# Integrated Multifunctional Graphene Discs 2D Plasmonic Optical Tweezers for Manipulating Nanoparticles

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

_{f}= 0.6 eV under incident intensity I = 1 mW/μm

^{2}, which has a very low incident intensity compared to other plasmonic tweezers systems. The optical forces on the nanoparticles can be controlled by modulating the position of LSPR excitation. Controlling the position of LSPR excitation by bias voltage gates to configure the Fermi energy of graphene disks, the nanoparticles can be dynamically transported to arbitrary positions in the 2D plane. Our work is integrated and has multiple functions, which can be applied to trap, transport, sort, and fuse nanoparticles independently. It has potential applications in many fields, such as lab-on-a-chip, nano assembly, enhanced Raman sensing, etc.

## 1. Introduction

## 2. Structure Design and Modeling

^{2}. The material of the insulator substrate layer is designated as BaF

_{2}(n = 1.45), which has high transmittance in the mid-IR and has been proven to be applicable in graphene optical tweezer systems [23]. We choose polyethylene nanoparticles for research because dielectric nanoparticles are widely used in the biological and medical fields [48]. When the Fermi energy of a single graphene disk is 0.6eV, the component of the optical force on the particle (radius 50 nm refractive index 1.6, positioned at 20 nm above the interface) is displayed in Figure 1c. Continuous monolayer graphene can be generated using the chemical vapor deposition (CVD) technique and transferred onto the substrate by the graphene transfer method [49]. Monolayers of graphene layers can be patterned into graphene disk arrays using focused ion-beam (FIB) [50]. The challenge of processing is to deposit a fixed thickness of the insulating layer over the graphene disk on the substrate.

_{0}and ε

_{d}are the dielectric constant and the thickness of the gate oxide. ${V}_{0}$ is the offset voltage caused by natural doping. ${v}_{\mathrm{f}}$ ≈ 10

^{6}m/s is Fermi velocity. t

_{d}is the distance from the graphene disc to the bias voltage gate. We use the Maxwell stress tensor to calculate the time-averaged optical force on the nanoparticle [54,55].

**S**is the outer surface of the nanoparticle and

**n**is its discrete normal vector, $\langle \rangle $ represents the time average. 〈T〉 is the Maxwell stress tensor which can be governed as:

**E**and

**H**are the intensity vectors of the electric and magnetic fields on the surface of the particle, respectively. Through Helmholtz Hodge decomposition, the optical force can be decomposed into conservative force and non-conservative force. We only take the conservative force into consideration [56,57]. The trapping potential in each direction can be approximately calculated as the line integral of the component of the optical force on the nanoparticle [58]:

## 3. Results and Discussion

_{0}= 1 mW/µm

^{2}, which has much lower than other plasmonic optical tweezer systems [42,43]. The optical forces on the nanoparticles and the trapping potential are calculated by the time-averaged tensor (Equations (3) and (4)) and the line integral calculation (Equation (5)), respectively. Figure 3a shows the components of the optical force as a function of the nanoparticle centered along the x-axis, with y-center = 0 and z-center = 50 nm. When x-center = −160 nm~160 nm, F

_{z}is always negative (F

_{z}< 0 pN), which proves that the nanoparticle will be pulled to the interface between the dielectric material and the trapping environment. When x-center > 0, F

_{x}is a positive force (F

_{x}> 0 pN), x-center < 0, F

_{x}is a negative force (F

_{x}< 0 pN), x-center = 0, F

_{x}= 0 pN. This means that the nanoparticle will be trapped by the optical force in the center of the graphene disk along the x-axis. As the result shown in Figure 3c, by the line integral of the force along the x-axis, we calculate the trapping potential of the nanoparticle along the x-axis, which is greater than 10 K

_{B}T to ensure stable trapping. Corresponding to the graphene disks diagram at the top of the figure, the trapping region in the x-axis direction contains the entire area of the adjacent graphene discs. This demonstrates that, by dynamically configuring the Fermi level of graphene, the motion of nanoparticles can be controlled along the x-axis. Figure 3b shows the components of the optical force as a function of the nanoparticle centered along the y-axis, with x-center = 0 nm and z-center = 50 nm. F

_{y}is a positive force (F

_{y}> 0 pN), y-center < 0, F

_{y}is a negative force (F

_{y}< 0 pN), y-center = 0, F

_{y}= 0 pN. This means that the nanoparticle will be trapped by the optical force in the center of the graphene disk along the y-axis. As the result shown in Figure 3d, by the line integral of the force along the y-axis, we calculate the trapping potential of the particle along the y-axis, which is greater than 10 K

_{B}T to ensure stable trapping. Corresponding to the graphene disks diagram at the top of the figure, the trapping region in the y-axis direction contains the entire area of the adjacent graphene discs. This demonstrates that, by dynamically configuring the Fermi level of graphene, the motion of nanoparticles can be controlled along the y-axis. Figure 3e shows optical forces Fz on nanoparticles with different radius radii as the nanoparticle centered along the z-axis, with x-center = 0 nm and y-center = 0 nm. d is the distance from the bottom of the nanoparticle to the graphene disc. It can be found that the larger the radius of the nanoparticle the greater the optical force on it. By the line integral of the force along the z-axis, we calculate the trapping potential of the nanoparticle on the x-axis. The result is shown In Figure 3f. When the radius of the nanoparticle is less than 50 nm, the trapping potential is less than 10 K

_{B}T. This can be improved by appropriately increasing the incident light intensity.

_{f}= 0.6 eV) under an incident light intensity I

_{0}= 1 mW/µm

^{2}. Figure 5a shows the temperature distribution. Graphene discs generate less electromagnetic heat than metallic materials because of their two-dimensional molecular composition (with a thickness of 0.34 nm) and high thermal conductivity (${\kappa}_{\mathrm{Graphene}}=2000W{K}^{-1}{m}^{-1},{\kappa}_{\mathrm{Au}}=314W{K}^{-1}{m}^{-1}$) at room temperature [59]. As is shown in Figure 5a, maximum rise temperature ΔT = T

_{s}− T

_{e}. T

_{s}is the temperature at each point when the simulation reaches a steady state. T

_{e}is the operating room temperature. ΔT increases only 2 °C compared to room temperature. To estimate the thermal behavior in the proposed system, The thermally induced convective distribution is represented in Figure 5b, including the fluid amplitude distribution (color map) and the fluid vector (arrows). The maximum convection velocity vectors can reach v = 0.1 nm/s. Flowing water is a radial symmetrical cycle that travels outward and upward. Driven by gravity and buoyancy, a Rayleigh-Benard system is formed [60,61]. This can improve trapping efficiency and prevent particles from sticking to the graphene disk.

^{−5}${\mathrm{sm}}^{-1}$ is the dynamic viscosity of the water. The convective force on the nanoparticle can be described by Stokes’ drag force equation, in which v is the fluid convection velocity vector. The thermophoretic force on the nanoparticle in the temperature gradient can be calculated using the formula in the table. $\nabla T$, ${D}_{T}$, are the thermal field gradient and thermophoretic mobility at a steady-state, respectively. For nanoparticles with a radius of 50 nm and a refractive index of 1.6 (polyethylene) in water, we assume $\nabla T\approx 55.6K\mu {\mathrm{m}}^{-1}$, ${D}_{T}\approx 1.55\mu {\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ [43]. We also evaluate the gravity force of the nanoparticle assuming the density of $\rho =950\mathrm{kg}/{\mathrm{m}}^{3}$ (polythene). As the results are shown in Table 1. The other kinds of forces on the nanoparticle are less than two or three orders of magnitude compared to the optical force during trapping time, and they have no significant effect on the trapping process of our optical tweezers.

_{0}= 1 mW/µm

^{2}. Through the previous discussion, the graphene disk in the ON state can produce strong plasmonic resonance to trap the nanoparticles in the middle, but the OFF state cannot be enhanced. The function of the system is achieved by switching the graphene disc ON and OFF states. Here we use a simplified Langevin equation for the simulation [64]:

_{B}, and T are the time step, Boltzmann’s constant, and the temperature of the environment, respectively. $W(t)$ is a vector of Gaussian random numbers, whose average is 0 and its variance is 1. To simplify the calculation, the optical force is calculated by the dipole method, with the nanoparticle approximated as an electric dipole [55,63].

^{5}times. The nanoparticles are released at moment 0, and the initial positions are set at (−180 nm, 180 nm), (−180 nm, −180 nm) in the x-y plane (z = 50 nm plane), respectively. As is shown in Figure 6a,b, at the moment of 15 us, the two nanoparticles move to the trapping positions driven by optical forces. The motion of nanoparticles tends to steady-state at t = 30 us. The nanoparticles are stably trapped in two trapping potentials. As is indicated in Figure 6c,d, by controlling the switching state of the graphene disk (Fermi energy), the optical field and the trapping position of the system are readjusted. The nanoparticle moves toward the new trapping position (t = 45 us) and achieves a steady-state at t = 60 us. As is illustrated in Figure 6e,f, the motion of the nanoparticles quickly responded to the change of the electric field, and the blue and red nanoparticles move to the same trapping region at t = 90 us and converged to the steady-state. The trap, transport, and fusion of the nanoparticles are achieved. A larger array of graphene discs can be designed to manipulate the motion of more than two nanoparticles independently.

## 4. Conclusions

_{0}= 1 mW/µm

^{2}. Finally, we demonstrate its ability to trap, transport, and fuse nanoparticles with a 3 × 3 array of graphene discs by means of the Langevin equation. We anticipate that it will become a new tool for nanoparticle manipulation and open new directions for its wide range of potential applications in lab-on-chip, nano-assemble, enhanced Raman sensing, etc.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) 3D schematic diagram of Tunable graphene disks 2D plasmonic tweezers for trapping and transportation of nanoparticles. (

**b**) Schematic of the cross-section of single graphene disc structure for optical trapping of nanoparticles. (

**c**) MST optical forces on a dielectric nanosphere. (

**d**) Band structure of graphene electrodes under a voltage bias.

**Figure 2.**(

**a**) The normalized intensity enhancement map under different Fermi energy and incident wavelengths. (

**b**) The normalized intensity enhancement under different Fermi energy (E

_{f}= 0.1 eV, E

_{f}= 0.6 eV), and corresponding simulated amplitude distribution. (

**c**) Simulated amplitude distribution in the 50 nm x-y plane above the graphene disc. The arrow (white) indicates the direction of the current. (

**d**) Simulated amplitude distribution in the x-z plane after the introduction of the nanoparticle.

**Figure 3.**The components of the optical force on the nanoparticle (n = 1.6) and the corresponding trapping potential are calculated by the MST method. (

**a**,

**c**) The center of the nanoparticle is on the x-axis. (

**b**,

**d**) The center of the nanoparticle is on the y-axis. (

**e**,

**f**) The center of the nanoparticle is in the x-axis, z-center, r = 30, 40, and 50 nm, respectively.

**Figure 4.**The optical forces on nanoparticles of different materials in different environment materials, Δn = n

_{p}− n

_{e}. n

_{p}and n

_{e}are the refractive indices of the nanoparticles and the trapped environment, respectively.

**Figure 5.**(

**a**) Heat power dissipation density Around the graphene disk, (

**b**) the steady-state fluid velocity and fluid vector under the incident light intensity I

_{0}= 1 mW/µm

^{2}.

**Figure 6.**Trajectories of two nanoparticles (blue and red) in a 3 × 3 graphene disk array, and a color map of the trapping potential. The blue line and the red line are the real-time movement tracks. The circular array corresponds to the switching state of the 3 × 3 graphene disc array, the solid circle corresponds to the ON state, and the hollow circle corresponds to the OFF state. (

**a**–

**f**) are trajectories in different switch configurations and at different times, 15, 30, 45, 60, 75, and 90 us, respectively.

Force | Method | Maximum (pN) |
---|---|---|

Optical force | MST | 3.25 |

Brown motion force | ${F}_{B}=\sqrt{2{k}_{B}T\gamma}$ | 2.6126 × 10^{3} |

Drag force | ${F}_{D}=\gamma \upsilon $ | 8.378 × 10^{3} |

Thermophoretic force | ${F}_{T}=\gamma {D}_{T}\nabla T$ | 7.221 × 10^{2} |

Gravity force | ${F}_{g}=\frac{4}{3}\pi {r}^{3}\rho $ | 4.974 × 10^{3} |

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Yang, H.; Mei, Z.; Li, Z.; Liu, H.; Deng, H.; Xiao, G.; Li, J.; Luo, Y.; Yuan, L.
Integrated Multifunctional Graphene Discs 2D Plasmonic Optical Tweezers for Manipulating Nanoparticles. *Nanomaterials* **2022**, *12*, 1769.
https://doi.org/10.3390/nano12101769

**AMA Style**

Yang H, Mei Z, Li Z, Liu H, Deng H, Xiao G, Li J, Luo Y, Yuan L.
Integrated Multifunctional Graphene Discs 2D Plasmonic Optical Tweezers for Manipulating Nanoparticles. *Nanomaterials*. 2022; 12(10):1769.
https://doi.org/10.3390/nano12101769

**Chicago/Turabian Style**

Yang, Hongyan, Ziyang Mei, Zhenkai Li, Houquan Liu, Hongchang Deng, Gongli Xiao, Jianqing Li, Yunhan Luo, and Libo Yuan.
2022. "Integrated Multifunctional Graphene Discs 2D Plasmonic Optical Tweezers for Manipulating Nanoparticles" *Nanomaterials* 12, no. 10: 1769.
https://doi.org/10.3390/nano12101769