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Abstract
Taking advantage of highly confined evanescent fields to overcome the free-space diffraction limit, we show plasmonic tweezers enable efficient trapping and manipulation of nanometric particles by low optical powers. In typical plasmonic tweezers, trapping/releasing particles is carried out by turning the laser power on and off, which cannot be achieved quickly and repeatedly during the experiment. We introduce hybrid gold-graphene plasmonic tweezers in which the trap stiffness is varied electrostatically by applying suitable voltages to a graphene layer. We show how the graphene layer absorbs the plasmonic field around the gold nanostructures in particular chemical potentials, allowing us to modulate the plasmonic force components and the trapping potential. We show graphene monolayer (bilayer) with excellent thermal properties enables more efficient heat transfer throughout the plasmonic tweezers, reducing the magnitude of thermophoretic force by about 23 (36) times. This thermophoresis suppression eliminates the risk of photothermal damage to the target sample. Our proposed plasmonic tweezers open up possibilities to develop tunable plasmonic tweezers with high-speed and versatile force-switching functionality and more efficient thermal performance.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The idea of exploiting light to capture and manipulate tiny objects was proposed by Arthur Ashkin first. In his seminal work in 1970, he showed that the radiation pressure from laser beams could accelerate dielectric microspheres [1]. However, for stable trapping of microspheres, he had to use either a laser beam and the chamber wall or two counter-propagating beams with the same characteristics. Sixteen years later, Ashkin demonstrated the trapping of dielectric particles using a single laser beam focused on a high numerical aperture (NA) objective lens [2]. He ascribed this achievement to a force that depends on the gradient of the laser beam intensity — i.e., the gradient force. Shortly after this development, the trapping and manipulating of living organisms like viruses and bacteria was realized [3].
While conventional optical tweezers are efficient tools for manipulating micron-sized objects [4,5], they are incapable of trapping sub-wavelength particles for various reasons [6,7]: The gradient force scales with the third power of the particle’s radius. Thus, the gradient force exerted on nanoparticles smaller than several tens of nanometers becomes insufficient to create a stable trap. To compensate for this force reduction, one may either increase the input power or tighten the beam focus to increase the light intensity. However, this would result in massive heat generation that may be harmful to the trapped objects, especially the biological samples. Besides, focusing light beyond the diffraction limit is not possible using conventional optics. Moreover, smaller particles experience more intense Brownian fluctuations due to reduced drag force. Thus, they are more likely to escape from the trap.
Plasmonic tweezers [8–17], usually employing metallic nanostructures for enhancing optical fields confined to tiny volumes, are promising alternatives to overcome the limitations of conventional optical tweezers. Despite having their drawbacks, including detrimental photothermal effects caused by absorption of light in metallic nanostructures, plasmonic tweezers can confine light beyond the diffraction limit and trap particles as small as several nanometers with low laser power. The intensity of the laser beam determines the trapping stiffness in typical plasmonic tweezers. Hence, the target particles are trapped/released by switching the laser on/off, generally. During the experiment, one cannot vary the laser intensity rapidly and repeatedly, which leads to a slow and inefficient force switching mechanism.
In our most recent work [18], we demonstrated how the excitation of localized surface plasmons (LSPs) in hexagonal arrays of gold triangles could lead to a significant increase in optical force and trap stiffness. In this work, we add a graphene sheet whose electrochemical potential can be tuned by a gate voltage [19–22] beneath the plasmonic array. Graphene absorbs the plasmonic field from the gold triangles, minimizing the plasmonic force and the depth of the trapping potential that a particle may experience. On the other hand, one can convert graphene to a non-absorbing material and increase the trapping-potential depth by applying a higher voltage. Moreover, we show that graphene with excellent thermal properties [23] enables more efficient heat transfer throughout the device, leading to less considerable temperature rise around the hotspots and less likely damage to the sample. The proposed scheme opens the path for the advancement of electrostatically tunable chip-based plasmonic tweezers with potential applications in biophysics.
2. Proposed device and operating principles
The proposed structure, as shown in Fig. 1, consists of hexagonal arrays of gold triangles on top of a graphene layer, which is separated by a thin dielectric layer from an ITO-covered coverslip as the substrate. On top of the plasmonic structures, a sample chamber accommodates polystyrene microspheres floating in the water, and the device is illuminated normally by a coherent y-polarized plane wave propagating in –z-direction. The graphene layer is biased via the top metallic contact and the bottom ITO layer, separated from graphene by a thin layer of dielectric. Using a transparent material such as ITO as a contact makes it possible to visualize the sample in transmission mode, adding a degree of freedom in designing the optical trapping and imaging setup.
Fig. 1. (a) 3D and (b) cross-section schematic illustration of the proposed tunable plasmonic tweezers.
2.1 Hexagonal arrays of gold triangles
In our earlier work [24], we reported the fabrication of hexagonal arrays of gold triangles, using angle-resolved nanosphere lithography (ARNSL), as a simple and efficient method with broad area coverage comparing to other methods such as electron beam lithography. In this technique, we utilize polystyrene microspheres as the shadow masks for creating the gold triangles on the substrate. Moreover, the radius of the polystyrene spheres used in this technique determines the feature size (l) of the gold triangles, and hence the LSPs resonant wavelength (i.e., the device working wavelength). Based on a simple 2D geometric calculation, we exploited polystyrene microspheres of radii r = 450 nm to create triangles of feature size l = 240 nm and thickness h = 50 nm with LSPs resonant wavelength of λ ≈ 1064 nm. Thus, an Nd: YAG laser, frequently used in biological applications, would be the most suitable optical source in our study.
2.2 Optical tunability of graphene
One may describe the optical behavior of graphene by its surface conductivity given by the simplified Kubo formula in the absence of an external magnetic field [25]. One can modulate the graphene's optical conductivity and its effective permittivity via tuning the graphene chemical potential (μC). Figure 2 depicts the variations in the real and imaginary parts of the effective permittivity (ε’ and ε'') versus the incident light wavelength for different graphene chemical potentials. As can be observed from this figure, over a specific range of wavelength absorbing graphene varies to a non-absorbing material and vice versa — e.g., for μC = 0.6 eV (blue dashes), in particular, ε'' = 0 for λ > 1100 nm, and ε'' ≠ 0 otherwise. This alteration is associated with the fact that the interband transitions only occur when ħω > 2μC. Moreover, for a given wavelength, the absorption coefficient of graphene changes by modulating the chemical potential. Specifically, at our working wavelength (λ ≈ 1064 nm), varying μC in the range of 0.4 −0.7 eV results in a significant variation in the absorption coefficient of graphene.
Fig. 2. (a) Real part and (b) imaginary part of the effective permittivity of graphene as a function of illumination wavelength and chemical potential of graphene.
2.3 Biasing mechanism
Applying desired voltages to graphene is performed via a top metallic contact and a bottom ITO layer separated from graphene by a thin oxide layer. Exploiting the parallel plate capacitance model [26], we determine the voltage required to create the desired change in the graphene chemical potential. Based on this model, a voltage V applied to a monolayer of graphene induces a change in charge carrier density of graphene:
where εd and td are the dielectric constant and the thickness of the gate oxide. This leads to modulation of graphene chemical potential: where ℏ and vf ≈ 106 m/s are the reduced Planck’s constant and the Fermi velocity in graphene. From Eqs. (1) and (2), one can calculate the required voltage to create the desired change in graphene chemical potential:Using Eq. (3), we calculate the required voltages for achieving a change of ΔμC = 0.3 eV for 5-nm thick SiO2, HfO2, TiO2, and hBN nanolayers as the typical gate dielectrics. The electrostatic dielectric strengths and the dielectric constants of these materials and the required applied voltages to induce the desired change in the graphene chemical potentials are given in Table 1. The choice of the gate dielectric depends on the experimental facility and the material availability. Nonetheless, we use SiO2 in the following simulations due to its simple fabrication procedure.
3. Simulation results
3.1 Scattering spectra
To study the optical characteristic of the proposed structure, we used a finite difference time domain (FDTD) method and have obtained the scattering spectra for the array of gold triangular nanostructures with dimensions of l = 240 nm and h = 50 nm. In doing so, we presumed a y-polarized plane wave propagating along the z-direction is incident upon the array, probing the backscattered light via a 2D z-normal monitor located at z0 = 10 nm above the gold triangles. For the refractive indices of SiO2 and water, we assumed nd = 1.5 and nw = 1.33, while using the material dispersions for gold and ITO according to those given in [29,30].
We have obtained the spectra for three different cases. (i) In the absence of a graphene layer and the presence of (ii) a graphene monolayer and (iii) a graphene bilayer. Figure 3 illustrates the results, with an inset depicting the schematic top view of the triangular gold array formed on a layer of graphene. This figure represents the scattering spectra in the absence of graphene (green dashed), in the presence of a single layer (magenta), and the presence of a double layer (blue), with chemical potentials of μC = 0.4 eV (solid) and 0.7 eV (dots). Comparing the scattering spectra from the structures with and without graphene confirms that the plasmonic peaks appearing at λ ≈ 1064 nm are all due to the LSPs exited on the gold nanostructures. Moreover, from the magenta and blue dots, we can realize that the incident light energy (at the LSPs resonance) is insufficient for contributing to the interband transitions in graphene (μC = 0.7 eV > ħω/2), making the graphene monolayer and bilayer incapable of absorbing the plasmonic field around the gold triangles. Nonetheless, the solid magenta and blue curves confirm the graphenes monolayer and bilayer of μC = 0.4 eV < ħω/2, owing to the dominance of their interband transitions, have partially absorbed the plasmonic field around the gold triangles. The same also applies to the chemical potentials μC <0.4 eV. In other words, an increase of ΔμC = 0.3 eV in the chemical potential of graphene with μC = 0.4 turns graphene from an absorbing material into a non-absorbing material for the incident light wavelength of λ ≈ 1064 nm. Furthermore, comparing the solid spectra indicates that a graphene bilayer absorbs more plasmonic fields than a monolayer does, for μC = 0.4 eV.
Fig. 3. Scattering spectra from an array of gold triangles (inset) in the absence of graphene (green dashes), in the presence of a single layer (magenta), and double layers (blue) of graphene with chemical potentials of μC = 0.4 eV (solid) and 0.7 eV (dots).
The dependencies of the scattering spectra on incident wavelength (500 nm ≤ λ ≤ 1500 nm) and graphene chemical potential (0.2 eV ≤ μC ≤ 0.8 eV) are indicated in Fig. 4. Particularly, for the operating wavelength (λ ≈ 1064 nm), one can observe the variation of the scattering spectra when the graphene chemical potential varies between μC = 0.4 eV and 0.7 eV. However, for μC ≤ 0.4 eV and μC ≥ 0.7 eV, the scattering spectra are nearly constant.
Fig. 4. Scattering spectra from an array of gold triangles patterned on double layers of graphene for various incident wavelengths (λ) and graphene chemical potentials (μC)
3.2 Plasmonic force and trapping potential
To study the trapping operation of the proposed system, one should evaluate the components of the average optical force exerted on a target particle, using the surface integral [31]:
Fig. 5. Schematic of a polystyrene particle of radius r = 500 nm, moving along x-direction above the hexagonal array of gold triangular structures (inset) at a height of z0 = 10 nm.
Figure 6(a) illustrates the plasmonic force components (Fx and Fz) induced by illumination of wavelength λ = 1064 nm and intensity I0 ≈ 0.9 mW/μm2, in the presence of a graphene monolayer (magenta) and a bilayer (blue) with chemical potentials of μC = 0.4 eV (solid for Fx and dots-dashes for Fz) and 0.7 eV (dots for Fx and dashes for Fz). Notice, owing to the structural symmetry, the y-component of the optical force is zero (Fy = 0) and hence not shown in the figure. The data shown in this figure indicate that both plasmonic force components (Fx and Fz) are enhanced when the graphene chemical potential is increased from μC = 0.4 eV to 0.7 eV that is consistent with the observation made in Fig. 3. It is worth noting that while |Fx| and |Fz| for the tweezers made of a double layer of graphene with μC = 0.4 eV are smaller than those of the tweezers made of a single layer of graphene with the same chemical potential, no difference can be observed in those force components in the trapping systems made of a graphene monolayer and bilayer with μC = 0.7 eV — i.e., the blue and magenta dots coincide and the same is true for the blue and magenta dashes. Moreover, the blue and magenta dashes and dot-dashes in the figure indicate that the z-component of the plasmonic forces for both tweezers and both chemical potentials are negative (Fz < 0) over the entire region of the interest, pulling the microsphere toward the surface of the nanostructures. It is worth mentioning that forces such as thermophoresis, fluidic lift, and electrostatic mechanisms oppose the z-component of the plasmonic gradient force preventing the particles from reaching the array top surface. Thus, to take the effects of all these opposing forces into account in our numerical simulations and force calculations, we considered a 10-nm spacing between the target particle bottom and the array surface, which is somewhat longer than the Debye length [32,33]. Moreover, the sign of Fx changes from positive to negative at x = 0 (going from −x to + x) for both chemical potentials, as indicated by dots and solid curves.
Fig. 6. (a) Components of the plasmonic force (Fx and Fz) exerted on a polystyrene microsphere of radius r = 500 nm passing along the x-direction above the sample top surface at y = 0 and z0 = 10 nm in the presence of a single layer (magenta) and double layers (blue) of graphene with chemical potentials of μC = 0.4 eV (solid for Fx and dots-dashes for Fz) and 0.7 eV (dots for Fx and dashes for Fz). (b) Trapping potentials versus x for a single layer (magenta) and double layers (blue) of graphene with chemical potentials of μC = 0.4 eV (solid) and 0.7 eV (dots). I = 0.9 mW/μm.
Integrating over this component results in a potential well whose depth is the measure of the tweezers trapping capability:
Here, we take the −10kBT as the stable trapping threshold [15,17,22]. Using Eq. (6) and the data for Fx shown in Fig. 6(a), we have calculated the potential wells along the x-direction for a monolayer (magenta) and a bilayer (blue) of graphene with chemical potentials of μC = 0.4 eV (solid) and 0.7 eV (dots), as depicted in Fig. 6(b). Comparison of the data shown in this figure indicates that the potential well for the tweezers composed of a graphene bilayer with μC = 0.4 eV is shallower than that a particle feels in the tweezers made of a graphene monolayer with the same chemical potential. Comparison of the data shown in this figure indicates that the potential well for the tweezers composed of a graphene bilayer with μC = 0.4 eV is shallower than that a particle feels when in the tweezers made of a graphene monolayer with the same chemical potential. Neither tweezers for the given conditions can trap the 500-nm PS particle efficiently (i.e., |U| < 10 kBT), though the potential wells for both tweezers deepen when the graphene chemical potentials increase by ΔμC = 0.3 eV, enabling efficient traps (i.e., |U| > 10 kBT). These observations suggest that one can trap and release the target PS microsphere via electrostatic gating of the graphene monolayer (bilayer) through varying the applied voltage without the need to turn the laser on and off.
3.3 Photothermal effects
Light absorption by metallic nanostructures generates heat in plasmonic tweezers, hindering the stable trapping of target objects, albeit their numerous advantages over their optical counterparts. To mitigate the heat generation issue, one can utilize plasmonic materials with high thermal conductivity to quickly transfer the generated heat into the substrate. In our proposed device, we considered arrays of gold nanostructures above a graphene layer whose thermal conductivity is more than six times larger than that of gold, allowing fast and efficient heat transfer from the plasmonic hotspots into the substrate. Here, we aim to show the advantage of using a graphene layer beneath the gold triangular nanostructures. In doing so, we used the finite element method (FEM) to solve the steady state heat transfer equation in our proposed structure [34]:
where ρ, Cp, u, and T are the density, the specific heat capacity at constant pressure, the velocity vector, and the medium temperature, q = −k∇T represents the heat flux by conduction, k is the thermal conductivity, and Q is the heat sources including plasmonic heating. The parameters used in the FEM simulations are given in Table 2.Table 2. The mass density (ρ), thermal conductivity (k), and heat capacity (Cp) of the materials used in the FEM simulations.
Using Eq. (7), we evaluated the temperature rise within the device structure in the absence and presence of graphene, as illustrated in Fig. 7. The numerical results in Fig. 7(a) and 7(b), representing the temperature distributions in the x-y and y-z planes, reveal that the enhanced plasmonic fields at the hotspots elevate the heat by ∼45°C from the room temperature (293 K) when there is no graphene. Since we employed a y-polarized plane wave to excite the LSPs in the array, the plasmonic field, and hence the temperature is further enhanced on the upper and lower gold triangles along y-direction in the array (Fig. 7(a)). Notice, by appropriately rotating the polarization of the incident light in the x-y plane, one can transfer the plasmonic hotspots to the other pairs of triangles, as desired. On the other hand, as depicted in Fig. 7(c) and 7(d), in the presence of a graphene monolayer, the temperature elevation at the hotspots is only ∼11°C. In other words, a graphene monolayer suppresses the photothermal effects and lowers the risk of damaging the sensitive target objects. Moreover, results presented by Fig. 7(e) and Fig. 7(f) show that the presence of a graphene bilayer suppresses the heat even further, lowering the medium temperature rise down to ∼7°C.
Fig. 7. Spatial distribution of the temperature in the proposed device plotted in the x-y (a, c, and e) and the y-z (b, d, and f) planes without graphene (a, b), with a single layer (c, d) and bilayer (e, f) of graphene. The initial condition for the temperature is considered 20°C (or 293 K).
To assess the stability of the plasmonic trap, one can calculate the stochastic Langevin force, Fth(t), at any instant of time t, which is responsible for the Brownian motion of the particles, from its correlation function,
According to [33,39], a spatial gradient in the environment temperature (∇T) exerts a thermophoretic force, FT = −γvT, on a particle. Here, vT = − DT∇T represents the steady state thermophoretic velocity, and DT is the thermophoretic mobility that depends on the temperature [40]. The magnitude of the thermophoretic force exerted on a polystyrene microsphere of 500 nm radius in the system without graphene is about several hundreds of fN - i.e., comparable to the magnitude of the plasmonic force components shown in Fig. 6(a). Hence, this thermophoretic force hinders the stable trapping/releasing function of the tweezers. Nonetheless, the presence of a graphene monolayer (bilayer) can reduce the thermophoresis down to ∼31 (19.5) fN and enables a more reliable trapping/releasing system. To provide a better comparison, we reported the amplitudes of the Langevin and the thermophoretic forces as well as the plasmonic force components in Table 3.
Table 3. The amplitude of the thermal forces ($F_{\textrm{th}}$ and $F_{\textrm{T}}$) and the plasmonic force components (Fx and Fz) in the absence and presence of graphene monolayer and bilayer.
4. Conclusion
We proposed a tunable plasmonic tweezers system in which the optical force components and the potential depth can be switched between trapping and releasing conditions electrostatically. By varying an externally applied voltage, we can modulate the graphene chemical potential underneath the hexagonal arrays of gold triangles. For specific chemical potentials, graphene functions as an absorber for the plasmonic field scattered from the gold nanostructures, reducing the plasmonic force components and the trapping potential sensed by the target particle. One can restore the initial values of force and trapping potential by simply applying the initial bias voltage to graphene. This proposed structure enables the fabrication of tunable plasmonic tweezers in which tiny objects can be trapped and released electrostatically without switching the laser power on and off. Finally, we compared the photothermal effects in our proposed device with those in a similar device without the graphene layer(s). The comparison reveals the graphene layer(s) underlying the plasmonic nanostructures absorb(s)the generated heat and facilitates the transfer of the corresponding thermal energy into the substrate, reducing the risk of photothermal damage to the target particles. We calculated the stochastic Langevin force and the thermophoretic force to evaluate the influence of heat on the tweezers’ functionality. Our results show the amplitude of the stochastic Langevin force, both in the absence and presence of the graphene layer(s), is negligible compared with the calculated optical force peaks. On the other hand, the magnitude of the thermophoretic force in the absence of graphene is significant, impeding the tweezers’ ability. Nonetheless, the presence of a graphene monolayer (bilayer) reduces the magnitude of the thermophoretic force by ∼ 23 (36) times and guarantees stable trapping of the target particle.
Funding
Tarbiat Modares University (IG-39703).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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