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Abstract
We have investigated the effect of enhanced optical force via the acoustic graphene plasmon (AGP) cavities with the ultra-small mode volumes. The AGP mode can generate stronger field confinement and higher momentum, which could provide giant optical force, and has no polarization preference for the optical source. We have demonstrated that the trapping potential and force applied on polystyrene nanoparticle in the AGP cavities are as high as -13.6 × 102 kBT/mW and 2.5 nN/mW, respectively. The effect of radius of rounded corners and gap distance of AGP cavities on the optical force has been studied. Compared with an ideal nanocube, nanocube with rounded corners is more in line with the actual situation of the device. These results show that the larger radius of nanocube rounded corners, the smaller trapping potential and force provided by AGP cavities. Our results pave a new idea for the investigation of optical field and optical force via acoustic plasmon mode.
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1. Introduction
The optical force, produced by the exchange of momentum between light and object, can be used to manipulate miniature particles [1]. The optical tweezer technologies proposed by Ashkin used to capture particles [2], atoms and biological cells have been widely applied in a variety of fields, such as cellular biology [3], physical chemistry [4,5], MEMS [6,7] and other fields [8–10]. Unfortunately, traditional optical tweezers can only capture particles of the same magnitude as the wavelength due to laser intensity and diffraction limit [11]. Recently, the control, enhancement and confinement of surface optical field by surface plasmons (SPs) provide a golden opportunity for the development of optical tweezers technology, which break the diffraction limit and laser intensity restrict for particles captured, and advance the optical manipulation to the subwavelength scale [12–14]. Despite this, the application of SPs-based optical tweezers still posses many challenges, and weak intensity is one of them. The existing metal-based structures have difficulty for providing giant field enhancement [15], while the two-dimensional material waveguide structures are difficult to form effective hot spot, which are adverse to the capture of particles stably [16–18].
Graphene plasmons (GPs), the collective oscillations of electrons at the interface between the dielectric and graphene sheet, have attracted a great deal of attention owing to their giant field confinement, and electrical tunability [19–22]. GPs are widely implemented in the field of optoelectronics due to their impressive properties [23–26]. Additionally, GPs coupled with their mirror charges formed a highly confined asymmetric electromagnetic mode as the graphene sheet is placed close to a metallic surface, known as acoustic graphene plasmon (AGP) mode [27–29]. Compared with GPs mode, stronger field confinement and higher momentum could be generated by AGP mode [30]. AGP mode generated inside the gap between graphene sheet and metallic surface can confine the electromagnetic field to virtually 1/300 of its equivalent free space wavelength [31], and also has smaller damping than traditional GPs mode [32]. AGP mode is highly promising in the fields of medicine, biotechnology and security due to these capabilities [33,34]. Epstein et al. successfully fabricated single nanoscale AGP cavities with ultra-small mode volumes and high momentum in experiment, which does not require pattern of graphene, and with no dependence on the polarization of light [26]. Predictably, the strong field confinement and high momentum of the AGP mode can provide an effective way for the enhancement of optical force. However, there is no in-depth investigation based on AGP mode in the field of optical manipulation at present.
In this paper, the enhancement of optical force via AGP cavities consisting of single nanocubes and graphene sheet is investigated. The optical forces of AGP cavities are investigated by the gradient force (GF) theory and Maxwell’s stress tensor (MST) method, and the results of the two ways show good consistency. The total optical force produced by a single AGP cavity reaches the maximum of 12.9 nN/mW as the gap distance of AGP cavity g = 2 nm. A pair of hot pots are generated at either edge of the gap of AGP cavity, which provide stronger trapping forces and more effective and uneven trapping potentials that are more conducive to trap the nanoparticles. The maximum trapping potential Ux and force Fx applied on a single polystyrene nanoparticle reach -13.6 × 102 kBT/mW and 2.5 nN/mW as the gap distance g = 2 nm. The trapping potential and force generated by AGP cavity are two orders of magnitude greater than those previously reported for graphene-based devices [34–37]. Such an order of magnitude of trapping force and potential are sufficient to overcome the Brownian motion of the captured particles. Subsequently, we investigate the effect of radius of rounded corners on the optical force of AGP cavities. The results show that the trapping potential and force provided by AGP cavities decrease sharply with the increase of radius of rounded corners. The variations of trapping potential and force with gap distance between nanocube and graphene sheet are also investigated. Our results may pave a new idea for the investigation of optical field and optical force via acoustic plasmon mode.
2. Structure and analysis
Experimentally, patterning of the graphene into stripes over the source drain contacts by using Laser Writer lithography, a positively charged poly (allylamine) hydrochloride (PAH) polymer with a thickness around 1 nm is then used as the connecting spacer to facilitate the binding between the nanocubes and graphene/h-BN [26]. For computational convenience, we consider a simpler model. Acoustic graphene plasmon (AGP) cavities are created by dispersing metallic nanocubes, with random locations and orientations, over monolayer graphene sheet, as shown in Fig. 1(a). A air gap exists between the nanocube and graphene sheet. The structure has no preference for polarization, which also confirms the random nature of the nanocubes, meaning that the interaction between adjacent nanocubes is very weak [27]. Therefore, the electromagnetic field near the AGP cavity can be effectively confined within its gap. We concentrate chiefly on the scattering cross section and optical force spectra response of single AGP cavity in subsequent investigations.
Fig. 1. (a) Schematic of the AGP cavities, which is composed of monolayer graphene and metallic nanocubes. (b) Cross section diagram of a single AGP cavity. The side length of nanocube L = 75 nm.
Figure 1(b) shows the cross section diagram of a single AGP cavity. AGP cavity is excited by the light source from far field, and the polarization direction of the source is along the graphene sheet. The power of the incident light source is 1 mW. The gap distance between the metallic nanocube and graphene sheet is g, and the side length of nanocube L = 75 nm. The Drude model provides an instructive approximation to the conductivity of graphene [38]:
where the relaxation time τ = 0.5 × 10−12 s, Fermi energy EF = 0.6 eV. The spectral response of AGP cavity can be tuned by regulating the Fermi energy of graphene. The two-dimensional conductivity model of graphene is imported into finite-difference time-domain (FDTD) simulations. Perfectly matched layer (PML) boundary conditions are taken and the mesh sizes are chosen to be 1 nm × 0.2 nm.3. Results and discussions
Perfect electric conductor (PEC) can be substituted for metals (Ag, Au) within the wavelength range that is of interest to us [39,40], as illustrated in Fig. 2(a). The scattering cross section (left) and optical force (right) spectra of different AGP cavities (Ag: dash curve, Au: dot dash curve, PEC: solid curve) have been investigated with the gap g = 2 nm between nanocube and graphene sheet. The scattering cross sections of metals (Ag, Au) and PEC are perfectly consistent. Meanwhile, the optical force spectra of metals and PEC AGP cavities are nearly equal, and the resonance wavelength is λ = 14 µm. It is feasible to replace metals with PEC in the spectra range of 5–20 µm.
Fig. 2. (a) The scattering cross section and optical force spectra of AGP cavities with different metarials. (b) The distributions of electric field intensity |E|2, electric field component Ey and magnetic field components Hz at the resonance wavelength of 14 µm. (c) Calculated normalized mode areas of the AGP cavity in the MIR (blue ball-curve) for different gap distance g, and its comparison to the NCoM cavity in the visible spectrum (red ball-curve).
Then, the distributions of electric field intensity |E|2, electric field component Ey and magnetic field components Hz at the resonance wavelength of 14 µm are investigated, as depicted in Fig. 2(b). The electric field distributions and field enhancement of different AGP cavities are identical. It can be seen that the intensity of localized electric fields within the gap of AGP cavity reaches 5000, and hot spots are distributed at both edges of the gap. The magnetic field components Hz of different cavities show that the magnetic field of metal-AGP cavity is distributed in the gap and nanocube, while that of PEC-AGP cavity is completely confined in the gap. Compared with the enhanced electric field intensity, the magnetic field intensity is much smaller, so it can be considered that the magnetic field inside the metal nanocube does not contribute to the field enhancement of the gap. Moreover, it can be seen from the scattering cross section and optical force spectra that the magnetic field inside the nanocube has no influence on the results of our study. Thus, PEC can be substituted for metals (Ag, Au) within the wavelength range that is of interest to us. Since we are using a 2D FDTD simulation, therefore, the mode area is calculated from Am = Wm/max [W(r)], where Wm and W(r) are the integrated electromagnetic energy over the entire space and the local energy density at the position r, respectively [41]. The mode areas of the AGP cavity and Au nanocube-on-metal (NCoM) system have been calculated. The excitation wavelengths of the AGP cavity and NCoM cavity are λA0 = 14 µm, and λN2 = 0.55 µm, respectively. The refractive index for the background medium is n = 1. Figure 2(c) shows the calculated normalized mode areas Am/(λ0/n)2 of a single AGP cavity with different gap distance g in MIR spectrum (blue ball-curve) and the equivalent NCoM system where graphene is replaced by a gold surface in the visible spectrum (red ball-curve). As shown in Fig. 2(c), The mode area of the AGP cavity is three orders of magnitude smaller than that of the NCoM cavity. Compared with the traditional NCoM system, the AGP cavity has a ultra-small mode area.
The gradient force (GF) theory is considered to analyze the total optical force obtained by AGP cavity for the first time. The operating wavelength is 14 µm. The effective refractive index neff of AGP cavity can be obtained by solving the dispersion relation [42]:
To validate the theoretical results, Maxwell’s stress tensor (MST) method is employed to calculate the optical force [11]. The optical force generated by AGP cavity can be calculated by integrating the MST around any arbitrary surface S enclosing the graphene sheet. The optical force in the y direction is given by
Taking Fermi energy of graphene EF = 0.6 eV, we calculate the effective refractive index and optical force for different gap distances g, as depicted in Figs. 3(a) and 3(b), and the curves and balls denote the theoretical analysis via GF theory and numerical calculation by MST method. The theoretical results agree well with the results of numerical calculations. Both the real and imaginary parts of the effective refractive index decrease with the gap distance g increases, which means that both the confinement and loss of this cavity decrease, as shown in Fig. 3(a). Figure 3(b) shows the variation of optical force with gap distance g. The coupling between nanocube and graphene is weakened, and the confinement of the AGP mode decreases as the gap distance increases, resulting in an exponential attenuation of the optical force. The optical force has reached 12.9 nN/mW as the gap distance g = 2 nm.
Fig. 3. (a) Real and imaginary parts of the effective refractive index neff of AGP cavity with different gap distances g. (b) Optical force generated by AGP cavity with different gap distances g. The trapping potential and trapping force of (c) the coupled graphene strips system and (d) AGP cavity with gap g = 2 nm. The operating wavelength is 14 µm.
The giant electromagnetic energy localized in the gap region can effectively capture individual nanoparticle. The distribution of the electrical field gradient determines the magnitude of optical trapping force:
where Ux is the optical trapping potential, α is the polarizability of particle in the Rayleigh limit. nm = 1 refers to the background medium [41]. The radius and refractive index of polystyrene nanoparticle are rp = 1 nm and np = 1.59. To compare the difference between GPs mode optical tweezers and AGP mode optical tweezers, we calculate the trapping potential and trapping force of the coupled graphene strips system, where the width and Fermi energy of graphene strips are 75 nm and 0.6 eV, respectively. As shown in Fig. 3(c), the trapping potential Ux and force Fx generated by the coupled graphene strips system are -12 kBT/mW and 11 × 10−3 nN/mW as the gap distance g = 2 nm, respectively. As depicted in the inset of Fig. 3(c), the electromagnetic hot spots in the graphene strips system are located at both ends of the strip, where the trapping potentials are greatest. Unfortunately, the trapping potentials provided by the coupled graphene strips system are too narrow due to the high confinement of electromagnetic fields, which is not conducive to stable capture. The trapping potential Ux and force Fx generated by the AGP cavity as the gap distance g = 2 nm are calculated, as depicted in Fig. 3(d). The trapping potential Ux has a value around -13.6 × 102 kBT/mW, which is enough to trap the nanoparticle stably, as shown in Fig. 3(d). As shown in the inset of Fig. 3(d), the field distribution determines that the width of the two symmetric potential wells generated by AGP cavity is much larger than that of coupled graphene strips system. The maximum trapping force occurs at the position where the trapping potential gradient is largest. The trapping force Fx applied on single polystyrene nanoparticle reaches 2.5 nN/mW at the edge of AGP cavity. The trapping potential and force generated by AGP cavity are two orders of magnitude greater than those generated by the coupled graphene strips system [34]. Thus, trapping the same particle requires two orders of magnitude less excited optical power for AGP mode optical tweezers than graphene strips system. Stable trapping of a single particle can be achieved as long as Ux around x = 0 is much larger than the Brownian motion kBT [31]. The standard of potential required for trapping particle proposed by Ashkin is U/kBT = 10 [43]. The results indicate that particle can be stably trapped and restricted in the central region of the gap.Experimentally, most of metallic nanocubes have rounded corners, thus, the results closer to the experiments could be obtained by using the nanocube with rounded corners in FDTD simulation [44,45]. rθ is the radius of rounded corner. We investigate the trapping potential Ux as a function of the position P of captured particle and the radius of rounded corners rθ, as shown in Figs. 4(a)-(c). As the radius of rounded corners increases to 36 nm, the nanocube transforms into a nanosphere. With the increase of radius of rounded corners rθ, the trapping potential applied on single polystyrene nanoparticle decreases gradually, and the potential well becomes narrower significantly, as depicted in Fig. 4(b). The position of the largest trapping potential approaches P = 0 as the radius of rounded corner rθ increases. This is because the position of the maximum field enhancement is closer to the center of AGP cavity with the increase of radius of rounded corner rθ. But there are still two potential wells as the radius of rounded corners reaches its maximum. The inset of Fig. 4(c) shows the trapping potential as the radius of rounded corners rθ = 36 nm. We marked the trapping area with the blue dotted line box. In other words, particle trapping can still be realized when the radius of rounded corner is the largest, but with a significantly smaller trapping area. The maximum range of capture depends on the boundary of the nanocube near the graphene side.
Fig. 4. (a) Trapping potential Ux in dependence on the position P and radius of rounded corners rθ.The inset shows the nanocube with rounded corners. Trapping potential varies with (b) the position P and (c) radius of rounded corners rθ of AGP cavity. The inset shows the trapping potential as the radius of rounded corners rθ = 36 nm. (d)-(f) Trapping force Fx varies with the position P and radius of rounded corners rθ.
Figure 4(d)-(f) show the variation of trapping force Fx with position P and the radius of rounded corner rθ. Unlike the distribution of trapping potential, the variation of trapping force tends to be inversely symmetric along the position P = 0, as shown in Fig. 4(d). As can be seen from the variation trend of the trapping force, the force decreases, and the position of the maximum force is closer to the center with the increase of the radius of rounded corner rθ. The maximum trapping potential and force appear at the same position for the same radius of rounded corner. As a result, the effect of the radius of rounded corners on the trapping potential and force should be considered in the experimental process and theoretical analysis.
The variations of trapping potential Ux and trapping force Fx with different gap distances g (5∼40 nm) and positions P are investigated in Fig. 5. With the increase of the gap g, the trapping potential weakens, as illustrated in Fig. 5(a). The trapping potential generated by the AGP cavity has giant potential energy and wide well width, which is sufficient to confine nanoparticle into the trap. Two symmetric and identical trapping potentials are generated by AGP cavity, as depicted in Fig. 5(b). The position of the largest trapping potential is distributed at the edge of AGP cavity, and does not change with the gap distance. The interaction between neighbouring AGP cavities is very weak, which enables each cavity in the overall system to capture particles independently. According to the variation trend of trapping potential Ux with gap g, the coupling between the nanocube and graphene sheet becomes weaker as the increase of gap distance, the trapping potential attenuates in the form of exponent, and the trapping force weakens, as depicted in Fig. 5(c). The maximum trapping potential Ux has a value around -10 kBT/mW as the gap distance g = 20 nm, which is sufficient to trap the nanoparticle, as depicted in the inset of Fig. 5(c).
Fig. 5. (a)-(c) Trapping potential Ux in dependence on the position P of captured particle and gap distance g. The inset shows the trapping potential as the gap distance g = 20 nm. (d)-(f) Trapping force Fx varies with the position P and gap distance g.
In general, the AGP cavity can provide a trapping potential sufficient to overcome Brownian motion and keep the particle stable within the potential well. In Figs. 5(d)-(f), the variations of trapping force Fx with different gap distances g and positions P are investigated. Similar to the change of trapping force with radius of the rounded corners, the trapping force decreases sharply with the increase of gap distance. The maximum trapping force Fx = 0.14 nN/mW as gap distance g = 5 nm, and the maximum trapping force occurs where the trapping potential gradient is largest, as depicted in Fig. 5(e). With the increase of the gap g, the magnitude of trapping force generated at both edges of AGP cavity approaches to 0 at the same time, and the speed of variation slows down, as exhibited in Fig. 5(f). The trapping potential and force generated by AGP cavity are two or three orders of magnitude greater than those previously reported for graphene-based devices [34–37], suggesting that the AGP mode can provide new possibilities for the enhancement of optical force.
4. Conclusions
In summary, we have investigated the optcial force enhanced by AGP cavity consisting of single nanocubes and graphene via gradient force theory. The total optical force produced by AGP cavity reaches the maximum of 12.9 nN/mW as the gap distance g = 2 nm. The trapping potential and force reach maximum -13.6 × 102 kBT/mW and 2.5 nN/mW as the gap distance g = 2 nm, respectively. The trapping potential and force generated by AGP cavity are about three orders of magnitude larger than those previously reported for graphene-based devices. The trapping potential and force are sufficient to complete the strong trapping and stable confining of nanoparticles. The theoretical and simulation results agree well. We also investigate the effect of radius of rounded corners on the optical force of AGP cavities, which is more instructive for device fabrication. The results show that the trapping potential and force provided by AGP cavities decrease sharply with the increase of radius of rounded corners. Lastly, we study the tendency of trapping potential and force with gap distance. With the increase of gap distance, the magnitude of trapping potential and force decay exponentially. Our results may open up new avenues for the manufacture of ultra-compact optical tweezers and the development of lab-on-a-chip devices.
Funding
National Natural Science Foundation of China (11947062, 62205278); Natural Science Foundation of Hunan Province (2020JJ5551, 2021JJ40523).
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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