example, for superlattices composed of R=30 nm particles, the extinction cross-section increases and resonant peaks are substantially broadened. As has been shown in a series of articles by McPhedran [44

44. R. C. McPhedran and D. R. McKenzie, “Exact modelling of cubic lattice permittivity and conductivity,” Nature (London) **265**, 128–129 (1977). [CrossRef]

46. R. C. McPhedran, D. R. McKenzie, and G. H. Derrick, “The conductivity of lattices of spheres II. the body centred and face centred cubic lattice,” Proc. R. Soc. London Ser. A **362**, 211–232 (1978). [CrossRef]

27. I. Romero and F. J. Garcia de Abajo, “Anisotropy and particle-size effects in nanostructured plasmonic metamaterials,” Opt. Express **17**, 22012–22022 (2009). [CrossRef] [PubMed]

18. A. L. Fructos, S. Campione, F. Capolino, and F. Mesa, “Characterization of complex plasmonic modes in two-dimensional periodic arrays of metal nanospheres,” J. Opt. Soc. Am. B **28**, 1446–1458 (2011). [CrossRef]

**17**, 22012–22022 (2009). [CrossRef] [PubMed]

*μ*can be tuned from 0.7 to 1.1 at visible frequencies. Increasing the particle size to R= 50 nm particles increases the span of

*μ*from 1.3 to 0.4 for wavelengths between 570 nm and 730 nm (data not shown).

*dμ*/

*dλ*opposite to

*dε*/

*dλ*. Similar to the response of an infinite array of metallic wires in the microwave range [47

47. T. Koschny, P. Marko, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E **68**, 065602 (2003). [CrossRef]

48. R. Merlin, “Metamaterials and the Landau Lifshitz permeability argument: Large permittivity begets high-frequency magnetism,” Proc. Nat. Acad. Sci. USA **106**, 1693–1698 (2009). [CrossRef] [PubMed]

*λ*=350 nm (a,b) and

*λ*=650 nm (d,e) are considered. For both wavelengths, the electric field distribution (panels a and d), clearly exhibits strong localization in the inter-particle gaps. However, the electric field is enhanced at

*λ*=650 nm compared to

*λ*=350 nm, due to a strong electric resonant mode in array. This electric resonance results in a strong dispersion of the lattice permittivity, which in turn produces a magnetic antiresonance. As seen in panels b and e, the magnetic field within the lattice is enhanced at

*λ*=650 nm, where the magnetic permeability of the lattice is 0.7, compared to

*λ*=350 nm, where

*μ*=1. Magnetic behavior is also evidenced in the displacement current between layers, defined as −

*iωD*, shown as arrows in panels (c) and (f). As can be observed, enhanced circulation of the displacement current density is present for

*λ*=650 nm compared to

*λ*=350nm, pointing to the enhanced magnetic response of Au superlattices at this wavelength.

*x*or

*y*, the interparticle coupling can be tuned. Figure 5 illustrates

*n*,

*ε*, and

*μ*for three different lattice geometries: a square lattice (L

*=62 nm, L*

_{x}*=62 nm) and two rectangular lattices, (L*

_{y}*=162 nm, L*

_{x}*=62 nm) and (L*

_{y}*=62 nm, L*

_{x}*=162 nm). In all cases, the incident illumination is assumed to be normal to the lattice and the electric field is polarized along the*

_{y}*x*-direction.

*increases, the coupling between Au particles decreases. This reduced coupling is evidenced by the reduced magnitude of the resonance in permittivity and the near-unity permeability calculated for the lattice of (L*

_{x}*=162 nm, L*

_{x}*=62 nm), denoted in red in Fig. 5. The particles, however, are less sensitive to lattice deviations perpendicular to the polarization direction. For example, the (L*

_{y}*=62 nm, L*

_{x}*=162 nm) lattice maintains the strong coupling of the particles in the*

_{y}*x*-direction while decreasing the coupling in the

*y*-direction. As the particles are mainly coupled in the

*x*-direction, this lattice still exhibits a strong resonant permittivity hence a non-unity magnetic permeability.

*θ*. The upper panels present the retrieved parameters for an incident electric field polarized in-plane while the lower panels show results for an out-of-plane polarization. As Fig.6 reveals,

*n*,

*ε*, and

*μ*are nearly independent of incident angle for each polarization implying that the array has an angle independent behavior. Correspondingly, the results for each polarization exhibit similar trends, indicating that the lattice is also almost polarization independent. Such results are consistent with the symmetric and isotropic geometry of the lattice unit cell. Accordingly, unary Au superlattices are indeed nearly-isotropic bulk magnetic metamaterials.

### 3.2. Polyhedral and binary nanoparticle superlattices

2. J. Henzie, M. Grunwald, A. W. Cooper, P. L. Geissler, and P. Yang, “Self-assembly of uniform polyhedral silver nanocrystals into densest packings and exotic superlattices,” Nat. Mater. **11**, 131–137 (2011). [CrossRef] [PubMed]

10. A. Dong, J. Chen, P. M. Vora, J. M. Kikkawa, and C. B. Murray, “Binary nanocrystal superlattice membranes self-assembled at the liquid-air interface,” Nature (London) **446**, 474–477 (2010). [CrossRef]

2. J. Henzie, M. Grunwald, A. W. Cooper, P. L. Geissler, and P. Yang, “Self-assembly of uniform polyhedral silver nanocrystals into densest packings and exotic superlattices,” Nat. Mater. **11**, 131–137 (2011). [CrossRef] [PubMed]

2. J. Henzie, M. Grunwald, A. W. Cooper, P. L. Geissler, and P. Yang, “Self-assembly of uniform polyhedral silver nanocrystals into densest packings and exotic superlattices,” Nat. Mater. **11**, 131–137 (2011). [CrossRef] [PubMed]

_{2}-type packing. This lattice, also known as a Kagome lattice, has been the subject of many studies spanning solid state physics and photonics [49

49. J. L. Atweeod, “Kagome’ lattice: a molecular toolkit for magnetism,” Nat. Matter. **1**, 91–92 (2002). [CrossRef]

51. Q. Chen, S. C. Bae, and S. Granick, “Directed self-assembly of a colloidal Kagome lattice,” Nature (London) **469**, 381–384 (2010). [CrossRef]

52. A. V. Giannopoulos, C. M. Long, and K. D. Choquette, “Photonic crystal heterostructure cavity lasers using kagome lattices,” Electron. Lett. **44**, 38–39 (2008). [CrossRef]

53. X. Li, J. Zhou, Q. Wang, X. Chen, Y. Kawazoe, and P. Jena, “Magnetism of two-dimensional triangular nanoflake-based kagome lattices,” New J. Phys. **14**, 033043 (2012). [CrossRef]

_{2}-type lattices composed of gold and silica nanoparticles. Each unit cell has one R=30 nm silica particle surrounded by six R=30 nm gold nanoparticles. As before, particles are assumed to be embedded in a medium with index n=1.5 and are separated by 2 nm.

*x*-polarized planewave. Although not shown,

*y*-polarized illumination excites the same spectral response. For visible wavelengths, the permittivity

*ε*varies between −7 and 20 while the permeability

*μ*is tunable between 0.5 and 1.2. An AB

_{2}array of R=50 nm gold and silica nanoparticles exhibits even stronger tunability, with

*ε*ranging from −6.5 to +30 and

*μ*between 0.2 and 1.7 at visible frequencies (data not shown). Upon illumination, two distinct types of plamonic modes are excited: ‘surface’ modes propagating in the plane of the lattice, and ‘gap’ modes propagating normal to the lattice plane between the particles. While surface modes carry power in the transverse lattice planes, gap modes carry power through the lattice in

*z*via the interparticle spacings. Figures 7(d) and 7(e) compare the electric field profiles of the lattice at wavelengths of

*λ*=300 nm (away from the resonance) and

*λ*= 613 nm (on resonance). As seen, the electric fields are mostly confined between the metallic particles. However, the

*xy*-field cross section on resonance is significantly enhanced, due to strong coupling to the electric resonant mode. Further, the electric field distribution in

*xz*and

*yz*-cross sections at each wavelength reveals substantial field transmission at

*λ*=300 nm (corresponding to excitation of ‘gap’ modes) and nearly-suppressed transmission at

*λ*= 613 nm (corresponding to excitation of ‘surface’ modes). Engineering the excitability of surface versus gap modes should provide further optical tunability, selectively exploiting the waveguided or resonant character of nanoparticle superlattices.

## 4. Outlook and conclusions

## Acknowledgments

## References and links

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43. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

44. | R. C. McPhedran and D. R. McKenzie, “Exact modelling of cubic lattice permittivity and conductivity,” Nature (London) |

45. | R. C. McPhedran and D. R. McKenzie, “The conductivity of lattices of spheres I. the simple cubic lattice,” Proc. R. Soc. London Ser. A |

46. | R. C. McPhedran, D. R. McKenzie, and G. H. Derrick, “The conductivity of lattices of spheres II. the body centred and face centred cubic lattice,” Proc. R. Soc. London Ser. A |

47. | T. Koschny, P. Marko, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E |

48. | R. Merlin, “Metamaterials and the Landau Lifshitz permeability argument: Large permittivity begets high-frequency magnetism,” Proc. Nat. Acad. Sci. USA |

49. | J. L. Atweeod, “Kagome’ lattice: a molecular toolkit for magnetism,” Nat. Matter. |

50. | U. Schlickum, R. Decker, F. Klappenberger, G. Zoppellaro, S. Klyatskaya, W. Auwarter, S. Neppl, K. Kern, H. Brune, M. Ruben, and J. V. Barth, “Chiral Kagome’ lattice from simple ditopic molecular bricks,” J. Am. Chem. Soc. |

51. | Q. Chen, S. C. Bae, and S. Granick, “Directed self-assembly of a colloidal Kagome lattice,” Nature (London) |

52. | A. V. Giannopoulos, C. M. Long, and K. D. Choquette, “Photonic crystal heterostructure cavity lasers using kagome lattices,” Electron. Lett. |

53. | X. Li, J. Zhou, Q. Wang, X. Chen, Y. Kawazoe, and P. Jena, “Magnetism of two-dimensional triangular nanoflake-based kagome lattices,” New J. Phys. |

**OCIS Codes**

(160.4760) Materials : Optical properties

(240.6680) Optics at surfaces : Surface plasmons

(160.3918) Materials : Metamaterials

(160.4236) Materials : Nanomaterials

(250.5403) Optoelectronics : Plasmonics

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Metamaterials

**History**

Original Manuscript: April 16, 2012

Revised Manuscript: June 16, 2012

Manuscript Accepted: June 19, 2012

Published: June 27, 2012

**Citation**

Hadiseh Alaeian and Jennifer A. Dionne, "Plasmon nanoparticle superlattices as optical-frequency magnetic metamaterials," Opt. Express **20**, 15781-15796 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15781

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