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 AB2 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

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2012 (2)

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]

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]

2011 (5)

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]

S. Campione, S. Steshenko, and F. Capolino, “Complex bound and leaky modes in chains of plasmonic nanospheres,” Opt. Express 19, 18345–18363 (2011).
[CrossRef] [PubMed]

S. N. Sheikholeslami, A. Garcia-Etxarri, and J. A. Dionne, “Controlling the interplay of electric and magnetic modes via Fano-like plasmon resonances,” Nano Lett. 11, 3927–3934 (2011).
[CrossRef] [PubMed]

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]

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]

2010 (4)

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]

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Monaharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticles clusters,” Science 328, 1135–1137 (2010).
[CrossRef] [PubMed]

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]

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

2009 (5)

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

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]

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]

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]

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]

2008 (5)

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

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

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

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]

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. 130, 11778–11782 (2008).
[CrossRef] [PubMed]

2007 (5)

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]

D. V. Tapalin, E. V. Shevchenko, C. B. Murray, A. V. Titov, and Petr Kral, “Dipole-dipole interactions in nanoparticle superlattices,” Nano Lett. 7, 1213–1219 (2007).
[CrossRef]

C. Rockstuhl, F. Lederer, C. Etrich, T. Pertsch, and T. Scharf, “Design of an artificial three-dimensional composite metamaterial with magnetic resonances in the visible range of the electromagnetic spectrum,” Phys. Rev. Lett. 99, 017401 (2007).
[CrossRef] [PubMed]

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/Ag2Te thin films,” Nat. Mater. 6, 115–121 (2007).
[CrossRef] [PubMed]

A. Tao, P. Sinsermsuksakul, and P. Yang, “Tunable plasmonic lattices of silver nanocrystals,” Nat. Nanotechnol. 2, 435–440 (2007).
[CrossRef]

2006 (4)

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]

A. Alu and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission line,” Phys. Rev. B 74, 205436 (2006).
[CrossRef]

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]

A. Alu and N. Engheta, “Optical nanotransmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes,” J. Opt. Soc. Am. A 23, 571–583 (2006).
[CrossRef]

2004 (2)

A. L. Rogach, “Binary superlattices of nanoparticles: self-assembly leads to metamaterials,” Adv. Funct. Mater. 43, 148–149 (2004).

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]

2003 (2)

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]

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]

2002 (3)

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

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]

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]

2001 (2)

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]

N. A. El-Masry, M. K. Behbehani, S. F. LeBoeuf, M. E. Aumer, J. C. Roberts, and S. M. Bedair, “Self-assembled AlInGaN quaternary superlattice structures,” Appl. Phys. Lett. 79, 1616–1618 (2001).
[CrossRef]

2000 (1)

1999 (1)

A. Taleb, V. Russier, A. Courty, and M. P. Pileni, “Collective optical properties of silver nanoparticles organized in two-dimensional superlattices,” Phys. Rev. B 59, 13350–13358 (1999).
[CrossRef]

1998 (1)

Z. L. Wang, “Structural analysis of self-assembling nanocrystal superlattices,” Adv. Mater. 10, 13–30 (1998).
[CrossRef]

1996 (1)

R. P. Andres, J. D. Bielefeld, J. I. Henderson, D. B. Janes, V. R. Kolagunta, C. P. Kubiak, W. J. Mahoney, and R. G. Osifchin, “Self-assembly of a two-Dimensional superlattice of molecularly linked metal clusters,” Science 273, 1690–1693 (1996).
[CrossRef]

1995 (1)

1983 (1)

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]

1978 (2)

R. C. McPhedran and D. R. McKenzie, “The conductivity of lattices of spheres I. the simple cubic lattice,” Proc. R. Soc. London Ser. A 359, 45–63 (1978).
[CrossRef]

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]

1977 (1)

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

1972 (1)

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

1970 (1)

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. 19, 377–379 (1970).
[CrossRef]

Alu, A.

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

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

A. Alu and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission line,” Phys. Rev. B 74, 205436 (2006).
[CrossRef]

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]

A. Alu and N. Engheta, “Optical nanotransmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes,” J. Opt. Soc. Am. A 23, 571–583 (2006).
[CrossRef]

Andres, R. P.

R. P. Andres, J. D. Bielefeld, J. I. Henderson, D. B. Janes, V. R. Kolagunta, C. P. Kubiak, W. J. Mahoney, and R. G. Osifchin, “Self-assembly of a two-Dimensional superlattice of molecularly linked metal clusters,” Science 273, 1690–1693 (1996).
[CrossRef]

Atakaramians, S.

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]

Atweeod, J. L.

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

Auguie, B.

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

Aumer, M. E.

N. A. El-Masry, M. K. Behbehani, S. F. LeBoeuf, M. E. Aumer, J. C. Roberts, and S. M. Bedair, “Self-assembled AlInGaN quaternary superlattice structures,” Appl. Phys. Lett. 79, 1616–1618 (2001).
[CrossRef]

Auwarter, W.

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. 130, 11778–11782 (2008).
[CrossRef] [PubMed]

Bae, S. C.

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

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D. V. Tapalin, E. V. Shevchenko, C. B. Murray, A. V. Titov, and Petr Kral, “Dipole-dipole interactions in nanoparticle superlattices,” Nano Lett. 7, 1213–1219 (2007).
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Figures (8)

Fig. 1
Fig. 1

Schematic of a three-dimensional unary superlattice composed of nanoparticles. Lx and Ly are the periods of the lattice in the x and y directions, respectively. The illumination is assumed to be a planewave with incident angle θ and wavevector k. The scattering coefficients of a single unit layer are defined by f 1 +, f 1 , f 2 +, and f 2 .

Fig. 2
Fig. 2

Normalized extinction cross sections for N= 1, 2, 3 and 4 layers of gold nanoparticles of varying radii R in a superlattice. Particle radii are (a) 5 nm, (b) 10 nm and (c) 30 nm. Illumination at normal incidence is assumed.

Fig. 3
Fig. 3

Effective refractive index, normalized impedance, relative permittivity and permeability of gold nanoparticle superlattices with radii of (a)5 nm, (b)10 nm and (c)30 nm. The red dots indicate the effective parameters for a semi-infinite array while the lines show the parameters for one layer of the lattice. Note that the scales are different for each case and are also expanded around the permeability resonance in panel (b).

Fig. 4
Fig. 4

Electric (a, d) and magnetic (b, e) field profiles and displacement current quiver plots (c, f) of Au nanoparticle superlattices composed of R=30 nm particles. Wavelengths of λ=350 nm (a–c) and λ= 650 nm (d–f) are included.

Fig. 5
Fig. 5

Effective refractive index and relative permittivity and permeability of 30-nm-radii gold nanoparticles arranged in rectangular lattices with periodicities L in the x and y directions. The incident electric field is polarized along the x-direction. The effective parameters of the square array are also included for comparison.

Fig. 6
Fig. 6

Effective refractive index and relative permittivity and permeability of R=30 nm gold nanoparticles arranged in a square lattice with Lx=Ly=62 nm as a function of incident angle, θ, and wavelength. The upper panels show the parameters for out-of-plane polarization and the lower panels show the results for in-plane polarization.

Fig. 7
Fig. 7

Effective parameters of 50 nm octahedral gold nanoparticles arranged in a cubic lattice with Lx=Ly=52 nm, illuminated at normal incidence.

Fig. 8
Fig. 8

(a) The optical properties of an AB2-type binary superlattice composed of R=30nm gold and silica nanoparticles. (b,c) Ex field profiles of the lattice in xy, xz and yz cross sections at λ= 300 nm (b) and 613 nm (c).

Equations (18)

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E ( r + L ) = E ( r ) exp ( i k . L ) H ( r + L ) = H ( r ) exp ( i k . L )
E ( r ) = m , n = E m n ( z ) exp ( i [ ( 2 m π L x + K x 0 ) x + ( 2 n π L y + K y 0 ) y ] ) H ( r ) = m , n = H m n ( z ) exp ( i [ ( 2 m π L x + K x 0 ) x + ( 2 n π L y + K y 0 ) y ] )
ε ( r ) = m , n = ε m n ( z ) exp ( i ( 2 m π L x x + 2 n π L y y ) )
d d z ( E x ) = i ω μ 0 [ ( [ K x ] [ N ] 2 [ K x ] I ) ( H y ) + [ K x ] [ N ] 2 [ K y ] ( H x ) ] d d z ( E y ) = i ω μ 0 [ [ K y ] [ N ] 2 [ K x ] ( H y ) + ( [ K y ] [ N ] 2 [ K y ] I ) ( H x ) ] d d z ( H y ) = i ω ε 0 [ ( [ K y ] 2 [ N ] 2 ) ( E x ) [ K y ] [ K x ] ( E y ) ] d d z ( H x ) = i ω ε 0 [ ( [ K x ] [ K y ] ) ( E x ) + ( [ K x ] 2 [ N ] 2 ) ( E y ) ]
[ K x ] = [ ( 2 m π L x + K x 0 ) 0 0 0 K x 0 0 0 0 ( 2 m π L x + K x 0 ) ] [ K y ] = [ ( 2 n π L y + K y 0 ) 0 0 0 K y 0 0 0 0 ( 2 n π L y + K y 0 ) ]
[ N ] 2 = [ e 0 e 2 N e 2 N e 0 ] .
( I ) = [ ( H y ) ( H x ) ] ( V ) = [ ( E x ) ( E y ) ]
d d z ( V ) = i ω μ 0 [ L ] ( I ) d d z ( I ) = i ω ε 0 [ C ] ( V )
[ L ] = [ [ K x ] [ N ] 2 [ K x ] [ 1 ] [ K y ] [ N ] 2 [ K x ] [ K x ] [ N ] 2 [ K y ] [ K y ] [ N ] 2 [ K y ] [ 1 ] ] [ C ] = [ [ K y ] 2 [ N ] 2 [ K x ] [ K y ] [ K y ] [ K x ] [ K x ] 2 [ N ] 2 ]
1 η 0 [ ( E x ) ( E y ) ] = [ [ k z m n 2 + β n 2 k z m n ] [ α m β n k z m n ] [ α m β n k z m n ] [ k z m n 2 + α m 2 k z m n ] ] [ ( H y ) ( H x ) ]
α m = k x 0 + m λ L x β n = k y 0 + n λ L y k z m n = [ ε α m 2 β n 2 ] 1 / 2
[ f 1 + f 2 ] = [ S 11 S 12 S 21 S 22 ] [ f 1 + f 2 + ]
[ [ 1 ] S 11 0 S 21 ] [ f 1 + f 1 ] = e i γ d [ S 12 0 S 22 [ 1 ] ] [ f 1 + f 1 ]
[ f n + f n ] = A [ B n + B n ]
[ B n + B n ] = [ e ind γ + 0 0 e ind γ ] [ B 1 + B 1 ]
[ B 1 B 1 + ] = A 1 [ f incident f reflected ] .
cos ( n k d ) = 1 2 T ( 1 R 2 + T 2 ) Z = ( ( 1 + R ) 2 T 2 ( 1 R ) 2 T 2 ) 1 / 2
μ = n Z ε = n / Z

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