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Abstract
Electrically tuneable, guided self-assembly of plasmonic nanoparticles (NPs) at polarized, patterned solid–liquid interfaces could enable numerous platforms for designing nanoplasmonic optical devices with new tuneable functionalities. Here, we propose a unique design of voltage-controlled guided 3D self-assembly of plasmonic NPs on transparent electrodes, patterned as columnar structures—arrays of vertical nanorods. NP assembly on the electrified surfaces of those columnar structures allows formation of a 3D superstructure of NPs, comprising stacking up of NPs in the voids between the columns, forming multiple NP-layers. A comprehensive theoretical model, based on quasi-static effective medium theory and multilayer Fresnel reflection scheme, is developed and verified against full-wave simulations for obtaining optical responses—reflectance, transmittance, and absorbance—from such systems of 3D self-assembled NPs. With a specific example of small gold nanospheres self-assembling on polarized zinc oxide columns, we show that the reflectance spectrum can be controlled by the number of stacked NP-layers. Numerical simulations show that peak reflectance can be enhanced up to ∼1.7 times, along with spectral broadening by a factor of ∼2—allowing wide-range tuning of optical reflectivity. Smaller NPs with superior mobility would be preferable over large NPs for realizing such devices for novel photonic and sensing applications.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
9 September 2019: A typographical correction was made to the author affiliations.
1. Introduction
The excellent combination of optical transparency and electrical conductivity has made transparent conducting oxides (TCOs) popular choice for numerous optoelectronics applications, e.g. LCDs, LEDs, solar panels, digital displays and touch screens, just to name a few [1–5]. On the other hand, recent progress made in tuneable nanoplasmonic metamaterials has seen a few exciting new realizations of in-situ voltage-controlled optical devices — based on directed self-assembly of sub-wavelength-sized plasmonic nanoparticles (NPs), forming reversible dense/sparse monolayers, at electrified liquid–liquid or solid–liquid interfaces [6–11].
An alliance of the two worlds possesses immense possibilities towards realizing in-situ electrically tuneable nanoplasmonic devices [12] on transparent electrodes, e.g. Fabry–Perot interferometers, optical filters, etc. — based on controllable density of NPs forming monolayers on flat TCO electrodes used as NP-substrates [2]. Formation of multi-layered structures or superlattices of NPs have been demonstrated previously [13–17], but those schemes lack reversibility and a theoretical framework which can guide designing application-specific systems. Development of reversible assemblies, consisting of controlled multilayer formation by plasmonic NPs on templated substrates [18], such as transparent electrodes, could give rise to many exciting new optical and photonic effects. A comprehensive theoretical framework is, therefore, needed for exploration of different possible scenarios to drive further research and experiments in this new area.
The present work theoretically investigates the optical responses of a guided self-assembly of plasmonic NPs on columnar transparent electrodes, allowing formation of three-dimensional (3D) superstructures of NPs. Such electrodes can be realized in the form of vertically aligned nanorod arrays [19,20] of ITO, ZnO, or other TCOs. By now, the columnar TCO nanostructures have found diverse applications, typically in photocatalytic water splitting for energy conversion [21,22], as anti-reflective coating [23–25], for photoluminescence and lasing [26,27], photoelectrodes [28] for solar cells, and sensors [29,30] etc., to name a few. Here, we propose to use specifically patterned TCO columnar electrodes as substrates for guided self-assembly of plasmonic NP layers for tuneable optical applications.
A theoretical framework is developed to study the optical response of such systems. Based on it, we propose to use vertically aligned ZnO nanorod arrays as ‘columnar electrodes’ for controlled layer-by-layer self-assembly of Au-NPs. We focused on ZnO as an electrode material, because it is more stable than ITO in electrochemical environment [31]; also more cost efficient [32]. The theoretical framework, however, is equally applicable to any other materials for electrodes and NPs, suitable for such structures.
2. System outline
The proposed system is designed to act as an electro-tuneable nanoplasmonic reflector of light, with controlled multilayer formation through self-assembly of NPs (otherwise dispersed in aqueous solution) on polarized ZnO columnar electrodes. Negatively functionalized NPs can assemble in multiple stacks on the surface of the positively polarized electrodes, filling-in the voids between the columns, to form a 3D NP lattice. Such 3D assembly of metallic NPs promises to increase optical reflectance of the incident light with controlled NP-stacking.
Figure 1 depicts a cartoon of the proposed scheme, where ZnO columns with precise structural dimensions and orientation, grown from a base ZnO plate form an electrode that could be polarised positively to assemble a 3D lattice of negatively functionalized NPs (in aqueous electrolytic environment [7]) to strongly reflect incident light [Fig. 1(a)], or polarised negatively to disassemble the NP lattice into the aqueous solution and allow transmittance of incident light with minimal reflection [Fig. 1(b)]. This is an alternative approach to constructing an electro-tuneable optical reflector [6,33,34].
Fig. 1. Voltage-guided 3D assembly/disassembly of negatively charged NPs into/from columnar-structured TCO electrodes in electrolytic solution. (a) ‘mirror’ state, with NP stacks assembled in the intercolumnar voids of positively charged electrodes which, as we show below, strongly reflect incident light; (b) ‘window’ state, with NPs repelled away to disperse in the bulk of the solution from negatively charged electrodes, allowing incident light to pass through.
3. Theory
This section presents a theoretical model to describe optical properties of a system of N-layers of NPs assembled at solid–liquid interfaces (SLI), around ZnO columnar structures, by extending our effective medium approach [35] with the use of multi-layer Fresnel’s reflection theory. To begin with, a few simplifying assumptions are made. The ZnO columns are first neglected, and a system consisting of just two layers of NPs is considered to demonstrate how a model describing NP monolayer [36] can be extended. We then present further extension to an N-layer system. Finally, the ZnO columns are introduced into the model by recalculating the dielectric function of the medium surrounding the NPs, using the Maxwell–Garnett formula [12], to obtain a comprehensive theoretical framework, which is verified against full-wave simulations.
3.1 Building up the stack — Two layers of nanoparticles
A five-layer-stack theoretical model, capable of estimating the optical response of a monolayer of NPs assembled at SLI, within quasi-static dipolar approximations, was described in detail in our previous work [35]. In order to introduce a second layer of NPs into that theoretical framework, a new parameter must be added to account for the separation between the NP layers, LNPs. Figure 2 illustrates how a system with two layers of NPs can be adapted using a six-layer-stack model. To estimate optical response of a monolayer of NPs assembled at a SLI, the image dipole interactions between the NP layer and its underlying substrate are included in the calculation through the effective quasi-static polarizability of each NP in that monolayer, ${\beta ^{({\parallel , \bot } )}}(\omega )$, see Eq. (2) in [35]. For the two layers’ case, we consider image interaction only between the NPs in the bottom layer (layer 4) and the substrate TCO electrode [layer 6, Fig. 2(b)]. Image interactions of the NPs in the top layer (layer 2) are not considered, as they are expected to be insignificant due to large NP–substrate separation [37–39]. Therefore, the effective quasi-static polarizability of individual NPs (estimated in the presence of all other neighbouring NPs) in each of those two monolayers are not equal, which can be calculated as [35]
Fig. 2. Modelling of two layers of assembled NPs at a solid–liquid interface using a six-layer stack model. (a) Original structure and (b) equivalent theoretical model. Variables — h : distance of the central plane of the first layer of NPs from the interface, R : NP radius, a : lattice constant, k : wave vector of the incoming light with incident angle, $\theta $, d : thickness of the pseudo-NP layer and L : inter-NP layer separation. The bottom panels show the mapping of ‘L-parameters’ between the original system and the equivalent model, where surface-to-surface (S2S) separation between NP-layers is adjusted while the center-to-center (C2C) distance remains the same
Based on those effective polarizabilities of individual NPs in each assembled monolayer, the parallel ($\parallel $) and perpendicular ($\bot $) components of the permittivity of the equivalent films representing those monolayers can be obtained as
With the knowledge of permittivity of each layer of the six-layer-stack model, a transfer matrix, ${\tilde{{\textrm M}}}$ can then be obtained, which allows calculation of optical reflectance, transmittance, and absorbance of the whole system. The equations for obtaining the transfer matrix, optical reflectance and transmittance are given in Appendix A.
3.2 Extending to N layers of nanoparticles
Now we can extend the model described above to N-layers. We no longer number the layers, but instead, label them. The label of each layer in the generalized model is shown in Fig. 3. All the layers denoting surrounding medium will be assumed to have the same dielectric constant, ${\varepsilon _{\textrm{med}}}$, but the wavevectors and phase shifts associated with them must be considered separately in calculation.
Fig. 3. Generalised multilayer Fresnel scheme for modelling N-layers of assembled NPs at a solid–liquid interface. (a) Original system and (b) adapted theoretical model with the labels for each layer. The labels are ‘TCE’ (transparent conductive-oxide electrode); ‘med,h’ (surrounding medium, between the final NP layer and the TCE); NPfilm, N’ (final NP layer, which includes dipolar image interaction with the TCE); ‘med’ (surrounding medium between each NP layer); ‘NPfilm’ (pseudo-NP films with no image charges); and ‘med,1’ (surrounding medium above the NP stacks).
The isotropic polarizability of an NP dispersed in an embedding medium of permittivity ${\varepsilon _{\textrm{med}}}$, (c.f. Eq. (3) in [35]), and the anisotropic permittivity of each pseudo-NP film read:
Whereas, for the n = N layer, which includes the image terms, the effective quasi-static polarizability is given by
With estimates of the permittivity of each layer of the N-layer system, one can obtain a transfer matrix using the coefficients of reflection and transmission at each interface along with the phase shifts in each layer. From the transfer matrix, optical properties such as reflectance, transmittance, and absorbance of the N-layer system can be calculated (see Appendix B for the equations used in the model).
It is important to realise that the theory becomes inaccurate when the underlying assumptions break down. At inter-NP separations of below 2 nm, the dipolar approximations begin to break down, as hybridized multipolar modes emerge and become significant, which are not included in our quasi-static dipolar approximation. Moreover, when NPs are very large, typically beyond 40 nm, again that quasi-static dipolar approximation for individual NPs may become inaccurate, with need for incorporating effects such as dynamic depolarization and radiation damping [40–42]. However, since in our proposed system, the NPs are typically functionalised with ∼2 nm ligands, their inter-NP gap is rarely less than 4 nm, and we focus on NPs much smaller than 40 nm, the quasi-static dipolar approximation is expected to remain valid. The accuracy of our model is checked against full-wave simulations in the subsequent section.
3.3 Introducing dielectric properties of ZnO columns in theoretical calculations
A 3D stack of N-layers of NPs can be obtained through the process of guided self-assembly of NPs using an array of polarized ZnO vertical nanorods (columnar electrodes), which act as a supporting structure or template for the NPs to assemble around. A specifically patterned array of ZnO nanorods, grown in the form of a triangular lattice, can support packing of NPs in a square array in the voids between the nanorods, as shown in Fig. 4. The NPs may slot in between those polarized nanorods, forcing NPs to lie directly on top of each other to build up vertical stacks of NPs, which extends in both lateral dimensions. The result is a 3D lattice of NPs, with characteristic lattice constants determined by the design of the template, the ZnO-nanorod array.
Fig. 4. Schematics of NP-assembly around ZnO nanorod array. (a) 3D view and (b) cross sectional view of ZnO nanorod arrays supporting assembly of NPs (each capped with ligands of length l) in the gaps around the nanorods to form a stack of monolayers with NPs assembled in a square lattice in each of those. Note: NP-capping ligands are not explicitly shown in (a), however in (b) the thickness of the layer of ligands around each NP is depicted, which will be later useful for calculation. Note: the nanorod array is typically grown on top of a substrate, which is not explicitly shown here.
In order to incorporate the optical effects from ZnO columns into the theoretical model, the volume fraction of ZnO material in the aqueous phase must be determined. The dielectric function of the surrounding medium of the NPs can then be adjusted to account for the presence of the ZnO columns.
For NPs to assemble into a square lattice, the relationship between the lattice constant of the ZnO columns, aZnO and the lattice constant of the NPs, aNP can be determined as
and This allows construction of an appropriate ZnO columnar array structure to achieve a desired inter-NP gap, gNP, where For square packing of NPs, the volume fraction of the ZnO material in a unit cell (see, dashed square in Fig. 4) can be calculated as4. Results and discussion
Figure 5 shows a comparison of the optical reflectance spectra calculated by our theoretical model (solid curves) and those obtained from full-wave simulations (dashed curves) using COMSOL Multiphysics, for different numbers of NP-layers stacked on top of each other. The excellent agreement between the two sets of data indicates that approximations of our theoretical model are accurate enough for describing the optical characteristics of the 3D NP lattice. Since the system under study is meant to act as a smart reflector, the analysis here focuses on the visible and near-infrared region of the electromagnetic spectrum, over wavelengths between 400 nm and 900 nm.
Fig. 5. Comparison of theoretical, T, (solid) reflectance spectra with those obtained from full-wave simulation, S, (dashed) for different numbers of NP-layers. Parameters: NPs of radius, (a) R = 6 nm and (b) R = 10 nm, ${\varepsilon _{\textrm{TCE}}} = 3$; ${\varepsilon _{\textrm{med}}} = $ 1.78; g = 4 nm.
The effect of increase in the number of layers of NPs is depicted for two different NP sizes, R = 6 nm in Fig. 5(a) and R = 10 nm in Fig. 5(b). A quick comparison tells that, as expected, the model predicts the optical properties more accurately for smaller NPs. In Fig. 5(a), the reflectance of the system increases dramatically when a second layer is deposited on top of the first monolayer. Not only the peak reflectance, but also the spectral linewidth gets broader. The trend of increase in peak reflectance and widening of the peak continues for the 3rd layer. However, upon adding the 4th layer, the peak splits into a high- and a low-energy mode, where the maximum reflectance in this spectrum gets lowered, but reflectance over long wavelength regime increases due to the emergence of that low-energy peak. With further addition of the 5th layer, the high energy mode exhibits blue shift and the low-energy mode undergoes red-shift, with peak reflectivity getting reduced for both modes. This peak splitting may be attributed to the phase differences between different layers of NPs, giving rise to anti-bonding (high energy) or bonding (low energy) type of plasmonic coupling between the layers of NPs. The splitting effect emerges beyond three layers for small NPs of radius R = 6 nm, when the accumulated phase difference between different layers becomes significant, as the entire stack of NPs cannot be excited simultaneously by the incident light.
For larger NPs, here with radius R = 10 nm in Fig. 5(b), the peak splitting appears much earlier — with deposition of the 2nd NP-layer itself. With the 2nd layer of NPs the peak reflectance increases to some extent. But with deposition of a 3rd layer, the high energy peak blueshifts and the low-energy peak redshifts. The overall peak reflectance reduces; however, reflectance increases towards long wavelengths due to the presence of the low-energy peak. The overall trends seen are like those observed for the small NPs in Fig. 5(a). With larger NPs, reflectivity is naturally enhanced; although the peak reflectance is relatively less dramatically affected by the number of stacked NP-layers, the relative changes in the long wavelength region are more pronounced.
Reflectance from the vertically-stacked NP assembly increases (decreases) when there is constructive (destructive) interference between light waves reflected by different NP layers. With increase in number of layers and NP size, the accumulated phase difference between light reflected from different layers grows, which results into wider splitting between the peaks. Besides widening of spectral gap between the peaks, increase in absorption of light, by larger number of NP-layers, leads to gradual reduction in the reflectance maxima for both high- and low energy modes.
In order to assess how the peak reflectance evolves in a 3D stack of NPs, we further plotted the maximum achievable reflectance while considering up to seven layers of NPs, for three different sets of NP sizes — radius of 5 nm, 8 nm, and 12 nm. Figure 6 shows that for each specific set of the NPs, there is an optimum number of layers for which the system achieves its maximum reflectance, beyond which the peak reflectance value cannot be further increased. The optimum number of layers is found to be 3, 2, and 1 layer(s) for NPs of radius 5 nm, 8 nm and 12 nm, respectively.
Fig. 6. Maximum reflectance of the system with the different numbers of NP layers, for NPs of three sets of different radius: R = 5 nm (triangles), R = 8 nm (circles), and R = 12 nm (crosses).
To appreciate this finding and to see how the reflectance spectra evolved in each case, we plot Fig. 7 containing reflectance spectra for up to four layers of NPs in each case. It shows the trend of peak reflectance getting mostly saturated beyond a few layers of stacking of NPs. Notice that with increase in NP size, the role of the accumulated phase difference between different NP-layers in peak-splitting gets bigger. This effect becomes prominent for 4 NP-layers of NPs with radii R = 12 nm in Fig. 7(c), where the high energy peak further splits into two. In this case, the usual low energy peak extends beyond 900 nm and hence, cannot be seen. The beginning of this new peak-splitting can be observed from the case of 3 NP-layers, where a shoulder peak (towards red) near the usual high energy peak can be seen. For the 4 NP-layers, that shoulder peak gets further red-shifted and is clearly seen, while the usual high energy peak undergoes a minor blue-shift.
Fig. 7. Effect of number of NP layers assembled in the voids of the ZnO columnar electrode structure on the reflectance spectrum, shown for different NP radii: R = 5 nm (a), 8 nm (b), R = 12 nm (c).
Figure 8 depicts electric field distribution patterns obtained from full-wave simulations, which allow deeper understanding of the observed changes in the peak reflectance with stacking up of NP-layers. The top-view plots highlight a common trend of peak splitting, shown in Fig. 7. Indeed, a comparison between a monolayer and a 3 NP-layers stack shows that the single peak (‘S’), seen in the optical reflectance spectrum of 1 NP-layer, gets split into two peaks with stacking up of 3 NP-layers, where plasmonic interactions among NPs, seen through the intensification of electric field in the hotspots, are found to be much stronger at the frequency of the high-energy peak (‘Hi’) as compared to the one of the low-energy peak (‘Lo’). This trend gets more pronounced for larger NPs.
Fig. 8. Normalized electric (E) field distribution patterns shown for 1 NP-layer and 3 NP-layers assembled in the voids between the transparent columnar electrodes (see Fig. 4), obtained for different NP radii: R = 5 nm (a), 8 nm (b), R = 12 nm (c). All E- field patterns are shown for a unit cell (shown in a perspective view with 3 NP-layers stacked around a columnar electrode grown on a substrate — see the red dashed box), which is repeated periodically in xy –plane, to emulate the characteristics of the proposed structure. Incident light propagates along –z axis with polarization along x-direction. In each layer, NPs are arranged in a square lattice. E-field patterns are obtained from top- and side views at the wavelength of the single peak (abbreviated here as ‘S’) for 1 NP-layer, and at the wavelengths of high-energy peak (‘Hi’) and low-energy peak (‘Lo’) for 3 NP-layers stack [all peak positions extracted from Fig. 7]. Top-view plots are calculated along an xy plane passing through the centres of the NPs in the top layer. Side-view plots are along an xz plane passing through the centres of the NPs. Following parameters are used in simulation: ${\varepsilon _{\textrm{TCE}}} = 3$; ${\varepsilon _{\textrm{med}}} = $ 1.78; g = 4 nm; LNP = 4 nm and l = 1 nm.
The side-view plots approve a few simplifying assumptions made in our theoretical model: (i) the interaction between different NP-layers in a stack is negligible (we see it via the low intensity of electric field in the gaps between the NPs of the adjacent layers, as compared to a clear enhancement of the field in between neighbouring NPs of the same layer). (ii) the need of incorporating image-charge interaction effects only for the bottom NP-layer (some weak confinement of the electric field, tiny yellow dots, can be seen only between the NPs of the bottom layer and the substrate of the columnar electrodes). The comparison among the relative strengths of inter-NP coupling in monolayers for different NP sizes also correlates with the fact that reflectance increases dramatically with the size of NPs. For a fixed inter-NP gap, larger NPs show stronger plasmonic coupling, giving rise to larger optical reflectance.
The side-view plots also reveal another trend that with increase in NP radius, the relative difference grows in plasmonic coupling at frequencies of the high- and low energy peaks (manifested in the magnitude of the electric field confined in the gaps between NPs). This substantiates our theoretical findings, that peak splitting widens with larger NPs; besides, the difference in optical reflectance between the high- and low energy peaks also increases. For a stack of multiple NP-layers, relatively large changes in electric field confinement between different layers (seen by comparing the magnitude of local field) also suggests that larger phase differences accumulate along the stack with increase in NP size. This leads to wider splitting of a reflectance peak (seen for a monolayer) into high and low energy modes with stacking up of multiple layers.
Interestingly, the maximum relative change in the peak reflectance value is found to be 69% for the NPs of 5 nm radius (along with spectral broadening by a factor of ∼2), and 11% for the NPs with 8 nm radius. Stacking of smaller NPs results in overall weaker reflection [the smallest considered size of R = 5 nm still have a sufficient plasmonic oscillator strength (‘plasmonic mass’) for noticeable reflection], but they demonstrate the strongest effect of the number of layers on the reflection peak. For the considered larger NPs, the number of NP-layers appears to have little effect on the peak reflectance. This may be due to a possible shadowing effect of the impinging light by the top layer, over the layers of NPs that lie beneath, or simply that a single layer of large NPs already reflects a large portion of incident light, and so there is little to be gained from additional NP-layers. Not shown here, we have also investigated different other inter-NP layer separations and NPs with radii greater than 12 nm, however, those did not lead to any significant increase of reflectance. If the rates of electrosorption/electro-desorption of NPs at the electrodes are determined by diffusion, smaller NPs do it faster and thus using them may be beneficial also in the context of fast switching with changing the electrode potential (see the note below).
There are a few more points to be noted, related to the density of the 3D-structures of NPs in the columnar electrode architectures. The distance between NPs within each layer is controlled by the lattice constant of the ZnO nanorod ‘electrode’ array. But the vertical distance between the arrays (i.e. the distance between NPs in each void-row) can be ligand-to-ligand densely packed. Although the NPs are functionalized by negatively charged ligands — to prevent their agglomeration in the bulk and to get attracted to the positively polarized electrode surfaces — they may come very close to each other inside the voids between the columnar structures of the electrodes. This is because of the screening of Coulomb interactions by electrons of the conducting material; for a cylindrical void in an ideal conductor the screening is exponential, with a characteristic decay length of ∼(pore radius)/2.4 [45,46]. In such pores, only a small number of charges of ligands on the neighbouring NPs will experience repulsion from each other, whereas the majority will get strongly attracted to the conductor’s walls. Of course, the voids between ZnO columns are not cylindrical pores enclosed from all sides, but qualitatively this kind of screening effect will be there. Hence, it may be easier to create here dense assemblies of NPs than on a flat surface. Electronic screening of Coulomb repulsion between NPs ‘electrosorbed’ in the neighbouring voids, will also undermine the propensity of zig-zag localization in favour of packing flat layers, so that we may not need to consider more complicated disposition of NPs than in the system of equidistant planes.
In this work we described the optical signals of the NP structures as formed. We did not in any way discussed the dynamics of the formation of these structures, or the values of electrode potentials we need to apply to warrant the desired electrosorption/desorption. These questions were studied in detail in previous publications in the context assemblies of NPs from solutions on flat interfaces [6,7,11]. There it was basically controlled by random diffusion of NPs from the bulk of electrolyte to the electrode near which they could feel its electric field (otherwise screened by electrical double layers keeping electrolyte bulk electroneutral). Estimates made there have shown that miniaturization of the electrochemical cells (building optical microchips) can bring the response time from hours to milliseconds. The same story will refer to the systems described in the present paper: the electrodes described above must be compartmentalized in microelectrochemical cells for achieving a reasonable switching time. Moreover, the electrode columns must not be taller than needed to accommodate those number of NPs after which the saturation of optical signal takes place. This will help to avoid unnecessary extra delay in ‘transmission-line-charging’ [47] of the inter-columnar voids with NPs.
In a nutshell, the assembled stack of NPs in form of multiple layers stacked on top of one another, supported by transparent ZnO columnar electrodes, give rise to a reflector or mirror state with controllable reflectance spectrum. The disassembled state of NPs, where NPs are randomly dispersed in the aqueous solution, enacts a transparent ‘window’, as the optical reflectance from bare ZnO columnar electrodes in aqueous solution is negligible, with peak reflectance < 1% (figure not shown), and hence, they are almost completely transparent. Therefore, the proposed system can be used as a reversible voltage-controlled mirror/window device.
For devices operating at terahertz, infrared, and microwave frequencies, post-fabrication tuning of absorption properties can be achieved using graphene-based plasmonic structures, where electrical tuning of graphene’s chemical potential [48] allows to vary the central absorption frequency. One can also realize thermally induced tunability in terahertz metamaterials-based devices, e.g. by using a specially designed nematic liquid crystal mixture [49]. In this work we focus on achieving tuneable optical properties mainly in the visible spectral range. Tunability schemes using graphene and liquid crystal properties mentioned above may not be as effective in the frequency range of our interest. In short, our proposed system offers an alternative scheme for obtaining electrical tunability of the optical properties of plasmonic structures in the visible spectrum, where the system benefits from inexpensive bottom-up self-assembly process and energy efficiency with sub-volt range tuning of electrode potential [7].
5. Summary
We propose a new class of electrically reversible 3D superstructure of plasmonic NPs. It is based on voltage-controlled guided self-assembly of functionalized metallic NPs, stacking in the voids between polarized surfaces of transparent electrodes patterned as an array of vertical nanorods embedded into electrolytic solution. The electrosorbed state of NPs enacts a reflector (mirror), while the electro-desorption of NPs into the solution switches the system to a transparent window, allowing incident light to pass through it with minimal extinction.
A comprehensive theoretical framework is developed, within the realms of quasi-static effective medium theory clubbed with multi-layer Fresnel reflection scheme, for calculating the optical responses, viz. reflectance, transmittance, and absorbance, from a multilayer assembly of NPs on these patterned electrodes grown from a TCO substrate. Verification of the model results against full-wave simulations shows excellent agreement, adding confidence in applicability of the developed theory for the description of the optical properties of such structures.
The design presented in the paper shows that significant increase in peak reflectivity can be observed with initial increase in the number of NP-layers, especially for small NPs. This would allow one to achieve a variety of reflective states from the same system, via in-situ electrical control of the number of self-assembled NP-layers. Small NPs are more responsive for the reflectivity tuning.
With further exploration, devices based on this principle may find new exciting applications in photonics, catalysis, and biosensing.
Appendix A: Calculation of optical response from a system with two layers of nanoparticles
The reflection and transmission coefficients, $r_{i,j}^{\textrm{s},\textrm{p}}$ and $t_{i,j}^{\textrm{s},\textrm{p}}$, at any interface between layers i and j, for s- and p-polarized light, are given as
As seen from the Eqs. (A1) and (A2), reflection and transmission coefficients depend on the wave vectors of the light in each medium. So starting from the dispersion relation of light (incident at an angle $\theta $) propagating in a medium with permittivity ${\varepsilon _1}$, the wavevectors in each layer of the six-layer stack model can be described as
With reflection and transmission coefficients known for each interface, a transfer matrix ${\tilde {\textrm M}}$ for the entire system can be evaluated as
After obtaining ${\tilde{\textrm M}}$, the coefficients of reflection and transmission of the entire system, $\tilde{r} = \; \frac{{{{{\tilde{\textrm M}}}_{21}}}}{{{{{\tilde{\textrm M}}}_{11}}}}$ and $\tilde{t} = \; \frac{1}{{{{{\tilde{\textrm M}}}_{11}}}}$, multiplied by their complex conjugates give the system’s reflectance and transmittance. The knowledge of reflectance and transmittance, immediately gives the absorbance of the system [35].
Appendix B: Calculation of optical response from a system with N layers of nanoparticles
Analogous to the two-layer case, the reflection and transmission coefficients of each interface can be taken from Eqs. (A1) and (A2), respectively, with the generalised form of the wavevectors given by
The generalised phase shifts for each layer, ${\delta _i}$ are calculated from the wavevector of the layer, ${k_i}$ multiplied by the layer thickness, ${t_i}$, given by ${\delta _i} = {k_i}{t_i}$. Therefore, the phase shits for each layer of the model are given by
For a model with N-layers of NPs, the transfer matrix of the whole system (including all NP layers, surrounding media, and TCE) can be calculated from the product of the transfer matrices for each layer in the system, given by
This reads asThe first product term in Eq. (B4) refers to the transfer matrix of the interface between the bulk aqueous solution and the top NP layer. The second and third terms in parentheses represent the two interfaces between each NP layer and its surrounding medium, and are multiplied by the number of such NP layers in the system. The fourth term accounts for the transfer matrix of the interface between the last NP-layer without image forces and the medium below. The following two terms (fifth and sixth) refer to the two interfaces of the bottom-most NP layer. The last, seventh term, is the transfer matrix for the interface of the last surrounding medium layer (of thickness, h) and the TCE.
As mentioned before, the reflection and transmission coefficients for the overall system can be found from $\tilde{r} = \frac{{{{{\tilde{\textrm M}}}_{21}}}}{{{{{\tilde{\textrm M}}}_{11}}}}$ and $\tilde{t} = \frac{1}{{{{{\tilde{\textrm M}}}_{11}}}}$. Once again, these values are squared with a complex conjugate to obtain the percentage reflectance and transmittance, now for an N-layer system.
Funding
European Commission (Marie Skłodowska-Curie individual fellowship S-OMMs).
Acknowledgments
The authors are thankful to members of Electrochemical Nanoplasmonics team at the Chemistry Department of Imperial College (in the first place –Joshua Edel, Anthony Kucernak, and Ye Ma) for useful discussions.
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