1. Introduction

Lab-on-a-chip optical tweezers are powerful and inexpensive tools for particles manipulation with low power consumption and non-contact and non-invasive nature that have attracted biologists, medical scientists, and nanotechnologists’ attention [1–6]. However, the conventional optical tweezers, due to the diffraction limit, are incapable of manipulating and trapping nanoparticles [7]. To overcome this obstacle, some research groups have used the strong gradient fields in near-field optical tweezers [8–11], one-dimensional photonic crystal nano-resonators [12–15], microring-resonators and microdisk-resonators [16–19], and the plasmonic tweezers [20–24]. The optical field enhancement in a microring resonator at the resonance condition makes it a suitable tool for manipulation and stable trapping of particles [25]. Knowing this, three different research groups have used ring resonator-based add-drop filters for trapping, sorting, tuning the trapping position, and releasing the polystyrene (PS) particles [25–27]. By coupling two coherent beams into a ring resonator in opposite directions and evoking a standing wave, one group has developed a tunable optical trapping system for nanoparticles [26]. The other group utilized an electronic logic ring-assisted Mach–Zehnder interferometer (MZI) and proposed a silicon-based lab-on-a-chip system [27]. In this system, a carrier injection can slightly modify the refractive index and hence tune the phase delay in the ring resonator devised in one of the MZI arms, switching the light beam into the corresponding output.

Evoking the field confinement at the metal/dielectric interface in plasmonic structures has recently attracted attention for the development of plasmonic force switches and tweezers for particle trapping/manipulating [20,28–36]. Moreover, optical tweezers based on the plasmonic ring resonator have been proposed recently, using gold [37]. Moreover, graphene has received significant attention as an alternative for gold, resulting in the development of tunable plasmonic structures for various applications like force switches [32,33], bandpass filters and tweezers [38–40]. As compared to Si waveguides, a graphene waveguide in addition to the capability of handling higher field intensity, lower loss, higher thermal conductivity, benefit from electrostatically tunable surface conductivity that can make them capable of supporting and guiding plasmonic waves in the THz and mid-infrared frequencies. Their plasmonic properties have made them attractive to applications such as lab-on-a-chip optophoresis for particles and nanoparticles manipulation [41,42].

Combining the advantages of the tunable graphene-based plasmonic waveguides and ring resonators, for the first time, we proposed a tunable plasmonic optophoresis system, in this paper. The proposed structure and its functionality will be elaborated in Section II.

2. Proposed structure and operation principle

Figure 1 illustrates a three dimensional schematic of the proposed lab-on-a-chip plasmonic optophoresis that consists of three input and two output microfluidic channels. In the optophoresis mid-section, there is a graphene nano-ring resonator of inner and outer radii r and R acting as a tunable plasmonic band-pass filter coupled to a graphene plasmonic waveguide of width W. The underlying substrate is a Si wafer covered with a thin layer of thermally grown SiO₂. As shown in Fig. 1, PS nanoparticles of various diameters, D, suspended in water are injected into the microfluidic “Inlet” by a syringe similar to those of [43,44]. Moreover, the two sheath flows force the injected particles to line-up in a single row about the waveguide midline, where they can continue to move by the fluid driving force. To control the chemical potentials of the graphene waveguide and ring, independently, they are connected via two different graphene microribbon interconnects to two externally biased parallel capacitance gate schemes outside the microfluidic system. The tapered geometry of the graphene microribbon that is connected to the ring and its joint position is chosen to minimize the ensuing plasmonic loss in the ring. The geometrical dimensions are given in Table 1.

 

Fig. 1 A three dimensional schematic of the proposed lab-on-a-chip optophoresis.

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Table 1. Geometrical Dimensions.

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The graphenes nanoribbons forming the waveguide and the ring resonator are biased to maintain equal chemical potentials (μ_C) sufficient to turn on the surface plasmons at their interfaces with SiO₂ layer. Consider a p-polarized single-mode THz source of intensity I₁ = 14.78 mW/µm² and frequency centered about the resonance of the ring resonator, passing via a grating configuration similar to that suggested by [45] and later we used in [46], for excitation of the surface plasmons polaritons on the graphene waveguide surface. Use of a dipole source close to graphene surface has also been suggested for stimulation of graphene surface plasmons [39,47,48].

The plasmonic wave, while propagating along the waveguide, is coupled to the ring resonator. This coupling can be turned ON an OFF, by tuning the chemical potential of the plasmonic waveguide/ring resonator, electrically. It is worth noting that energy of the optical phonons in the given graphene nanoribbon and ring resonator is much larger than that of the given THz input signal [49,50], and hence the plasmon-phonon interaction is absent. The gradient force excreted on the PS particles due to the highly confined plasmonic field in the y-z plane near the graphene surface can trap the particles if their diameters are large enough to satisfy the trapping condition — i.e., the trapping potential depth exceeds 10k_BT, in which k_B and T are the Boltzmann constant and temperature. Otherwise, the PS particles are unaffected by the plasmonic force and continue to move out from outlet 1, under the influence of the fluid driving force. Nonetheless, when a PS particle is trapped within the coupling region the plasmonic force due to the coupled plasmonic wave divert it towards the ring resonator. Then, by reducing the ring resonator chemical potential enough to turn OFF the plasmonic wave on its surface, coupling condition is no longer preserved, releasing the trapped particle and let it move towards the outlet 2.

3. Simulation method

We have used the three-dimensional finite difference time domain method to solve Maxwell equations, numerically. For these calculations, the perfectly matched layers boundary conditions along the x-, y- and z-directions have been utilized. The average plasmonic force exerted on a particle is [32],

Moreover, to evaluate the optical conductivity of the graphene nanoribbon and the nanoscale ring resonator for investigating their plasmonic behavior, we have neglected the quantum confinement effects to simplify the electromagnetic calculations intensively with acceptable accuracy, according to [51]. There, the authors have compared the numerical results for graphenes nano-ribbons/disks utilizing the tight-binding approach and random phase approximation method with those evaluated by the classical approach and have shown that the differences in the resultant surface conductivities are negligible. In the classic approach, the graphene optical surface conductivity for the mid-infrared to THz signals where ℏω≪k_BT and in absence of an external magnetic field can be dominated by the intra-band transitions of the simplified Kubo formula [39], if μ_C>k_BT:

4. Result and discussion

To obtain the simulation results given in this section, in addition to the parameters given in Table 1, we have used 1-nm mesh size for the numerical simulations.

First, we evaluate the dispersive behavior of the proposed structure changing the chemical potentials of the ring (μ_CR) and waveguide (μ_CW), simultaneously. The real and imaginary parts of the effective index — i.e Re (n_eff) and Im (n_eff) — versus the input signal frequency, f, are represented by the solid and open symbols in Figs. 2(a) and 2(b), for μ_CR = μ_CW = 0.3 (circles), 0.4 (squares), 0.5(diamonds), and 0.6 eV (triangles). The maximum chemical potential can be achieved, applying the gate voltage of 2.4 V to a parallel plates capacitor with a 5-nm thick SiO₂, as the gate dielectric. From Fig. 2(b), one can easily see that the surface plasmons polaritons (SPPs) propagation loss in the waveguide is smallest at μ_C = 0.6 eV. Our calculations show that the SPPs propagation length (LSPP) at the ring resonance frequency (f = 10.38 THz) in the range of 0.4≤μ_C≤0.6 eV varies almost linearly with μ_C. A linear fit shows that LSPP (μm) ≈8.5μ_C (eV) −1.6.

 

Fig. 2 (a). Real and (b). Imaginary parts of the effective refractive index (n_eff) versus frequency for various graphene chemical potentials.

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To investigate the coupling between the plasmonic waveguide and the ring resonator, at the ring resonance frequency, we have calculated the transmission coefficient through the waveguide versus μ_CW and μ_CR. Figure 3(a) displays the simulated profile. As can be seen from this profile, for μ_CW = μ_CR = 0.6 eV the transmission through the waveguide is insignificant (~2.8%), meaning that the resonance between the waveguide and the ring resonator is nearly perfect. Henceforth, we call this the “resonance condition”. Moreover, the same profile reveals that for μ_CW = 0.6 eV and μ_CR<0.48 eV the transmission maximizes (T~98%), for which the resonance between the waveguide and the ring resonator is insignificant and henceforth we call this the “off-resonance condition”.

 

Fig. 3 Transmission profile versus (a) μ_CW and μ_CR at an input signal frequency of f = 10.38 THz, (b) μ_CR and f for μ_CW = 0.6 eV, and (c) μ_CW and f for μ_CR = 0.6 eV

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To elaborate more on the resonance and off-resonance conditions that are crucial to the trapping and releasing the nanoparticles, we keep μ_CW = 0.6 eV fixed and simulate the transmission profile next while varying μ_CR and f. The simulated profile is depicted in Fig. 3(b). As shown in this figure, the dark blue ribbon, representing the loci of insignificant transmissions (resonance conditions) in the f −μ_CR plane, reveals that when μ_CW = 0.6 eV there are numerous pairs of μ_CR≤0.6 eV and f ≤10.38 THz that satisfy the resonance conditions. However, the smaller the μ_CR, the larger the plasmonic loss in the ring (Fig. 2(b)) and hence the weaker the resonance strength. On the other hand, as μ_CR decreases in accordance with the dispersion curve shown in Fig. 2(a), Re (n_eff) in the ring region increases and the ring resonance is redshifted. Furthermore, we keep μ_CR = 0.6 eV fixed and vary μ_CW and f and calculate the transmission. Figure 3(c) shows that the variations in the resonance conditions for various μ_CW are all centered about the same frequency (f = 10.38 THz). However, as μ_CW decreases, according to Fig. 2(b), the plasmonic loss in the waveguide increases and hence the resonance strength becomes slightly weaker.

Figure 4 compares the distribution of the normalized power intensity across the surface of the designed structure in the strongest resonance condition at f = 10.38 THz and μ_CR = μ_CW = 0.6 eV with that of a specific off-resonance condition at the same frequency (μ_CR = 0.3 eV and μ_CW = 0.6 eV). Figure 4(a) shows that the coupling between the waveguide and the ring resonator in the resonance case is so strong that prevents the plasmonic wave to reach the waveguide output. On the contrary, Fig. 4(b) shows that in the given off-resonance condition, the redshift in the ring resonance frequency together with the increased plasmonic loss on the ring surface weaken the coupling considerably, leading to a significant transmission through the plasmonic waveguide, consistent with Fig. 3(a). The data have been shown, so far, reveal that the proposed structure can serve as a plasmonic force switch that is suitable for tapping and sorting nanoparticles, at f = 10.38 THz.

 

Fig. 4 Normalized power distribution across the designed structure surface at f = 10.38 THz for (a) μ_CR = μ_CW = 0.6 eV (resonance) and (b) μ_CR = 0.3 eV and μ_CW = 0.6 eV (off-resonance).

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The coupling coefficient, determined by the gap size (g), is a vital parameter concerning the sorting and trapping application in resonance condition. Figure 5 shows the transmission spectra for various gap sizes (10≤g≤40 nm), under the same biasing condition as for Fig. 4(a). An obvious observation from this figure is the negligible effect of the gap size on the resonance frequency. Moreover, the solid curve, representing the spectrum for g = 10 nm, indicates that the resonance transmission for the smallest gap is less than 0.1%. Whereas, the spectra depicted by the dashes-dots, dashes, and dots show that the transmissions for g = 20, 30, and 40 nm are about 4.5%, 17%, and 25%.

 

Fig. 5 Transmission spectra for μ_CR = μ_CW = 0.6 eV and g = 10, 20, 30, 40 nm.

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To investigate the trapping and sorting capabilities of the proposed structure with the given gap sizes one must consider another vital parameter, in addition to the coupling coefficient — i.e., the y-component of the gradient force (F_y), determined by the gradient of the power intensity, above the surface in a y-z plane. Recalling Fig. 4(a), one can easily see that the gradient of the power intensity, at the resonance condition along the y-direction, is significant at about x≈−150 nm and x≈0 (i.e, about the center of coupling region). Knowing these, we used Eqs. (1)–(3) to numerically calculate the plasmonic force components, F_x (scattering), F_y (gradient) and F_z (gradient) force exerted on a given PS nanoparticle versus y coordinate at x≈−150 and z = 5 nm, first. Figure 6 illustrates the numerical results obtained for a nanoparticle of diameter D = 30 nm. As can be observed from the dots, the scattering force is insignificant, F_x(y)<0.5 pN. The dots-dashes show that the out-plane component of the gradient force is negative (i.e., F_z(y)<0). However, various opposing forces, originating from thermophoresis, fluidic lift, gravity, and electrostatic mechanisms balance this gradient component, keeping the nanoparticle above the surface [20]. Moreover, the dashes show that F_y(y) is fluctuating between some positive and negative values, resulting in a negative potential well that confines the nanoparticle on the waveguide surface along the y-direction. The depth of this potential well (0>U_y(y)>−6k_BT) is shallower than the depth required for efficient trapping (i.e.,−10k_BT). Nonetheless, as we have explained in Section 2, the fluid driving force pushes the particle to move along the waveguide (x-direction) until it arrives at the coupling region, where it may be trapped efficiently. It is worth noting that our calculations show that the scattering force along the waveguide (−350 nm≤x≤0) at y = 0 and z = 5 nm is also insignificant.

 

Fig. 6 The dots, dashes, and dots-dashes (left axis) represent F_x, F_y, F_z exerted on a PS of particle of diameter D = 30 nm, whose centered moves along y-direction at x = −150 nm and z = 20 nm; and the solid curve (right axis) denotes U_y(y) in terms of k_BT; for μ_CW = μ_CR = 0.6 eV. f = 10.38 THz, and I₁ = 14.78 mW/μm².

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Then, using the same approach as before we have investigated the effect of the gap size on the F_y excreted on a PS nanoparticle and the corresponding potential versus the particle y coordinate, U_y(y), above the surface in the y-z plane at x = 0 and z = 5 nm. Figure 7 compares the numerical results. The solid curve in Fig. 7(a) (seen from the left axis) illustrates the profile of F_y(y), exerted on a PS particle of diameter D = 30 nm, positioned at x = 0 and z = 5 nm along the y-axis, in the proposed system with the gap size g = 10 nm, under the same resonance condition as that of Fig. 4(a). The solid curves in Figs. 7(b) and 7(c) represent similar data for structures with gaps g = 20 and 30 nm. The dashes in each figure (seen from the right axis) represent the corresponding potential energy, U_y(y). The two vertical dashes-dots, in each figure, depict the y coordinates of the plasmonic waveguide edges, while the two vertical dots represent the y-coordinates of the outer and inner edges of the ring nearest to the waveguide. The horizontal dots represent the −10k_BT that is the reference for the efficient trapping potential. The input source power intensity of I₁ = 14.78 mW/μm² is considered for these calculations.

 

Fig. 7 The solid curve (left axis) represents F_y(y) and the dashes (right axis) denotes the corresponding U_y(y) in terms of k_BT, when the center of a PS particle of diameter D = 30 nm is positioned at x = 0 and z = 20 nm, for g = (a) 10 nm, (b) 20 nm, and (c) 30 nm. μ_CW = μ_CR = 0.6 eV. f = 10.38 THz, and I₁ = 14.78 mW/μm².

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Comparison of the solid curves in Fig. 7 shows that as g increases the coupling coefficient decreases so that the force strength within the coupling zone along the y-direction (F_y) weakens significantly, specifically around the right edge of the waveguide, within the gap, and around the outer and inner edges of the ring. Note that the weaker the F_y, the weaker the driving force for the particle toward the ring, and the weaker the chance for the particle to be trapped by the ring while passing through the coupling region. Furthermore, comparison of the corresponding potential energies shows that only for g = 10 nm the PS nanoparticle experiences a potential well much deeper than 10k_BT, over the vast range of 17.44 nm <y <62.08 nm, where it can be trapped effectively. However, for the other two gaps, the potential energies are shallower than 10k_BT all along the y-direction. Henceforth, we use g = 10 nm that is small enough to provide a complete resonance condition, leading to a potential well deep enough to trap a 30 nm diameter PS nanoparticle, efficiently, under the given illumination and biasing condition.

Note that at the instance that the particle is trapped about the coupling region, if V_R is turned to the off-resonance condition, the trapped particle will be released and moved out of the microfluidic channel through outlet 2 by the fluidic driving force. Otherwise, the scattering force — i.e, the tangential component of the plasmonic force in the polar coordinates, F_ϕ = F_xcos ϕ−F_y sinϕ, where ϕ = tan⁻¹(F_y/F_x) — can move the trapped particle around and above the ring. In that case, the radial component of the plasmonic force in the polar coordinate — i.e., F_r = F_x cos ϕ + F_y sinϕ — acts as the gradient force that may trap the particle somewhere above the ring. The gradient potential along the ring radius at any given ϕ is hence given by U_r (r) = −∫F_rdr. On the other hand, from Fig. 4(a), one can see that the mode intensity around the ring edges at ϕ ~0 (x~0; −15 nm<y< + 65 nm and + 180 nm<y< + 240 nm) and ϕ ~90° (60 nm<x<110 nm; y = 125 nm) experiences significant gradients. Figure 7(a) represents the U_r1(ϕ = 0:x = 0; 17.44 nm<y<62.08 nm) We have also calculated U_r2(ϕ = 0; + 180 nm<y< + 240 nm) and U_r3(ϕ = 90°: 60 nm<x<110 nm; y = 125 nm). The numerical results show that the two latter potential depths do not satisfy the stable trapping condition — i.e., |U_r2|<|U_r3|<10k_BT. Nonetheless, in both cases, U_r(r) confines the particle along the y-direction above the ring, while F_ϕ forces it to move around it.

Besides achieving the most desirable physical and geometrical parameters required for designing plasmonic tweezers capable of trapping a nanoparticle, Fig. 3(a) has revealed that for the biasing condition μ_CW = 0.6 eV and μ_CR≤0.5 eV the transmission through the plasmonic waveguide T ≥90%, approaching unity for μ_CR≤0.48 eV. Now, we would like to find the most appropriate off-resonance condition for which the proposed tweezers release the trapped particle completely. Hence, we calculate U_y(y) for μ_CR = 0.3 eV (solid curve), 0.35 eV (dashes), 0.4 eV (dots-dashes), 0.45 eV (dots), and 0.5 eV (thick solid curve), as illustrated in Fig. 8. Comparison of these off-resonance potential energies with that of resonance condition shown in Fig. 7(a), reveals that the smaller the μ_CR the shallower and narrower the potential well, while moving towards the −y-direction. Furthermore, for μ_CR≤ 0.35 eV, the potential energy flattens to insignificant values within ± 0.1k_BT, over a vast region covering a larger portion of the ring resonator. Hence, when the μ_CR is switched from 0.6 eV to values ≤0.35 eV the trapped particle can be undoubtedly released and flow towards the outlet 2. This 250 meV change in the graphene ring resonator chemical potential can be achieved by a 2.029 V change in the bias applied to the corresponding gate. Moreover, using the data taken from Fig. 3, the switch ON/OFF ratio at the waveguide output ratio is about −15.52 dB.

 

Fig. 8 U_y(y) experienced by a 30-nm diameter PS nanoparticle positioned at the coordinates (x = 0, z = 5 nm) when f = 10.38 THz, and I₁ = 14.78 mW/μm², μ_CW = 0.6 eV, and for off-resonance μ_CR = 0.3, 0.35, 0.4, 0.45, and 0.5 eV.

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To investigate the trapping/sorting resolution of the proposed optophoresis system concerning the particle size, we study the effect of particle size on the potential depth at the resonance condition, next. Figure 9 illustrates the calculated potential energies versus y-position, using PS nanoparticles of diameters D = 18, 22, 26, and 30 nm. Thick solid line, dashes, dashes-dots, and dots respectively depict the corresponding U_y(y) for the source intensity of I₁ = 14.78 mW/µm². As can be observed from these curves only the thick dots cross the −10k_BT line. In other words, under the given illumination the proposed plasmonic tweezers are incapable of sorting/trapping the nanoparticles of D<30 nm, efficiently. Nonetheless, by increasing the input signal intensity to I₂ = 29.22 mW/µm², we have repeated the calculations for the smaller particles. The thin solid line and dashes represent the potential wells for nanoparticles of D = 18 and 22 nm, under the intensified illumination. These new results show that the proposed tweezers under the new illumination become capable of sorting/trapping the nanoparticles of D≥ 22 nm, efficiently.

 

Fig. 9 Profiles of U_y(y) for PS nanoparticles of diameters D = 18 nm (thick solid curve), 22 nm (thick dashes), 26 nm (thick dashes-dots), and 30 nm (thick dots), for I_{1 =} 14.78 mW/µm². The thin solid line and dashes represent the similar profiles for D = 18 and 22 nm, for I₂ = 29.22 mW/µm².

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Finally, we investigate the effect of nanoparticle refractive index on the sorting/trapping functionality of the proposed plasmonic tweezers. The thick solid line, dashes, dashes-dots, and dots in Fig. 10, respectively, represent the potential energy sensed by the nanoparticles of the same diameters D = 30 nm and refractive indices of n = 1.45, 1.55, 2, and 2.5, when the input signal intensity is I₁ = 14.78 mW/µm². These data show that the system is capable of effectively sorting/trapping the nanoparticles of given diameters with refractive indices n>1.45. Nonetheless, the thin solid line in the same figure reveals that by increasing the input signal intensity to I₂ = 29.22 mW/µm², the system becomes capable of effectively sorting/trapping the nanoparticles with n = 1.45.

 

Fig. 10 Profiles of U_y(y) for nanoparticles of the same diameters D = 18 nm but different refractive indices n = 1.45 (thick solid curve), 1.55 nm (thick dashes), 2 (thick dashes-dots), and 2.5 (thick dots), for I_{1 =} 14.78mW/µm². The thin solid line represents a similar profile for n = 1.45, for I₂ = 29.22 mW/µm².

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5. Conclusion

We have proposed a lab-on-a-chip plasmonic force switch, composed of a nano-scaled ring resonator coupled to a waveguide both made of graphene nanoribbons. Simultaneously biasing the ring and waveguides at 2.4 V, via two parallel plate capacitors with 5-nm SiO₂ layers as their gate dielectrics, turns the switch on making it capable of efficiently trapping a PS nanoparticle of diameter D≥30 with an input signal of intensity 14.78 mW/µm² and center frequency 10.38 THz. Solely reducing the applied bias to the ring resonator by about 2.029 V (i.e., decreasing the ring chemical potential by 250 meV), the potential energy of U_y≤−10k_BT, over a vast region covering a large portion of the ring resonator, drops to ± 0.1k_BT, letting the trapped particle to be completely released. The equivalent ON/OFF ratio of this plasmonic switch is −15.52 dB.

For the proposed lab-on-a-chip optophoresis system to trap the PS nanoparticles of diameters as small as 22 nm, the input signal intensity should be increased to 29.22mW/µm². Moreover, the same input signal intensity is required for trapping the nanoparticles of a refractive index as small as 1.45 and diameter 30 nm.