1. Introduction

Advances in self–assembly have enabled development of nanocrystal superlattices - metallic, semiconducting, or dielectric nanoparticles arranged in periodic arrays [1–11]. Such lattices are typically composed of one, two, or three types of nanocrystals, forming unary, binary, and ternary superlattices, respectively. By tuning the interplay of electrostatic interactions with entropic, van der Waals, steric, and dipolar forces, nanocrystal superlattices can exhibit a structural diversity transcending that of natural lattices. For example, more than fifteen distinct binary nanocrystal superlattice arrangements have been realized, many with no isostructural intermetallic counterparts [8]. More recently, self-assembly of ternary nanocrystal superlattice bilayers has been shown to yield arrangements unique from any three-dimensional lattice [11]. To date, nanocrystal superlattice assemblies have been achieved over millimetre-scales, both substrate-supported and free-standing [8, 10].

The structural diversity of nanocrystal superlattices gives rise to intriguing electronic and magnetic properties distinct from their constituent nanoparticles. For example, binary superlattices of PbTe and Ag₂Te nanoparticles in AB₂ lattices have been shown to exhibit strongly enhanced conductance, with over a 100-fold increase in conductivity compared to unary PbTe or Ag₂Te superlattices [9]. Fast and efficient optical modulators have also been demonstrated based on enhanced electro-optical interactions and nonlinearities in self-assembled nanocrystal superlattices [12, 13]. Since the emergent properties of nanocrystal superlattices are highly dependent on the nanoparticle size, shape, composition, and the lattice stoichiometry, they may provide the framework for new, bottom-up metamaterials at visible frequencies [7]. In particular, electromagnetic coupling between superlattice constituents could enable tunable electric permittivities, magnetic permeabilities, and refractive indices spanning positive, negative, and near-zero values. Combined with the unique optical properties of quantum-confined nanostructures, such tunability could lead to profound applications for superlensing, cloaking, or optical communications [14, 15].

In recent years, periodic nanoparticle-based structures have received increasing attention [17–27]. For example, at microwave frequencies, cubic lattices of large-permittivity polaritonic spheres have been shown to exhibit a negative permittivity and permeability [28, 29]. At optical frequencies, metallic nanoparticle lattices have been shown to support propagating modes characterized by tunable positive refractive indices and permittivities [18,21,23,24,27]. Close-packed trimer, quadramer, and heptamer-based plasmonic assemblies have also been shown to support optical-frequency magnetic resonances and Fano resonances of electric and magentic modes [30–33]. In this paper, we theoretically determine the effective permittivity and permeability of close-packed nanoparticle superlattices. Attention is given to unary Au and binary Au/silica nanoparticle superlattices composed of varying nanoparticle size, shape, and lattice geometries. We base our superlattice structures on lattices that have already been experimentally demonstrated, or that promise to exhibit a bulk permeability at optical frequencies based on the unit-cell geometry [30]. To determine their optical properties, we use a three-dimensional, analytic algorithm based on rigorous coupled wave analysis (RCWA) [34]. We modify and generalize the method so that it is unconditionally stable for two- and three-dimensional periodic arrays with constituents of arbitrary shape and material.

The method generates the full electromagnetic fields of the lattice as well as its scattering parameters, which in turn can be used to extract the superlattice effective refractive index, normalized impedance, permittivity, and permeability [35–37]. Our results indicate that the optical properties of superlattices can be broadly tuned at visible frequencies via variation of nanoparticle size, separation, shape, and lattice structure. Notably, superlattices of non-magnetic particles exhibit emergent paramagnetism, which to the best of our knowledge is the first report of optical magnetism in plasmonic nanoparticle superlattices. Furthermore, the optical properties of nanoparticle superlattices are largely angle- and polarization-independent, suggesting their potential as isotropic, three-dimensional magnetic or negative index metamaterials.

2. Theoretical framework

Rigorous coupled wave analysis is a well-known method for solving Maxwell’s equations in periodic media. To date, it has been successfully employed in the analysis of surface relief gratings, corrugated surfaces, and photonic crystals [38–41]. Compared with other numerical techniques such as finite difference time domain, finite difference frequency domain and finite element methods, RCWA provides an accurate analytical, full-vectorial solution to Maxwell’s equations in periodic structures. It is therefore well-suited for analysis of nanocrystal superlattices. Here, we briefly present our extended and generalized RCWA formulation for two and three dimensional periodic nanoparticle arrays.

2.1. Two-dimensional superlattices

Consider an infinite two dimensional array of arbitrarily shaped objects periodically arranged in the xy-plane. The materials are assumed to have a linear electromagnetic response. According to Bloch’s theorem [42], the eigensolutions of the array are pseudo-periodic given by the product of a periodic function and a plane wave with wavevector k. Assuming a complex exponential time dependence of the form exp(−iωt), the pseudo-periodic electric and magnetic fields can be written as:

Here, ε_mn(z) is the mn^th Fourier coefficient of the dielectric constant at any position z orthogonal to the xy-plane. For numerical implementation, the series must be truncated. Including up to the N^th harmonic in each direction, Maxwell’s equations can be written in terms of the unknown expansion coefficients of the electric and magnetic fields:

To simplify Eq. (4), the components of the four coupled first order differential equation can be grouped into (V) and (I) vectors:

Matrices μ₀[L] and ε₀[C] have dimensions of inductance and capacitance per unit length, respectively, and are given by:

Outside of the superlattice, in the homogeneous media surrounding the array, a Rayleigh expansion can be employed to describe the fields. For a pseudo-periodic source of wavelength λ and normalized parallel wavevector (k_x0, k_y0), the transverse components of the electric and magnetic fields of the mn^th diffraction order are related to each other by:

For any illumination, (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

In general, a three-dimensional periodic structure can be modeled as stacks of two-dimensional arrays along the 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.

Figure 1 illustrates the method. For one layer of nanoparticles in a lattice, f refers to the expansion coefficients of the tangential components of electric or magnetic fields (e.g. [V]). The incident and reflected waves at each interface are related to each other by the scattering matrices S_ij as:

 

Fig. 1 Schematic of a three-dimensional unary superlattice composed of nanoparticles. L_x and L_y 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 $f_2^{-}$.

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If the lattice is composed of n stacked layers in the z-direction, the field expansion coefficients of the n^th interface ( $f_n^{\pm }$) can be determined from the expansion coefficients of the top layer ( $f_1^{\pm }$). In terms of the Bloch modes $B_n^{\pm }$ propagating upwards(+) or downwards(−) at the n^th interface,

Note that $f_n^{-}$ denotes the transmission coefficient T of the lattice while $f_1^{+}$ denotes the reflection coefficient R. These quantities can be directly used to extract the superlattice optical parameters, as described below.

2.3. Permittivity, permeability, and refractive index retrieval

Assuming the lattice is characterized by subwavelength periodicities, only the 00^th diffracted order will be a propagating mode. Accordingly, the nanocrystal superlattice will exhibit only specular reflection and will tend to scatter as a uniform slab in the far-field. The optical properties of superlattices can therefore be determined using the well-developed parameter retrieval procedure [35,36]. For a homogeneous slab with thickness d illuminated by a normal-incidence planewave, the refractive index of n and normalized impedance of Z are related to the reflection and transmission coefficients, $R(=f_n^{+})$ and $T(=f_n^{-})$ via:

The imaginary component of the refractive index, 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:

3. Results

3.1. Unary nanoparticle superlattices

First, we consider arrays of gold spherical nanoparticles, as illustrated in Fig. 1. Three superlattices are explored, composed of nanoparticles with radii 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]. Using the method presented in Section 2, good convergence is obtained with 21 harmonics.

Figure 2 presents the extinction cross section (1−P_T) of Au nanoparticles superlattices composed of 1, 2, 3, and 4 layers. The transmitted power P_T 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 x-direction.

 

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.

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As seen in Fig. 2, for all particle radii, extinction cross sections increase with the number of layers. For superlattices composed of R=5 nm particles, the extinction spectra exhibits a dominant resonance peak around 528 nm. The response is not unlike that of an individual R=5 nm Au particle, which predominantly radiates as an electric dipole. However, the superlattice resonance is ∼36 meV lower in energy than an isolated Au particle (which peaks at 520 nm), due to energetically-favorable dipole-dipole interactions in-plane. As the number of lattice layers increases, the resonance energy of the superlattice slightly increases. This blue-shift can be attributed to the interaction of retarded multipolar modes in different 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 [44–46] the inclusions in a tightly coupled array cannot be described merely by dipoles, even for deeply subwavelength particles.

Figure 3 shows the calculated optical parameters of each unary superlattice, including the refractive index, normalized impedance, effective relative permittivity, and effective relative permeability. Optical properties are calculated for 1 through 4 unit layers, in addition to the semi-infinite, half-space case (i.e, a lattice with an infinite number of layers in the lower half-space). For simplicity, only results for one layer and the semi-infinite case are presented. Notably, the results show striking agreement between the parameters - consistent with recent results of spherical and elliptical plasmonic nanoparticle lattices [27].

 

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).

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As seen in the upper panels, the real component of n can be readily tuned via different particle sizes. Similarly, the imaginary part of n monotonically increases with increasing particle size; this superlattice behavior is not unlike that of individual Au particles, which exhibit increased absorption as their size increases. Note the peaks in the extinction cross section of Fig. 2 are strongly correlated with the peaks of n″.

Through calculations of the lattice index and impedance, shown in the upper panels, the lattice permittivity and permeability can be determined via Eq. (18). Interestingly, despite having constituents with large negative permittivity, Au superlattices composed of R=5 nm spheres have a positive relative permittivity throughout visible frequencies. As particle size increases to R=30 nm, the permittivity can be tuned from −1.5 to +10 at visible frequencies [18, 27]. More interestingly, as particle size increases, the array begins to exhibit emergent magnetic behavior. For lattices of R=5 nm particles, the effective permeability is unity. However, as particle size increases to R = 30 nm, the array exhibits paramagnetic behavior where 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).

Surprisingly, this magnetic response arises even if the lattice consists of only one layer of Au nanoparticles, where any magnetic dipolar moment would not be expected to align with the driving magnetic field. Such a response is actually a magnetic antiresonance, characterized by dμ/dλ opposite to dε/dλ. Similar to the response of an infinite array of metallic wires in the microwave range [47], these visible-frequency magnetic antiresonances arise from large dispersion in the electric permittivity near the resonance frequency [48].

Anti-resonances can be understood by considering the allowed wavevectors of a periodic structure. In particular, the wavevectors of a periodic structure are confined within the Brillouin zone. For simplicity, assume a one-dimensional periodic structure with period a. Then, the magnitude of the largest wavevector in the array is $\frac{\pi }{a}$. Note that the Brillouin zone edge is inversely proportional to the lattice period. So for lattices with very small periods, like natural crystals, it is valid to ignore the boundedness of the wavevector and relate k and ω with a constant refractive index as k = nω over a wide range of frequencies. However, as the frequency increases, or when artificial crystals with larger lattice periods are considered, the boundedness of the wavevector can no longer be ignored. In these cases, the finiteness of the wavevector requires that the assigned refractive index of the material be bounded as well. Since the refractive index and the effective permittivity and permeability are related to each other by n²(ω) = ε(ω)μ(ω), any resonant feature in one parameter must be accompanied by an anti-resonant feature in the other parameter. In nanoparticle superlattices, for example, a resonance in the effective lattice permittivity gives rise to an antiresonance in the permeability, allowing the retrieved index to remain bounded. Note that the relation between the real and imaginary parts of the parameters is determined via Kramers-Kronig consistency. Hence, magnetic antiresonances are often characterized by μ″ < 0. The negative sign of μ″ does not imply that the structure is active; indeed, the passiveness of the structure requires only that n″ and Z′ remain positive.

To further elucidate the emergent electric and magnetic properties of unary Au nanoparticle superlattices, Fig. 4 illustrates a time snapshot of the electric and magnetic fields of a R=30 nm nanoparticle superlattice composed of two layers. Wavelengths of 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.

 

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.

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Since emergent magnetism relies on the coupling of electrically-resonant elements, variations of the lattice geometry should change the emergent permeability. To investigate this dependence, we consider rectangular lattices of R=30 nm particles. By varying the lattice dimensions along 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_x=62 nm, L_y=62 nm) and two rectangular lattices, (L_x=162 nm, L_y=62 nm) and (L_x=62 nm, L_y=162 nm). In all cases, the incident illumination is assumed to be normal to the lattice and the electric field is polarized along the x-direction.

 

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.

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Compared to the square array, denoted with black curves in the figure, rectangular lattices exhibit a smaller real and imaginary refractive index, due to the increased void space between the particles. As L_x 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_y=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_x=62 nm, L_y=162 nm) lattice maintains the strong coupling of the particles in the 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.

To serve such as bulk metamaterials, superlattices should be fairly insensitive to the angle and polarization of the incident light. Figure 6 illustrates the spectral behavior of the effective optical parameters for various illumination angles 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.

 

Fig. 6 Effective refractive index and relative permittivity and permeability of R=30 nm gold nanoparticles arranged in a square lattice with L_x=L_y=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.

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3.2. Polyhedral and binary nanoparticle superlattices

In recent years, considerable attention has been given to the synthesis of more complex superlattice structures - including superlattices of non-spherical particles and binary nanocrystal superlattices [2,10]. In this section, we illustrate how these more complex superlattice structures can enable enhanced optical parameter tunability.

First, we consider the the behavior of lattices of octahedral gold nanoparticles packed in a cubic geometry. The particle dimensions are assumed to be 50 nm on each side and are separated by 2 nm, as shown in Fig. 8. Such octahedral nanoparticle superlattices have recently been synthesized by Henzie and colleagues [2]. To accurately model the tips and edges of the constituent nanoparticles, each unit cell is divided into 50 layers. Figure 7 illustrates the effective parameters of this lattice when illuminated with a normally-incident plane wave. The effective parameters show similar trends as the spherical particle lattice, but with a red shift in the resonance frequency. This shift can be attributed to the larger coupling between the edges of the adjacent octahedral particles compared to the adjacent spheres. This larger coupling also leads to a larger range of values for the effective permittivity and permeability: the effective permittivity varies between −8 and 23 while the effective permeability swings between 0.4 and 1.4 at optical frequencies. Interestingly, these octahedral constituents can also be packed in more diverse lattices, including geometries with a chiral axis [2]. The emergent optical character of these lattices is the subject of on-going and future work, where properties like optical chirality and dichroism are anticipated.

 

Fig. 7 Effective parameters of 50 nm octahedral gold nanoparticles arranged in a cubic lattice with L_x=L_y=52 nm, illuminated at normal incidence.

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Fig. 8 (a) The optical properties of an AB₂-type binary superlattice composed of R=30nm gold and silica nanoparticles. (b,c) E_x field profiles of the lattice in xy, xz and yz cross sections at λ= 300 nm (b) and 613 nm (c).

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In addition to non-spherical particle lattices, considerable attention has been given to the synthesis of compound nanocrystal superlattices composed of more than one type of material. Among them, the simplest binary lattice is characterized by AB₂-type packing. This lattice, also known as a Kagome lattice, has been the subject of many studies spanning solid state physics and photonics [49–51]. For example, the flat band structure of Kagome lattices renders them a promising candidate for lasers, as they provide a platform for coherent lasing in all directions [52]. Further, these lattices are known to develop weak long-range ferromagnetic order from paramagnetic building blocks [53]. Here, we analyze the emergent optical magnetism of binary AB₂-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.

Figure 8(a)–8(c) shows the effective parameters of one unit layer of a gold/silica nanoparticle superlattice, illuminated at normal incidence by an 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 AB₂ 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

We have developed a generalized and stable RCWA method for the analysis of two and three dimensional periodic structures, and applied the method to the analysis of plasmonic nanocrystal lattices. To our knowledge, this is the first report determining the calculated refractive index, electric permittivity and magnetic permeability of superlattices using analytic, full-vectorial solutions to Maxwell’s equations. Our results indicate that the effective permittivity of superlattices can be tuned throughout a broad range of large positive and negative values. Furthermore, this tunability can yield a paramagnetic response of the lattice, despite all constituents being non-magnetic. The emergent electric and magnetic response is highly dependent on the constituent particle size, shape, separation and lattice arrangement, but is largely independent of illumination angle and polarization. It is predicted that combination of magnetically resonant elements with metallic nanoparticles in unary or binary arrays will generate simultaneous resonances in the electric permittivity and magnetic permeability. Such materials could form the basis for new, negative index metamaterials at visible frequencies, enabled by advances in colloidal chemistry and self-assembly.