1. Introduction

The property of noble metal nanostructures to exhibit surface plasmon resonance, the collective oscillation of the electron cloud on the metal surface in response to the oscillating electric field of the incoming light, avails them novel applications for sensing and detection at the nanoscale. The optical properties of metal nanoparticles [MNPs] have been explored for enhancing different processes such as surface enhanced Raman spectroscopy [1], metal enhanced fluorescence [2], near-field microscopy [3] etc. Moreover, the interaction of MNPs with light leads to the polarisation of electric charge on the nanoparticle surface, enabling it to induce an electrostatic force on a nearby charged nanoparticle. Depending on the nature of the polarisation induced, MNPs can exert an attractive or repulsive force on adjacent nanoparticle(s) in vicinity. The potential of these electrostatic forces, also termed as optical forces has been explored for various applications such as optical manipulation [4], optical tweezing [5], and optical trapping of molecules and other nanoscale objects [6]. In most cases for optical trapping of molecules and smaller entities the excitation condition demands for a tightly focused laser beam since the distribution of optical forces due to field gradient is limited to the diffraction limit of light. This often sets a bottleneck on the sizes of the particle that can be trapped optically. MNPs can address this limitation due to their ability to confine the incoming light below its diffraction limit through local field enhancement of the electric field [7]. MNPs strongly enhance the incoming field around their edges and in the gap regions for an ensemble of nanoparticles, avoiding rigorous procedures carried for optical trapping such as beam steering.

Optical forces realised in MNPs are usually very small in magnitude to realise any motion in bulk systems. However, the opto-mechanical force balance at the nanoscale allows unique opportunities for leveraging these forces to drive the mechanics in nanomechanical systems. The integration of optical properties of MNPs with the mechanical properties of nanomechanical resonators opens up new possibilities for optically designing and controlling the motion in these hybrid systems, while at the same time benefiting from the miniaturization for their effortless integration with next-generation technological platforms. So far the utilisation of optical forces has mostly been implemented in studies focused on the trapping of molecules and similar entities. However, their potential for independently driving motion at the nanoscale has not been well explored. Here, we present a numerical study for the case of a plasmomechanical nanopillar-antenna system, which consists of a singly clamped nano-pillar dimer structure with metal nanoparticle on top of it, and report on design considerations of metal nano-antennas with the aim of maximising electrostatic forces between nanoparticles in a dimer for prospects of optically driving motion in these systems. The proposed nano-pillar antenna configuration offers the possibility to combine the free-space adressability of metal nanoparticles with three dimensional motion of nano-pillar resonators, which avails this system advantages over other nanomechanical structures such as beams, bridges or strings. The actuation and transduction of nanomechanical resonators by optical forces generated in MNPs can be utilised for multiple purposes such as ultra-sensitive force and mass sensing applications [8], optical modulation and reconfigurability [9] etc. A recent study reported the generation of optical forces in a lateral configuration in gold nanorod based system, and consolidated the key experimental demonstration of motion driven at the nanoscale by optical forces originating from plasmonic interactions between the nanorods [10].

2. Methods

The numerical study was performed using a commercial electromagnetism solver tool COMSOL Multiphysics (Version 5.5) [11]. We used the wave optics module to solve Maxwell equations in our simulation domain for the wavelength range 500–1100 nm. The dielectric function for gold was taken from Johnson and Christy [12]. The nanoparticle dimer was placed in the middle of a hollow multi-layered core-shell structure Fig. 1, which consists of the modelling region. Air were used as the background index and perfectly matched layer absorbing boundary conditions were applied to prevent reflections at the simulation boundaries. Optical properties of the gold dimer were calculated by performing electro-magnetism (EM) calculations in the scattered field formulation, which has yielded reliable results for optical properties of plasmonic nanoparticles [13]. The background EM field was taken into account in absence of the scatterer (gold dimer), and the scattered field was then solved for the presence of the scatterer in the modelling domain. The total field consisted of the background and the scattered field together. The scattering cross-sections of the nanoparticle dimer were calculated by integrating a sphere over it and applying scattering boundary conditions. The reliability of our model was verified through by comparison with Mie theory for the visible wavelength range (see Supplement 1, Note 1). The corners and edges of nanodisc and nanocube dimers were rounded by a radius of 5 nm to counter the EM field effect related artefacts, which can result due to sharp edges or corners of the nanoparticle. A blue-shift in the scattering spectra was observed as a consequence of rounding the edges of the nanocube dimer (see Supplement 1, Note 6), which has been found in agreement with existing studies detailing the effect of rounding on the scattering properties of nanocubes [14].

 

Fig. 1. Modelling domain and schematics of plasmomechanical nanopillar antenna system for the gold nanoparticle dimer.a The modeling environment consists of air (indicated in blue), which is truncated by PML absorbing boundaries. b Nanodisc antenna of diameter $d$ and thickess $t$ (a). Nanocube antenna of edge-length $h$ (b). Nanobar antenna of edge-length $h$, thickness $t$ and length $w$ (c). The entire pillar structure comprised of nanopillar of length $L$ and gold nanoparticle of thickness $t$ (d). Excitation conditions of the nanoantenna dimer system with electric field $E$ and wave vector $k$ (e). The bending of the pillars has been exaggerated here for better illustration.

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3. Results and discussion

For calculation of optical forces in plasmomechanical nanopillar system, light was perpendicularly incident on the gold nanoparticle dimer system with electric field aligned along the inter-particle axis of the dimer (see schematic in Fig. 1(b)). Optical forces were first calculated for a gold nanodisc dimer as it represents the simplest geometry for a plasmomechanical system (i.e. plasmonic nanoparticle on top of a cylindrical non-metallic nanostructure). The scattering properties were studied in gold nanodisc dimer of diameter $d$ for changes in the optical properties due to plasmonic coupling at varying inter-particle distances between the nanodiscs. Scattering efficiencies of nanodisc dimers of diameter 100 nm, 150 nm and 200 nm were calculated under plane wave excitation conditions for inter-particle distances ranging between 10–50 nm. The plasmon resonance peak associated with the scattering spectra red-shifted as the distance between the two nanodiscs decreased for all radii (see Supplement 1, Note 2). The Maxwell stress tensor method was used for calculation of the optical forces on the gold dimers. This method has been implemented multiple times for its reliability for calculation of optical forces in plasmonic nanostructures [15,16]. In contrast to the conventional approach of enclosing a box over the nanoparticle’s surface and calculating the forces on it, the force tensor components were integrated directly over the gold nanodisc surface. The surface integration allows capturing of arbitrariness related to the shape of the nanostructure, which can greatly influence the magnitude of the optical forces arising from it. This approach has previously been reported to estimate the optical forces to a greater accuracy on realistic nanoparticles and molecules [17]. Optical forces were studied between disc dimers for three different diameters 100 nm, 150 nm and 200 nm with a thickness $t$ of 50 nm. The forces were calculated along the inter-particle axis between the discs, which makes the x-component of the stress tensor $F_x$ relevant for our calculations. We kept a minimum distance of 10 nm between the nanostructures in our studies as multi-polar order plasmon coupling prevails at shorter distances [18,19], which could interfere with the nature of the attractive optical forces. In order to establish effective plasmonic coupling between the nanoparticles, the polarisation of the incident light was aligned along the inter-particle axis of the dimer. The excitation wavelengths were chosen for each case corresponding to their plasmon resonances, as local field enhancement maximises at resonant excitation.

As shown in Fig. 2, the optical forces grow in magnitude as the distance between the two nanodiscs decreases. Moreover, the influence of radius of the nanodiscs can be seen on the optical forces, as the optical forces grow further in magnitude with increasing radius of the nanodiscs for all separation distances. The largest attractive force of 5 pN/(mW/$\mu$m$^{2}$) was calculated for the nanodisc of radius 100 nm at 10 nm separation distance. The increase in optical forces with decreasing inter-particle distances follows as a consequence of greater plasmonic coupling between the discs, which has been previously observed for bipyramidal gold nano-dimers [20]. We calculated the local field enhancement of the electric field ($E=E_{np}/E_0$), where $E_{np}$ is enhancement of the nanoparticle and $E_0$ is the incident field enhancement, in 100 nm, 150 nm and 200 nm diameter nanodiscs respectively, to observe plasmonic coupling between the nanodiscs. A stronger plasmonic coupling can be seen to emerge at smaller gap distances between nanodiscs for all radii, which increases the local field enhancement (see Supplement 1, Note 3). To understand the nature of the optical forces exhibited by nanodisc dimers, we studied charge distribution on their surfaces. On exciting nanodiscs at their plasmon resonance conditions, opposite charged centers form on both nanodiscs on their upper and lower edges. The increase in the diameter of the nanodiscs leads to an increase in their polarisability, with a polarisation density of $4 \times 10^{-12}$ C/m$^{2}$ and $20 \times 10^{-12}$ C/m$^{2}$ in 100 nm and 200 nm diameter discs respectively, as observed from the polarisation density strength of the nanodiscs (see Supplement 1, Note 4). The distribution of charges on the nanodisc seems to be unaltered from the change in diameter. The increase in the dipolar strength with increasing diameter of the nano-dimer causes a stronger electrostatic force between nanodiscs of larger radii, resulting in a larger attractive force between them.

 

Fig. 2. Optical forces in gold nanodisc dimers of diameter 100 nm, 150 nm and 200 nm, with 10–50 nm inter-particle distance between the discs. Nanodisc dimers were excited under plane wave conditions with electric-field strength of 1 V/m. Each nanodisc dimer was excited at its respective plasmon resonance, deduced from its scattering spectra (given in Supplement 1, Note 2).

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The plane wave approximation condition often complies with a coherent source with an illumination area large enough to substantially exceeds the geometric scattering cross-sections of the scatterer under exposure. In real world scenarios, however, we often come across light sources with a Gaussian profile and a limited spot size. To draw relevance of our studies with excitation source conditions used in experimental set-ups, we calculated optical forces in the nanodiscs using Gaussian beam excitation conditions. We observed striking similarities in the inter-particle gap dependent trends and the overall magnitude of optical forces obtained by Gaussian beam excitation with that of optical forces calculated using plane-wave excitation conditions (see Supplement 1, Note 5). An attractive force of 4.57 pN/(mW/$\mu$m$^{2}$) was calculated in 100 nm radius nanodisc dimer under Gaussian beam excitation as compared to 5 pN/(mW/$\mu$m$^{2}$) under plane wave excitation. The observations provide a reasonable validation for exploring plasmonic enhanced optical forces under plane wave excitation conditions.

The next set of numerical studies on internal optical forces was performed on gold nanocube dimers as they display great potential for localisation and enhancement of the local electric field. Nanocube dimers of varying sizes and configuration have previously been explored for their ability to strongly confine the electromagnetic field around their corners and in the gap region between them [21,22]. Moreover, the localisation of the electric field enhancement in nanocubes can be externally tuned by varying the wavelength of the excitation source [23]. The nanocube’s edge length $h$ and inter-particle distance play an important role in realising the wavelength related tunability. The onset of wavelength based tunable effect was observed for nanocubes with edge-length exceeding 75 nm with inter-particle distance of 10 nm for a nanocube dimer arranged in a side-side configuration. We compared near-field enhancements in the nanocube dimers of edge length 100 nm and 150 nm with 10 nm inter-particle gaps. The maximum electric-field enhancement was achieved at resonant excitation wavelengths for both edge-lengths, with strong localisation of the electric-field present in the gap region between the nanocubes. On red-shifting the source wavelength from the plasmon resonance of the cube dimer, the hot spot between the adjacent faces of the nanocube dimer becomes inactive, leading to hot-spots present at the outer corners of the nanocube dimer becoming more prominent (see Supplement 1, Note 7). This change in the electromagnetic field enhancement occurs due to the density of oscillating dipoles becoming more localised at the outer edges of the nanocubes at off-resonant excitations.

To understand the effect of particle diameter on optical forces in nanocube dimers, we calculated optical forces in gold nanocube dimers by varying the edge-length, while keeping a constant separation distance of 10 nm. The excitation wavelength for each case was chosen corresponding to the plasmon resonance of the nanocube dimer. With increasing edge-length the optical forces between the nanocube dimers grow stronger in magnitude. An attractive force of 0.32 pN/(mW/$\mu$m$^{2}$) was present between nanocubes of 50 nm edge length, which further rises on increasing the edge-length to 150 nm as seen in Fig. 3. The optical forces realised in nanocube dimers are stronger in magnitude compared to nanodiscs. For nanodisc and nanocube dimer of same diameter 150 nm and 10 nm gap distance, attractive force of 3 pN/(mW/$\mu$m$^{2}$) in nanodiscs and 13 pN/(mW/$\mu$ m$^{2}$) in nanocubes was calculated respectively. Interestingly, optical forces in nanocube dimers were seen to exhibit a wavelength specific behavior unlike nanodiscs. The change in the optical forces vs wavelength (here now on referred to as spectral response of optical forces), in a nanocube dimer of 50 nm edge-length exhibits a broad peak which follows its scattering spectra as seen in Fig. 4. On increasing the edge-length of the nanocube dimer to 125 nm, the spectra of optical forces narrows down substantially, while the scattering spectra further broadens and exhibits additional peaks as seen in Fig. 4(b). This behavior in scattering properties is expected due to field retardation effects experienced for increase in the size of the nanoparticle. On the other hand the spectral response of the optical forces in nanodisc dimers follow the scattering spectra closely and does not substantially deviate from it, even for increasing radii of the nanodiscs (see Supplement 1, Note 8). The distribution of electric charges on the nanostructure’s surface could potentially be the prominent reason for such behaviour, which in case of a nanocube dimer is altered with variation in the edge-length. The increase in nanocube’s edge-length leads to the asymmetric distribution of charge over the nanocube’s surface, which causes narrowing of the spectral range of the attractive forces. To support our explanation, we studied the polarisation density for a nanocube dimer with an edge-length of 125 nm and 10 nm inter-particle separation to observe the charge distribution over it. The polarisation was seen to vary substantially within a small wavelength range around the plasmon resonance of the dimer, along with asymmetric distribution of the electric charges across the facing sides of nanocubes in the dimer configuration (see Supplement 1, Note 9).

 

Fig. 3. Optical forces in gold nanocube dimers of edge-length 50–150 nm at 10 nm inter-particle gap distance. The calculations were performed under plane wave excitation conditions using a field strength of 1 V/m. All cases of nanocube dimers were excited at their respective plasmon resonances

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Fig. 4. Spectral response of optical forces in nanocube dimers. a Response of optical forces with wavelength in nanocube dimers with 50~nm edge-length and 10~nm gap distance. b Response of optical forces with wavelength in nanocube dimers with 125~nm edge-length and 10~nm gap distance.

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We extended our studies of optical forces to the case of a nanobar dimer to gain a better quantitative understanding of the relation between charge distribution on the nanostructure’s surface and the spectral response of the resulting optical forces. Studying optical forces for a nanobar configuration allows the possibility to vary the length of the nanostructure while keeping the edge length and thickness constant. Choosing this geometry therefore offers insights into changes in distribution of charge on the nanobar dimer by varying its aspect ratio. The side-length $h$ and thickness $t$ of the nanobar dimer was fixed to 50 nm at 10 nm inter-particle separation, while the length $w$ was varied from 50 nm to 150 nm. As seen in Fig. 5, the optical forces increase quadratically with an increase in the aspect ratio, with an attractive force of 7.5 pN/(mW/$\mu$m$^{2}$) experienced by the nanobars of length 150 nm. This force is greater in magnitude in comparison to the strongest force observed in nanodisc 5 pN/(mW/$\mu$m$^{2}$). The stronger optical force in nanobar dimer stems from its higher dipolar strength of $26 \times 10^{-12}$ C/m$^{2}$, as compared to $20 \times 10^{-12}$ C/m$^{2}$ in nanodiscs, resulting in a stronger electrostatic interaction between the nanobars. The spectral response of the optical forces in a nanobar dimer exhibits a behaviour similar to what we observed in nanodisc dimers; it follows the scattering spectra closely, even for increasing aspect ratio of the nanobars (see Supplement 1, Note 8). The charges calculated across the nanobar dimer of highest aspect ratio 3, exhibit a dipolar like charge distribution with symmetric charge centers acquired by both nanobars (see Supplement 1, Note 10). Since varying the aspect ratio does not alter the side-length of the nanobars in our case, the symmetry of the charge distribution seems to be unaffected by changes made to the length. This observation reinforces our explanation that the narrow spectral response of optical forces follows as a consequence of the asymmetric distribution of the charges present on the nanostructure’s surface, which is uniquely displayed by nanocube dimers. Comparing the optical forces with the dipolar strengths in all three studied nanodimer systems, it is evident that higher polarisation densities result in larger optical forces (see Supplement 1, Note 11).

 

Fig. 5. a Optical forces in gold nanobar dimers of side length 50~nm, height 50~nm and length varied between 50--150~nm at 10~nm inter-particle gap distances. The calculations were performed under plane wave excitation conditions using a field strength of 1~V/m. All cases of nanobar dimers were excited at their respective plasmon resonances. b Optical forces in nanodisc dimer of radius 100~nm and nanocube dimer of edge-length 100~nm at 20~nm inter-particle distance. The calculations were performed under Gaussian beam excitation conditions in the non-paraxial approximation. Nanodiscs were excited at 625~nm and Nanobars at 670~nm. The spot size of the Gaussian beam was kept equivalent to the excitation wavelength for both cases respectively.

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The irradiance from the excitation source plays an important role in enhancing light-matter interactions in plasmonic nanostructures and can strongly influence the optical forces resulting from it, given the emitted power levels do not exceed the damage threshold of the structures under illumination. Metal nanostructures, in-particular gold, can resist changes to its optical properties upon radiant exposure upto a few hundred milli-watts [24], which makes it a promising candidate for studying effects of increasing power on the optical forces. Choosing an optimal geometry is equally important to fully explore the potential for boosting the optical forces in nanostructures. We compared the enhancement of optical forces in a gold nanodisc dimer of 100 nm diameter with a nanocube dimer of 100 nm edge-length, since the regions of electric field localisation differs in both cases due to geometry. The inter-particle distance for the dimer was kept 20 nm and the irradiance was controlled by using a Gaussian beam source. Usually the spot-size of Gaussian beams used in numerical studies are limited to micron sizes for visible wavelength ranges. Narrowing the beam spot to the order of the excitation wavelength offers the possibility for improving optical forces as a consequence of increase in the electromagnetic energy density. We resolved this constraint by employing the non-paraxial beam approximation condition for our Gaussian beam source, which has been introduced in recent version of COMSOL Multiphysics (Version 5.6), allowing 1:1 ratio of the beam spot to the excitation source wavelength. This formulation utilises the plane wave expansion methodology to generate a Gaussian beam with a tighter spot size (see Supplement 1, Note 12). The dimer nanoantennas were excited at their respective plasmon resonance conditions, which is 625 nm for the nanodisc and 670 nm for the nanobar. The deviation of the beam spot between the two cases of excitation was small enough to not externally influence the magnitude of the optical forces.

The relation of the attractive optical forces with power levels of the excitation source for a nanodisc and nanocube dimer can be seen in Fig. 5(b). It can be seen that in both type of nanostructure geometries the attractive forces increase linearly with the optical power. The attractive forces can reach upto 120 pN in a gold nanocube dimer and 100 pN in case of a gold nanodisc dimer for 100 mW power of the excitation source. A proportional increase in the optical forces is observed between the nanocube and nanodisc dimers, as a constant ratio of 1.167 in the optical forces is maintained between the two nanostructure geometries for all excitation power levels (1 mW to 100 mW). The local field enhancement in a nanocube dimer is stronger than in nanodiscs, which consequently leads to exertion of a stronger optical force in them. Such magnitudes of attractive optical forces show great potential for all-optical modulation such as actuation and transduction of plasmomechanical systems at the nanoscale.

The transduction of motion in nanopillars is decisive for their applications. Optical force acting on a plasmomechanical nanopillar can be approximated as a point load since the dimensions of a gold nanoparticle are quite smaller compared to the nanopillar itself. In the first bending mode of nanopillars, the maximum displacement amplitude lies at the free end of the pillar resonator [25], which goes well with placement of the gold nanoantenna on top of the nanopillar for producing maximum deflection. The stiffness of the nano-pillar is governed by its geometric moment of inertia, which depends on the pillar geometry. Since the motion of the nanopillars is along x-axis due to optical forces, the area moment of inertia for bending in $x$-direction is expressed as $I_y = \iint z^{2}$ dA. The deflection of pillars $x = F/k$ can be estimated from the ratio of the force applied $F$ to the stiffness of the pillars $k = 3 E I_y/L^{3}$, with the geometric moment of inertia $I_y = \pi d^{4}/64$ and $I_y = h^{4}/12$ of a disc and cube, respectively.

For estimation of maximum deflection in a nanopillar dimer, we compared nanodiscs and nanocube dimers of varying diameter at 10 nm inter-particle distance. Assuming silicon nanopillars with youngs modulus $E = 150$ GPa, we chose the length of pillars $L = 10~\mu m$ for our study, since a high dynamic range of 85–90 has been recently reported for pillars around this height [26]. As shown in Table 1, the magnitude of deflection in nanodisc dimers decreases with increase in the particle diameter, with a static deflection of 0.77 nm/(mW/$\mu$m$^{2}$) observed for 100 nm diameter pillars. A similar trend is also observed for nanocube dimers, where maximum deflection of 1.34 nm/(mW/$\mu$m$^{2}$) is observed for cubes with the shortest edge-length of 50 nm. Therefore, we realise stronger magnitude of displacement in nanocubes compared to nanodiscs. The observations reveal that rather than optical forces alone, the force-balance between the optical forces and the stiffness of the pillars determines the resulting deflection of the pillars. Our results indicate that designing slender pillars can produce maximum deflection of nanopillars at short inter-particle distances and favours the geometry of nanoantennas which can produce maximum local field enhancement. Our calculations reveal an improvement in deflection by 3 orders of magnitude over deflections estimated in pillar dimers actuated by electrostatic forces [27]. The results inform on the promising prospects of nanopillar antennas for plasmomechanical actuation and transduction, where the performance of the nano-pillars can be further optimised through improvement of their quality factors.

Table 1. Estimated deflection in silicon nanopillars with gold nanodimer on top and 10 nm inter-particle distance. Plane wave excitation conditions were used with 1 V/m electric-field strength.

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4. Conclusion

In summary, we presented detailed studies on optimising optical forces in gold nano-dimers. Our findings emphasize on the essence of electric-field enhancement for boosting optical forces arising from plasmonic interactions, and guide towards the design considerations which aid in maximising them. We observe the role of polarisability of gold nanoparticles as central in governing the nature of optical forces and show the strong influence of nanoparticle geometry on it. Using a novel numerical methodology, the influence of optical power was studied on the optical forces, signifying the role of the excitation source for amplifying their magnitude. Our results highlight the potential of optical forces realised in metal nanostructures for driving and reading out the motion of nanomechanical resonators, particularly when integrated with nanopillar structures. The all-optical actuation and transduction of nano-pillar antenna arrays has immense potential for ultra-sensitive mass and force sensing, optical modulation and reconfigurability, while offering low power demands and robust integration at the nanoscale.