1. Introduction

Surface plasmon (SP) aperiodic planar structures have been a subject of growing interest in the recent time [1,2]. With current advances in plasmonic biosensing, Surface Enhanced Raman Scattering and photovoltaics [3–5] the need for SP substrates capable of producing dense, high magnitude local field enhancements is ever more pressing. Complex SP substrates in use today are primarily based on periodic / isolated aperiodic morphologies, random structures on roughened surfaces and fractal colloids. Periodic substrates rely on simplistic arrangements of particles and voids of various geometries to produce fully local plasmon resonances with fixed locations of field enhancements [6,7]. Enhancement magnitude in these structures is moderate [1,8] and enhancements spectrum can be controlled by varying particle shape, interparticle spacing and material parameters. Unlike periodic morphologies continues random and fractal structures do not exhibit translational invariance which prevents convergence of running wave eigenfunctions and stipulates field localizations [9,10]. This effect leads to symmetry breaking and nonlocal transport properties that result in formation of random ‘hot spots’, anomalous absorption and wide inhomogeneous enhancement spectrum in visible and low infrared. Study of aggregates of this type has shown existence of delocalized modes with very large intensity$\left(I>{10}^5\right)$ and sub-wavelength field localizations which dramatically enhance various optical processes [11]. However, control of local electromagnetic properties is fundamentally limited by random, irreproducible nature of the structures which requires ensemble-average, mean-field and effective medium theories to describe plasmonic field enhancements.

Recently novel structures based on chaos dynamics and recursive mathematical rules have been introduced [1,12]. While possessing a degree of self-similarity deterministic aperiodic structures (DAS) do not exhibit translational symmetry and hence possess morphological properties of both random and periodic substrates. These structures are defined using a square lattice of discrete particles and voids with layouts based on deterministic set of rules. DAS are reproducible and can be manufactured using standard techniques such as E-beam lithography.

In this paper we introduce extension of Generalized Ohm’s Law (GOL) to the case of deterministic particle arrays. We apply the method to deterministic aperiodic morphologies and compare our local field simulations with published results obtained using discrete dipole approximation. Using GOL we analyze scaling affects, compare various deterministic morphologies and draw conclusions about field localization mechanisms and enhancement spectrum profiles. To gain better understanding of DAS enhancements we study non-local properties of the structures such as inverse participation ratio, eigenstate localization, field intensity statistics and absorption. Our results suggest that DAS can be a promising candidate for efficient, controllable SP substrates for today’s needs.

2. Method description

Interaction of particles in plasmonic DAS is complex electromagnetic problem that involves a fully vectorial solution of Maxwell equations of selfsimilar objects in planar geometry. When the number of DAS particles is small, self-similarity is not pronounced and DAS behaves as a random structure therefore simulations of large particle arrays are necessary. However, numerical intensity of simulation becomes limiting factor for structures with large number (>100) of discrete elements hence we must implement semi-analytical technique for EM analysis.

2.1 Generalized Ohm’s Law

To calculate resonant properties and field distributions in complex structures numerical methods such as T-matrix [13] and discrete dipole approximation (DDA) [14] have generally been utilized. Providing efficient analytical analysis and expanding treatment beyond traditional quasistatic approximation GOL approach has also been used to study optical characteristics in large random morphologies [15–17]. GOL methodology described by Eqs. (1)-(9) follows discussion of Ref. 17. At the base of this method lies assumption that local fields and far fields converge on virtual reference planes at a distance from thin substrate permitting local fields to have curl and thereby accounting for losses in planar structure (Fig. 1 ). In the limit of $D\ll {\lambda }_{inc}$, $D\gg d,l_0$ where D is particle diameter, ${\lambda }_{inc}$is incident wavelength, $l_0$ is a distance to the reference plane and d is film thickness relationship between the fields and currents becomes fully local and divergence free. Under this premise GOL allows to reduce a full set of coupled three-dimensional Maxwell equations to two uncoupled planar continuity equation for electric and magnetic current densities:

 

Fig. 1 The scheme used by Generalized Ohm’s Law approach. Under deterministic GOL extension particles and voids are represented as single metallic and dielectric bonds. Potential lattice sites coincide with points of particle contact in y direction.

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For analytical treatment KH matrix can be viewed as $H=H\text{'}+iH"$ with imaginary part $H\text{'}{\text{'}}_{ij}\ll H{\text{'}}_{ij}$ treated as perturbation [20]. Thus Eq. (6) can be solved for real part of the Hamiltonian as an eigenproblem:

2.2 Particle discretization and matrix definition

In order to analyze collective field enhancements we must adopt discretization method that conserves attributes of the particle contributing to modal coupling. It’s been shown that equivalent circuit representation leads to reduction of system complexity and allows to retain key EM properties [21,22]. Authors of [21] have successfully utilized this method to study simple nontrivial geometries and large periodic structures. In general discrete particles of arbitrary geometry require fine mesh to describe resonant and coupling behavior, however when we limit the treatment to single mode spherical or disk geometries a coarse mesh can adequately capture these effects. Following this premise, we describe a particle as a two-terminal element with all the physical properties included into parameters $u_m$ and $w_m$. Similarly dielectric lattice link is represented as a two-terminal element with parameters $u_d$and $w_d$. Then a particle in dielectric substrate can be viewed as dipolar bond connecting two neighboring sites in dielectric lattice. Using this assumption in the framework of GOL results in conservation of two-dimensional dipole radiation pattern $E(r)\propto \cos (\varphi )/{\left|r\right|}^2$where ϕ is the angle between field polarization and r. Resonant condition $\tilde{\epsilon }{\text{'}}_m(\omega )=-{\tilde{\epsilon }}_d$of disk polarizability is also preserved. Since direction of incident field is uniform it’s possible to describe deterministic particle array as arrangement of parallel dipolar bonds in the square lattice mapped according to deterministic pattern and coupled by constraints of Eq. (1). In terms of equivalent circuit models metal two-port element can be identified as series$R_m{L}_m$ network and dielectric element as capacitor$C_d$.

In our analysis we assumed zero neighboring particle separation in both longitudinal and transverse direction which allows analysis in extremity of maximum nearest neighbor coupling. We specify longitudinal and transverse permittivity matrices, $U^l$ and $U^t$, whose elements correspond to bonds parallel and perpendicular to direction of polarization. In previous studies of fully random structures [19,23] metal and dielectric tiles that constitute two-dimensional substrate were mapped directly to conductivity matrices with both $U^l$and $U^t$defined by percolation constant. In present deterministic approach $U^l$ is equivalent to binary deterministic pattern matrix with 1 replaced by ${\tilde{\epsilon }}_m$and 0 replaced by ${\tilde{\epsilon }}_d$while $U^t={\tilde{\epsilon }}_d$. Applying Eq. (4) at the lattice potentials on the point by point bases, using elements $U^l$ and $U^t$ as permittivity parameters between neighboring points produces KH matrix. Periodic boundary conditions were implemented by connecting lattice edge points left to right and top to bottom assuring current conservation.

The same considerations apply to permeability matrices and KH for magnetic current density, but since $H_0$ and $E_0$ are orthogonal, the pattern matrix is rotated by ${90}^{\circ }$. Similarly it is possible to obtain results for orthogonal direction of polarization,$E_0/\left|E_0\right|=x$, by translating potential lattice by $T=[a/2,a/2]$ and repeating procedure described above . Since $dk{n}_d\ll 1$and ${\epsilon }_d=1$ we assumed ${\tilde{\epsilon }}_d=D/2d+{\epsilon }_d$ and ${\tilde{w}}_d=i\omega (d+D/2)/4\pi$ as renormalized dielectric permittivity and permeability values [17]. ${\tilde{\epsilon }}_m$ and ${\tilde{w}}_m$ were calculated using Eq. (3) and Eq. (5).

3. Structures

As examples of DAS we used Fibonacci and Rudin-Shapiro patterns (Figs. 2 (a) and 2(b)) generated according to Ref. 1. Structures produced with this method result in trivial fractal geometries, $D_f=2$ in our case. Discrete Fibonacci Fourier transform and continues Rudin-Shapiro Fourier transform allow comparison of various degree of disorder: periodic and random-like.

 

Fig. 2 Deterministic structure pattern and corresponding Fourier transform (a) Fibonacci (b) Rudin-Shapiro (c) CCA Fractal (d) Random.

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Assembly of Cluster-Cluster Aggregate (CCA) fractal structures (Fig. 2(c)) mimics mechanism of metal colloid aggregation. In order to generate CCA structures random walk algorithm was used [24]. Under this method particles randomly distributed in cubic lattice move with equal probability in all directions. Boundary conditions are assumed periodic and particles leaving once side of the cube reappear on the other. Upon impact these particles form clusters which in turn move as single entities. In the end of assembly process random three-dimensional distribution of particle is reduced to a single cluster. Taking projection of this structure on the plane produces planar fractal structure. Dimensionality of fractals used in the simulation was computed using boxcount method to be $D_f=1.68$ .

Random deterministic structures (Fig. 2(d)) were generated using percolation algorithm according to which pattern matrix elements are assigned metal or void values with given probability p (percolation constant). In our study percolation threshold value $p_{perc}=0.5$ was used. Since structures are deterministic and $U^t={\tilde{\epsilon }}_d$, percolation thresholds is not identical to the case of fully random structures and as a packing fraction corresponds to$p=0.25$. Packing fractions for other simulated structures were: $p=0.25$(Rudin-Shapiro), $p=0.26$(Fibonacci) and $p=0.11$(CCA fractal).

4. Analysis and discussion

Deterministic GOL extension coding was performed using MATLAB. Robust code implementation allows fast (< 10 min.) simulations of particle arrays using standard desktop PC (2.2 GHz processor speed, 2 GByte RAM). Particle dimensions were taken as $D=30$ nm, and $d=10$nm. Deterministic structures were set to have square geometry with variable array size N. Plane wave excitation with and $\left|H_0\right|=\left|E_0\right|/Z_0=1/377$ was assumed.

In optical and infrared range metal permittivity can be described using Drude model:

4.1 Comparison of GOL and DDA results

In recently published paper [12] Fibonacci and Rudin-Shapiro structures were analyzed using modified DDA approach. Although we cannot compare results directly since in DDA case simulations were performed on particles having spherical geometry and finite neighboring particle separation, we can qualitatively analyze similarities of both results. Profiles of DAS maximum field enhancement spectrum (Figs. 3 (a) and 3(b)) have similar main features between two methods. In particular strong enhancements in$\lambda >{\lambda }_{res}^{plasmon}$ region, longitudinal and transverse mode splitting in Fibonacci arrays and broad, multi-peak Rudin-Shapiro spectrum are present in both GOL and DDA simulations. Enhancement spectrums obtained with GOL are blue shifted and broadened relative to DDA profiles as a result of mismatch in simulation conditions and discretization artifacts. Considering that the only common factor in these simulations is DAS morphology, similarities suggest that modal properties and resulting enhancement spectrum profile are features of specific DAS and are independent of constituent particle geometry or interparticle distance.

 

Fig. 3 Maximum field enhancement versus wavelength for structures of variable size ($N=44$black, $N=72$ red, $N=98$ blue) and variable morphologies (A) Fibonacci (B) Rudin-Shapiro (C) CCA Fractal (D) Random.

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4.2 Coupling mechanisms and resonant behavior in deterministic structures

Deterministic nature of DAS enables morphological analyses of large aperiodic structures. Examining DAS geometry (Fig. 2(a) and 2(b)) we observe that Rudin-Shapiro and Fibonacci contain discrete particle clusters with finite range of dimensions: $\left(1,1\right)$, $\left(2,2\right)$, $\left(2,3\right)$, $\left(3,2\right)$, $\left(3,3\right)$, $\left(3,4\right)$, $\left(4,3\right)$, $\left(4,4\right)$ and $\left(1,1\right)$, $\left(1,2\right)$, $\left(2,1\right)$, $\left(2,2\right)$ respectively. Each of the clusters is related to subset of ${\tilde{\epsilon }}_m\in U^l$, ${\tilde{\epsilon }}_d\in U^t$and corresponds to unique set of KH potentials. Potentials corresponding to clusters embedded in similar aperiodic medium correspond to single eigenvalue and since $H\text{'}$ matrix is nondegenerate constitute the same extended eigenstate. Eigenstates of the clusters that don’t share similar embedding morphology are localized.

When $\tilde{\epsilon }{\text{'}}_m/\tilde{\epsilon }{\text{'}}_d=1$ deterministic structure is in plasmon resonance regime (${\lambda }_{res}^{plasmon}=358$nm for silver in air) and each row in KH is composed of $\pm {\tilde{\epsilon }}_m$off -diagonal elements indicating uniform coupling magnitude. Ratio $\tilde{\epsilon }{\text{'}}_m/\tilde{\epsilon }{\text{'}}_d$ grows with wavelength which results in strong longitudinal coupling in the cluster and, to a lesser degree, increased transverse coupling. Coupling between the neighboring clusters remains unchanged. As wavelength increases cluster eventually becomes an isolated entity decoupled from the rest of the system and having its own natural resonant frequency. Figure 4 is an illustration of this behavior. Rudin-Shapiro longitudinal current density distribution has been evaluated at wavelengths corresponding to low frequency resonances. Increase in localization of field enhancements in the clusters with growing wavelength is clearly notable. Theoretical upper limit of transverse coupling can be derived by setting${\tilde{\epsilon }}_m\to \infty$ in which case longitudinally connected sites collapse onto each other and effective transverse coupling becomes ${\epsilon }_{eff}^t=n\cdot {\epsilon }_d$ where n is the number of columns in original cluster.

 

Fig. 4 Rudin-Shapiro array pattern (a) and corresponding current density distribution (b)$\lambda =422$ nm (c)$\lambda =432$ nm (d)$\lambda =452$ nm, $N=44$.

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Hence, while at long wavelength DAS field enhancements are localized in resonant eigenstates of simplistic clusters, at wavelength in the vicinity of Plasmon resonance particle clusters can contribute to long range resonant modes. Figure 3(b) can be used to confirm this hypothesis. Here long-wavelength resonances remain essentially unperturbed under scaling while coupling at short wavelengths is significantly modified. It is possible to utilize this property to shape long-wavelength part of DAS enhancement spectrum by proper selection of constituent particle clusters.

Resonant frequencies of individual clusters are determined by their geometry and are similar to characteristic vibrational frequencies of fractal “blobs” [26]. Single particle plasmon resonant peak corresponds to resonance condition $i\omega L_{eff}=-1/i\omega C_{eff}$ where $C_{eff}=C_d$ can be derived similar to resistance of infinite sheet [27]. In case of a chain of disks with $N_{row}$ elements, field enhancements spectrum exhibits two maximums: plasmon resonance at ${\lambda }_{res}^{plasmon}$and collective resonant mode at ${\lambda }_{res}^{coll}$ where ${\lambda }_{res}^{coll}>{\lambda }_{res}^{plasmon}$, similar to the chain of spheres [28]. Increasing the number of disks in the chain redshifts ${\lambda }_{res}^{coll}$since $L_{eff}\propto N_{row}$. At the same time it increases collective mode enhancement magnitude as current density induced by the wave ${\tilde{\epsilon }}_m(r)E_0$ incident on the particle grows with wavelength. The number of resonant peaks in the cluster is proportional to number of cluster columns since effective impedance of surrounding medium for individual columns varies and ${\omega }_{res}^n=1/\sqrt{L_{eff}{C}_{eff}^n}$. Keeping the number of rows constant, number of columns,$N_{col}$, corresponds to $N_{col}/2$ resonant peaks due to symmetry.

Examining enhancement spectrum it can be observed that the magnitude of field enhancements is larger in the structure with high degree of disorder (Rudin-Shapiro) (Fig. 3(b)) then in quasi-periodic structure (Fibonacci) (Fig. 3(a)). At the same time Rudin-Shapiro enhancement magnitude is similar and Fibonacci enhancement exceeds those of random structures at the DAS maximum enhancement wavelengths (Fig. 3). Deterministic random and fractal structures exhibit broad spectrum with maximum enhancements peaks redshifted with respect to DAS due to increased longitudinal dimension of constituent particle clusters. Resonant Q factor decreases at long wavelength as$R_m^{eff}$of the particle chains grows which leads to broadening of resonant peaks. At the same time density of local “hot spots” in random structures becomes progressively smaller since longitudinal dimension of resonant clusters grows and probability of random occurrence of the long cluster decreases. Maximum field enhancement wavelengths for simulated structures with array size$N=72$ were:$\lambda =388$ nm (Fibonacci), $\lambda =408$nm (Rudin-Shapiro),$\lambda =713$nm (Fractal), $\lambda =516$nm (Random).

Studying spatial intensity distribution (Fig. 5 ) we can see that random and fractal structure enhancement localizations are characterized by power law dependency resembling dipole intensity distribution$I\propto 1/r^4$while in DAS high enhancement density and local coupling destroys log-normal pattern. In Fibonacci structures apart from cluster field localizations there is transverse localization that occurs as a result of $U^t={\tilde{\epsilon }}_d$, $\left|{\tilde{\epsilon }}_d\right|\ll \left|{\tilde{\epsilon }}_m\right|$, and quasi-periodic nature of the sequence which is manifested by conductive longitudinal channels with relatively low transverse spreading (Fig. 5 (a)).

 

Fig. 5 Spatial intensity distribution at maximum field enhancement wavelength (a) Fibonacci (b) Rudin-Shapiro (c) CCA Fractal (d) Random.$N=72$in all cases.

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As it becomes evident, high degree of structural disorder leads to increase in magnitude of field enhancements accompanied by enhancement localizations. These simultaneous effects create a trade-off between magnitude of field enhancements and enhancement density. Various DAS morphologies can be used to tune and customize density/magnitude combination for particular application by selecting structure with appropriate degree of disorder. Disorder and periodicity can be quantified using spectral density of corresponding two-dimensional Fourier transform (Fig. 2, inserts).

4.3 Analysis of field enhancements using random medium techniques

Similar to quantum mechanical methods [29] field localization in deterministic structures was studied using inverse participation ratio defined by $IPR={\displaystyle \sum _i{\left|I_i\right|}^2}/{\left({\displaystyle \sum _i\left|I_i\right|}\right)}^2$where$I_i={\left(E_i-E_0\right)}^2$.For purely extended eigenstates $IPR\propto N^{-\dim }$ where dim is system dimensionality ($\dim =2$in our case) and for strongly localized states $IPR\propto N^0$ [30]. The log-log plot IPR slope of DAS (Figs. 6(a) and 6(b)) is size independent indicating extended and scalable field enhancements. On the other hand random and fractal structure results suggest existence of long range eigenstates at ${\lambda }_{\max }\gg {\lambda }_{res}^{plasmon}$throughout full scaling range indicated by size dependent IPR fluctuations (Figs. 6(c) and 6(d)). Minimum DAS size beyond which structures is considered fully extended is estimated from Figs. 6(a) and 6(b) as$N=30$. Below this limit DAS can be treated as random structures where self-similarity is not fully manifested. For IPR scaling analysis we used subsets of single random structure, and unique CCA fractals for each size step.

 

Fig. 6 Inverse participation ratio versus structure size (a) Fibonacci (b) Rudin-Shapiro (c) CCA Fractal (d) Random at $\lambda =358$nm (black squares) and at maximum field enhancement wavelength (red circles).

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Density of field enhancements was quantified using slope of intensity distribution function expressed as$X(I)={\displaystyle \sum _i\delta (I_i-I)dS}$calculated over the structure area. Intensity distribution log-log profiles (Fig. 7 ) are notably different depending on morphology. In particular at large intensity values where statistical dependency takes power form,$X\propto I^{-a}$ for wavelengths corresponding to maximum field enhancement critical exponents were estimated as:$a=1.38$(Fractal, Random),$a=0.19$(Fibonacci) and $a=0.52$(Rudin-Shapiro) confirming reduction in enhancement density with increased resonant wavelength. At the wavelength of Rudin-Shapiro maximum enhancement critical exponents in random structures are: $a=0.6$(Random), 0.41(Fractal) indicating similar enhancement density between DAS and non-deterministic morphologies.

 

Fig. 7 Intensity distribution function versus logarithm of intensity computed at maximum field enhancement wavelength (a) Fibonacci (b) Rudin-Shapiro (c) CCA Fractal (d) Random. $N=72$ in all cases.

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As a result of transverse localization statistical profile of Fibonacci intensity distribution (Fig. 7(a)) is unbalanced with steep transition between enhanced and attenuated local fields. Random and fractal structure plots (Figs. 7(c) and 7(d)), while showing a resemblance to dipole profiles [19], experience a peak at $\log (I)=0$,$\left|E_i\right|=\left|E_0\right|$indicating large number of metallic and dielectric links that don’t participate in current conduction which results in low density field enhancements. Rudin-Shapiro intensity profile (Fig. 7(b)), unaffected by $U^t={\tilde{\epsilon }}_d$, mimics truly random structure intensity distribution [19] and its smooth log(I) = 0 transition signifies isotropic average current density.

We can analyze eigenstate localizations and symmetry effects using eigenstate expansion of $H\text{'}$ (Eq. (10)). Localization length of each state is calculated using a gyration radius: ${\xi }^2={\displaystyle \int {\left(r-{〈r〉}_n\right)}^2{\left|{\mathrm{\Psi }}_n(r)\right|}^{2}dr}$ where ${〈r〉}_n={{\displaystyle \int r\left|{\mathrm{\Psi }}_n(r)\right|}}^{2}dr$ is mass center of the nth state. As a result of $U^t=const$ and $p\le 0.26$ power law scaling of localization length reported in Ref. 23 is no longer preserved and due to introduced non-local symmetry long range eigenstates in the interval $0\le \mathrm{\Lambda }\le 4$are allowed (Fig. 8 ). To form eigenstates with $\mathrm{\Lambda }<0$, $\mathrm{\Lambda }>4$eigenstates are required to contain fully metallic and dielectric nodes, therefore discontinuity in localization length arises at $\mathrm{\Lambda }=0$and $\mathrm{\Lambda }=4$in all but CCA fractal structures where large continues metal and dielectric areas dominate. Presence of low and medium range DAS eigenstates in the interval $0\le \mathrm{\Lambda }\le 4$confirms existence of enhancement localizations (Figs. 8(a) and 8(b)). These localizations are stipulated by local DAS symmetry in form of particle and void clusters. In random and fractal structures low-range eigenstates in the interval $0\le \mathrm{\Lambda }\le 4$ are not permitted due to absence of symmetry (Figs. 8(c) and 8(d)) which confirms our IPR conclusions. As wavelength increases negative eigenvalue range extends and resonant eigenstates make $\mathrm{\Lambda }=0$ transition with varying degree of localization. Nonmetallic eigenstates do not change their sign retaining positive eigenvalues.

 

Fig. 8 Eigenstate localization length versus eigenstate eigenvalue (a) Fibonacci (b) Rudin-Shapiro (c) CCA Fractal (d) Random. $N=72$, $\lambda =358$nm.

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4.4 Absorption

Far-field results computed using Eq. (9) show that DAS offer higher absorption and consequently better ohmic heating then Random and Fractal structures of the same size (Fig. 9 ) which can be beneficial for applications such as thermal cancer treatment and nanostructure growth [31,32]. Ohmic current dominates low-frequency absorption spectrum and displacement current plays major role in the Plasmon resonance region. Therefore there is a discrepancy between maximum field enhancement (Figs. 3(c) and 3(d)) and absorbance profiles (Figs. 9(c) and 9(d)) at high optical frequency in random and fractal structures since their field enhancements are dipolar in nature and enhancement density decreases with wavelength. Computed absorbance for large CCA (Fig. 9(b)) is in good agreement with results reported for silver colloid CCA [33]. Scattering contribution was neglected (small particles) and experimental extinction efficiency was directly related to absorption. Maximums of field enhancements coincide with absorption peaks in DAS and therefore can be identified by performing absorption crossection measurements.

 

Fig. 9 Absorbance versus wavelength for arrays of variable size ($N=44$black, $N=72$ red, $N=98$ blue) and variable morphologies (A) Fibonacci (B) Rudin-Shapiro (C) CCA Fractal (D) Random.

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5. Conclusions

In summary we conclude that single bond particle discretization applied to GOL permits semi-analytical solution of SP DAS electromagnetic problem adequately capturing effects of local coupling and modal properties. Resulting simulations demonstrate presence of fractal “blobs” in DAS - particle clusters with localized field enhancements and short range eigenstates convergent due to local symmetry. This effect implies that DAS enhancement spectrum can be partly controlled by selecting appropriate configuration of constituent particle clusters. Our study suggests existence of trade-off between density and maximum achievable field enhancements in deterministic structures which can be estimated based on degree of system disorder by corresponding Fourier transform. Large density of “hot spots”, high field enhancements and design controllability creates a strong case for utilizing DAS as viable SP substrate.