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Abstract
We investigate the plasmonic mode profiles along the depth of various vertical nanoholes to realize vertical nano-manipulation. Our numerical results show that by varying the incident wavelength, we can reposition the plasmonic hotspots along the sidewalls of the staircase and conical nanoholes without and with an oxide spacer. Using the Maxwell stress tensor and the three-dimensional finite difference time domain numerical method, we calculate the optical forces exerted on a polystyrene nanoparticle of 10-nm radius and the related potential energy along the sidewall, demonstrating the trapping sites corresponding to the plasmonic hotspots. Furthermore, the simulation results reveal that adding an oxide spacer of appropriate thickness adds a degree of freedom in controlling the position of the trapping site at the expense of a slight decrease in the trapping figure of merit. These results indicate that the proposed plasmonic conical nanoholes are potentially suitable for controlled vertical transportation of DNA strands.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recently, nanopore-based biosensors due to their rapid electrical detection and characterization of biomolecules attracted ample attention in detecting different bio-analytes such as DNA molecules [1,2], proteins [3–6], and in revealing DNA sequences [3,7,8]. Moreover, since nanopore technology promises affordable and fast genome sequencing by providing long read lengths (5 kbp) have recently gained considerable attention. Many types of nanopores have been proposed that are based on biological [2,4], dielectric [9], and plasmonic [10] membranes to form a single nanopore [6,11] or an array of nanopores [5,12].
The main idea is to pass a DNA molecule through a nanopore and read off each base when it is located at the narrowest part of the pore membrane. The identity of the base is probed by measuring the ion current passing through the nanopore [13], while the displacement of the DNA through a nano-aperture is driven by the applied electric field [14,15]. Reading the DNA bases in a nanopore has some difficulties due to the fast translocation of DNA through the nanopores and consequently contribution of several nucleotides to the sensed signal. In most previous studies, DNA molecules translocate through the channels with a speed of ∼1 base/µs. This speed is too high for accurate detection of nucleotides and needs to be controlled. It has been suggested that a reasonable translocation velocity should be as low as 1 nucleotide per millisecond (∼3 Å/ms), which is three orders of magnitude slower than the current speeds [16]. One way of slowing down the translocation is to couple the translocation with the optical forces [17]. The confined and enhanced electromagnetic field by the localized surface plasmons (LSPs) in a metallic nanohole has the potential to control analyte displacement in the nanohole. Meanwhile, horizontally nested double nanoholes (DNHs) have been reported as good nanoholes to enhance self-induced back action (SIBA)-mediated optical traps [18–20] with many applications including trapping nanoparticles [21,22] and single protein molecules [23]. Recently, a new method to trap nanoparticles by a DNH has been demonstrated [24], translocating them by the electrophoretic force induced by an ionic current.
In this paper, we propose plasmonic nano-holes with and without dielectric spacers. These structures could be used to trap nanoparticles or DNA strands along the nanoholes walls, via controlling the trapping sites along the wall by varying the incident beam wavelength. In these structures, different plasmonic modes with different vertical mode profiles can be excited in a controllable manner by tuning the input source wavelength. As a result, different vertical trapping sites will be created for each incident wavelength that can be used for nanometric particle movement along the nanoholes walls. The operation of the system is investigated by considering a polystyrene nanoparticle of 10-nm radius, as an example. The proposed plasmonic nanoholes have two main advantages: (i) they do not face the ionic current problems that have been reported by others [16,17]; (ii) the rates of trapping and displacing the nanoparticles by them are controllable.
2. Proposed structure and operating principles
Figure 1 illustrates the proposed structure and its operating principles, schematically. As can be observed from this figure, the basic structure is composed of a couple of gold (Au) nanolayers with nanoholes of different radii, separated by an oxide spacer. The nanoholes in the Au nanolayers can support the LSPs in the wavelength range of 300 ≤λ ≤ 2000nm appropriate for trapping nano-objects such as proteins and DNAs. This proposed nanostructure can be fabricated using the focused ion beam FIB method [24]. In this structure, immersed nanoparticles in water are placed on the top of the nanohole while various x-polarized plane waves with different wavelengths (λ) are illuminated in the –z-direction. Each intentionally chosen incident wavelength traps a particular nanoparticle in a specific vertical z-position, depending on the incident light intensity. As the incident wavelength is decreased the corresponding plasmonic hotspot (trapping site) moves downward. Moreover, the oxide spacer can control the trapping sites as will be discussed in the following sections. Furthermore, the transmitted signal changes when the particles move along the depth of the nanohole that can be used as a measure for sensing nanoparticles. It is worth mentioning that this structure can be used as a complementary method for nano-object translocations.
Fig. 1. Two-dimensional schematic (x-z plane) view of the proposed structure with vertically nested nanoholes and its operating principles.
3. Simulation method
To simulate the structure, we have employed the three-dimensional (3D) finite-difference time-domain (FDTD) method and the total field-scattering field method, considering an x-polarized plane-wave normally incident upon the nanohole from the top (i.e., along the −z-direction). Perfectly matched layers (PMLs) have been considered along all directions in these simulations. After obtaining the LSPs field distributions, the Maxwell stress tensor (MST),
Meanwhile, this potential energy formulation for the movement of a spherical particle of radius r and dielectric constant εd that is surrounded by a fluid of dielectric constant εf, at a point x where the LSPs mode intensity is I(x) $\varpropto$ |E(x) |2 simplifies to [28,29],
4. Results and discussions
4.1. Cylindrical nanoholes
First, we consider a single Au nanolayer of thickness TAu through which a nanohole of radius R is supposedly ion-milled [24]. Then, varying TAu and R, we investigate the decencies of the light transmission through the nanohole on its dimensions (Fig. 2(a)). As can be observed from Fig. 2(a), each transmission spectrum contains two types of peaks. The first type of peaks in all spectra occur at the same wavelength (i.e., λ ∼517 nm). These peaks are related to the interband transitions (i.e., ∼2.4 eV) of electrons in the Au nanolayer [30], solely depending on the material optical properties and cannot be tuned. As the energy of the photons increases from 2.4 eV (λ<517 nm), numerous interband transitions occur, significantly increasing the extinction. The minimum extinction occurring at the onset of the interband transitions [30] leads to a maximum in the transmission spectrum (the first type of peaks). Moreover, below 2.4 eV (i.e., λ>517 nm) the metal behaves very much like a free-electron metal and the dispersion relation can be described by a Drude dielectric function. The extinction spectrum in this regime decreases by increasing the energy of the incident photons, resulting in the second type of peak, in each spectrum that corresponds to the resonance wavelength (λres) of the metal LSPs which in turn depends upon the nanohole dimensions (R and TAu). As can be observed from these peaks, for a given TAu, as R increases, the peak is red-shifted due to the longer distance between the opposite surface charges in the larger nanoholes that require lower restoring force (or lower incident energy) to be excited. Nonetheless, for a given R, by an increase in TAu, the peak is blue-shifted. The larger the radius, the easier for the light to pass through the aperture, and hence the larger transmission peak [31], for a given λ through a known thickness. Conversely, the larger the TAu, the thicker the blocking layer [32], and hence the smaller the transmission, for a given λ passing through the aperture of a given radius. To get more insight into the behavior of the nanoholes geometry in designing the structure, we’ve calculated the dependency of LSPs resonant wavelength on the R/TAu ratio for given thicknesses and illustrated the results in Fig. 2(b). As can be observed from this figure, this dependency for any given thickness is almost linear — i.e., λres $\varpropto$(R/TAu), — similar to that reported in [33].
Fig. 2. (a) Transmission spectrum of a single cylindrical nanohole of different thicknesses TAu = 20, 60, 100 nm and radius R = 50, 100, 125 nm. (b) LSP resonance wavelength versus radius-to-thickness ratio for different TAu = 20, 40, 60, 80, 100 nm.
4.2. Staircase cylindrical nanoholes
Knowing the dependency of the transmission spectrum on the geometrical dimension of a single nanohole, next, we investigate the transmission spectrum and the mode intensity profile of (vertically nested) cylindrical nanoholes with two different radii (R = 50 and 100 nm) devised into two Au layers of the same thicknesses (TAu), separated by an oxide (SiO2) spacer of thickness Tox. This spacer, with the same lateral dimensions as those of the top metal layer, can decouple the LSPs on the walls of the nanoholes in the top and bottom Au layers. Figure 3(a) illustrates the transmission spectrum through the nanostructure with different Tox(nm) = 0 (solid line), 10 (dashes-dots), 50 (dot), and 75 (dashes), as can be seen, more visibly in the inset. For the sake of comparison, Fig. 3(a) also shows the transmission spectra through two single nanoholes of radii R = 50 (thin solid line) and 100 nm (thin dashes), each devised in an Au layer of thickness TAu=40 nm. As can be observed from this comparison, the resonant wavelengths (λres) of the staircase nanoholes exhibit blueshift (redshift) to that of the single nanohole of radius R = 50 (100) nm. This is because the average radius of nested nanoholes is 50 nm < Rav < 100 nm. Moreover, as Tox increases the transmitted power also increases due to the dielectric loading effect [21,22]. To get more insight into the behavior of the staircase nanoholes, we’ve plotted the normalized mode intensity profiles (|E/E0|2) in the x-z plane at the resonance wavelength of the nested nanoholes (λres∼650 nm) and that of the larger single nanohole (λres∼ 877 nm). The results are illustrated in Fig. 3(b). As can be observed from this figure, similar to the results presented for the single cylindrical nanoholes, the larger wavelength excites LSPs in the larger (top) nanohole in the nested structure and vice versa. Moreover, as Tox increases the top and bottom LSPs modes become farther apart.
Fig. 3. (a) Transmission spectra through the single cylindrical nanoholes of R = 50 and 100 nm and nested cylindrical nanoholes separated by oxide spacers of Tox = 0, 10, 50, 75 nm. (b) Profiles of LSPs mode intensities (| E / E0|2) at the excitation wavelengths of λ=650 and 877 nm.
Furthermore, by examining the transmission spectra through the vertically nested nanoholes, we can conclude that the transmission peak wavelength follows the behavior of the smaller nanohole. Although we have performed 3D simulations, due to the x-polarized incidence and the circular symmetry of the nanoholes in the x-y plane (at any given z), we have depicted the mode intensity profiles in the x-z plane at y=0, throughout the manuscript.
4.3. Conical nanohole
Practically, a conical nanohole can be made by the FIB method [24]. The smoothness of the sidewall of a conical nanohole can be defined by the number of the steps of the incremental decrease in the diameter of the ion-milled area moving downward. The minimum size of each incremental step (maximum smoothness of the nanohole sidewall) is limited to the Au atomic scale. Hence, one can reasonably neglect any scattering due to the sidewall roughness. Here, we consider a single conical nanohole of given height and top and bottom radii (diameters) Rt (dt) and Rb (db) devised in an Au nanolayer of thickness TAu.
Here, we set the conical nanohole dimensions to TAu (the height of the conical hole) = 200 nm and the top and bottom radii Rt = 150 and Rb = 50 nm. The corresponding transmission spectrum is plotted in Fig. 4(a). As shown in this figure, this transmission spectrum follows the same trend as those of the nested cylindrical nanoholes. Figures 4(b)–4(i) illustrates the side (x-z) view of the conical hole at y = 0. The dashes parallel to the nanohole sidewall indicate the path along which the trapping force component and the corresponding potential energy are going to be calculated. For this purpose, via an appropriate rotational translation, the new coordinates x′-z′ is made such that the x′ axis is parallel to the sidewall. Figures 4(b-ii)–4(b-vi) depict the profiles of the normalized LSPs mode intensities on the sidewall, calculated for five x-polarized incident light signals at the sampling wavelengths of λ = 544, 630, 650, 743, and 861 nm, represented by the five vertical lines in Fig. 4(a). These five wavelengths are those for which the LSPs mode intensity is maximized at a particular position along the sidewall. In other words, as can be seen in Fig. 4(b) at these five different wavelengths, localized hot spots are created, each leading to a potential well that may be a suitable trapping site for the targeted nanoparticle. A comparison of these five profiles reveals that as the incident wavelength decreases, the LSPs mode (hotspot) is displaced downward along the nanohole sidewall. This is a similar phenomenon to that observed in Fig. 3. This behavior can be used to tune the coordinates of the trapping site with nanometer resolution as desired, employing an appropriate commercial tunable optical source.
Fig. 4. Transmission spectrum (a) through the conical hole of dimensions TAu = 200 nm, Rt=dt /2 = 150, Rb =db/2 = 50 nm, whose side (x-z) view is depicted in (b-i), with the normalized LSPs mode intensity profiles excited at the sampling wavelengths of λ=544 (b-ii), 630 (b-iii), 650 (b-iv), 743 (b-v), and 861 nm (b-vi), corresponding to the vertical lines in (a). Profiles of the corresponding potential energies versus the x and z coordinates (x′-axis) are depicted in (c).
Due to the x-polarized incident lightwave and the symmetry of the conical nanohole about y(y′)-direction, a particle inside the hole experiences two plasmonic force components (Fx and Fz), in the x-z plane at y = 0 that can be translated to Fx′ (parallel to the sidewall) and Fz′ (perpendicular to the sidewall) in the same plane, while Fy=Fy′ = 0. For a hotspot localized about point x′0≡ (x0, z0) to be an efficient trapping site, the component Fx’ should change sign at x0′ and at the same time Fz′(x0′) < 0, pushing the particle toward the sidewall and meanwhile U (x0, z0) ≤−10kBT. To satisfy these conditions simultaneously when a polystyrene particle of 10 nm radius is present, we have set the intensities of five the sampling light signals to be Iinc (mW/μm2) = 300 (λ=544 nm), 634 (λ=630 nm), 792 (,λ=650 nm), 616 (λ=743 nm), and 880 (λ=861 nm). These five intensities are chosen to guarantee for achieving the −10kBT criterion [27,29,34], for the depth of the potential well at each trapping site related to each sampling wavelength, as depicted in Fig. 4(c). We have found that by increasing the nanoparticle radius to 20 nm, the range of minimum required light intensity for the stable trapping (maintaining the −10kBT criterion) reduces down to 18 < Iinc < 191 mW/µm2, similar to the observations we have reported elsewhere [27–29]. Furthermore, for slowing down DNA (with the length per base of 50 nm for dsDNA) while transporting through the nanohole, it is not required to maintain the −10kBT criterion, and hence smaller input intensities are required.
It is noteworthy to mention that different forces such as thermophoresis and electrostatic mechanisms can oppose the z-component of the gradient optical force and may prevent the particles from sticking to the sidewall of nanohole. Thus, to take the effects of all these opposing forces into account in our numerical simulations and force calculations carried by the MST method, we have considered a fixed spacing of ∼20 nm between the particle center and the sidewall along x-direction that is somewhat larger than the Debye length [35].
Now, using Eq. (4), for each sampling signal, we calculate the profile of the trapping potential energy, U(x’) ≡ U(x, z), that a spherical polystyrene (PS) nanoparticle of radius r = 10 nm feels while moving along the x′-direction, near the sidewall. Figure 4(c) illustrates the numerical results. Translating the coordinate x′ back to the corresponding pair of x- and z-coordinates, we have labeled the horizontal axis of Fig. 4(c) by the (x, z) coordinates. The vertical dots in Fig. 4(c) represents the (x0, z0) coordinates of efficient trapping sites (hotspots). As can be observed from this figure, the two signals with the shortest (544 nm) and the longest (861 nm) wavelengths, each has only one efficient trapping site, respectively occurring at the coordinates (x0, z0) ≅ (−30, 0) and (−126, 192). In other words, for the incident signal of λ=861 nm, when the target PS nanoparticle enters the nanohole from the top, it can be trapped at a site ∼8 nm below the top nanohole edge, whereas for the signal of λ=544 nm the nanoparticle is trapped around the bottom edge. Moreover, for each of the other three signals, there are two efficient trapping sites somewhere between the top and bottom edges. The coordinates (x01, z01) ≅ (−122, 184), (−99, 138), and (−92.5, 125) represent the positions of the first minima for signals of λ=743, 650, and 630 nm, whereas the second respective efficient trapping sites are located at the coordinates (x02, z02) ≅ (−35, 10), (−32, 4), and (−32, 4). These results emphasize that by employing an appropriate commercial tunable optical source, one can move the coordinates of the trapping site along the sidewall of the nanohole. It is evident that by turning the input source off, the trapped particle is released, moving out the nanohole from the bottom. Moreover, these results reveal that the first four efficient trapping sites are allocated within the top half of the conical nanohole (i.e. z>100 nm). Knowing this and taking the advantage of the capability of an oxide spacer that can decouple the neighboring LSPs modes, we will consider a conical nanohole devised in two Au nanolayers separated by an oxide spacer, in the following subsection. This provides an extra degree of freedom in controlling the vertical position of the trapping site. Adding several oxide spacers and choosing the Au layers thicknesses appropriately one can create a desired number of hotspots (trapping sites) around the edges of the SiO2/Au interfaces at the predefined coordinates, as desired.
4.4. Conical nanohole with an oxide spacer
In this subsection, to exemplify the conclusive remark we have drawn at end of the previous subsection, we consider two Au/SiO2/Au multilayers with the same total thickness of Tt=Tt-Au+TS+Tb-Au = 200 nm, with Tt(b)-Au=80 (80) nm and 140 (20) nm representing the thickness of the top (bottom) Au nanolayers and TS = 40 nm being the spacer thickness, both having conical nanoholes of the same dimensions Rt = 3Rb=150 nm supposedly ion-milled through the multilayers. Here, we illuminate the nanohole, in each multilayer inserted in the water, with two incident optical signals of appropriate λinc and Iinc, from the top. These signals are chosen to create plasmonic hotspots around the top and bottom edges of the Au nanolayers as shown in Figs. 5(a)–5(d). Figure 5(e) illustrates the profiles of the potential that a PS nanoparticle of 10-nm radius feels along the x′ coordinate near the nanoholes sidewalls, under the presumed illuminations. The rectangle enclosed by the dots (dashes) represents the oxide spacer in the symmetric (asymmetric) multilayer. The observation from the solid circles (triangles) indicates that among the four hotspots created by the signal of λinc = 672 (856) nm and Iinc = 740 (1056) mW/μm2 in the symmetric multilayer {Figs. 5(a) or 5(b)}, three (two) sites positioned at the coordinates {x0, z0} ≅ {−95, 130}, {−74, 88}, and{−34, 8}, ({−124, 188} and {−94, 128}) can trap the given PS nanoparticle efficiently. Similarly, the open squares (diamonds), depicting the potential profiles along the sidewall of the nanohole in the asymmetric multilayer under the incident signal of λinc = 716 (851) nm and Iinc = 317 (845) mW/μm2 {Figs. 5(c) or 5(d)}, indicate that only two (three) of the four hotspots positioned at {x0, z0} ≅ {−64, 68} and {−34, 8} ({− 124, 188}, {−64, 68}, and {−34, 8}) can be efficient trapping sites for the given PS nanoparticle. Comparison of the potential profiles in Fig. 5(e) with those depicted in Fig. 4(c) reveals that the addition of an oxide spacer symmetrically (asymmetrically) to the nanostructure with a conical nanohole has adds two rings of hotspots at the predefined coordinates (edges of the Au/SiO2 interfaces) along the nanohole sidewall.
Fig. 5. Distribution of the normalized LSPs mode intensity throughout the conically nested structure under the illumination of (a) λinc = 672 nm and Iinc = 740 mW/μm2, (b) λinc = 856 nm and Iinc = 1056 mW/μm2 (both with an oxide spacer poisoned at 80 ≤ z ≤ 120 nm), (c) λinc = 716 nm and Iinc = 317 mW/μm2, and (d) λinc = 851 nm and Iinc = 845 mW/μm2 (both with an oxide spacer poisoned at 20 ≤ z ≤ 60 nm). (e) The potential energy profiles that a 10-nm radius PS nanoparticle at the x and z coordinates near the nanohole sidewalls feel, corresponding to the LSPs mode profiles shown in (a) solid circles, (b) solid triangles, (c) open squares, and (d) open diamonds.
All these data demonstrated for the conical nanoholes without and with spacers, either symmetric or asymmetric multilayers can be utilized for the transportation of DNA strands through a nano-aperture, in a controllable manner. This can be made plausible by trapping a few parts of a DNA strand at predefined coordinates along the nanohole sidewall and releasing the strand from the trapping sites, by changing the incident wavelength or turning the input source off.
4.5. Figure of merit
The trapping capability of a particular nanohole, irradiated by an optical signal of given Iinc and λinc can be quantified by a figure of merit defined by the percent variance of transmission through the nanohole [36,37],
where T0 represents the light transmissions through the nanohole when solely filled by water with no nanoparticle present. T is a similar transmission when in addition to water there is a truncated hollow cone of equal thickness and height (=2r) to substitute the PS nanoparticle of diameter 2r (i.e., 20 nm for this work) just around the hotspot. The outer diameters of the top and bottom bases of the hollow cone, for each hotspot, should be adjusted for the cone outer face to become in contact with the nanohole sidewall. This substitution is made to take the dielectric loading effect into account. Otherwise, the presence of a PS nanoparticle has little effect on the transmission.Using Eq. (5), we have calculated ΔT for the single conical nanohole (without an oxide spacer) with the dimensions and five illumination conditions given in Fig. 4(b). The open symbols in Fig. 6 represent the figures of merit for nd=1.56 (squares), 1.7 (circles), 1.85 (triangles), and 2 (diamonds). The solid symbols shown in the same figure represent similar data (i.e., ΔT) obtained for the light passing through nanohole with an oxide spacer in the symmetric multilayer with the same dimensions and all four illumination conditions given in Figs. 5(a)–5(d). The rectangle enclosed by the dots specifies the (x, z) coordinates of the oxide layer.
Fig. 6. The figure of merit (ΔT) vs. coordinates (x, z), calculated for a single conical nanohole in an Au nanolayer with no spacer (open symbols) and for a conical nanohole with a spacer in the symmetric multilayer Au/SiO2/Au (solid symbols), when a truncated hollow cone of equal thickness and height (20 nm) is substituted for the PS nanoparticle around each hotspot.
A general observation from both sets of data is that the larger the material refractive index, the greater the dielectric loading effect, and hence the larger the trapping figure of merit. For example, ΔT for nd=2 is about 3-4 times larger than that for nd=1.56, depending on the illumination conditions or the hotspot intensity. Moreover, a comparison of the data depicted by solid symbols with those being represented by the open symbols reveals that the presence adds degree freedom in repositioning the trapping sites along the sidewall at the expense of reducing the trapping figure of merit by a few percent.
Moreover, by comparison of the data represented by the solid symbols, one infers that the largest trapping figure of merit (ΔT∼11.3-36.3%), for any given refractive index for the nanohole with spacer, belongs to the trapping site positioned near the bottom SiO2/Au interface — i.e., (x0, z0) ∼ (−74, 88). A similar comparison for the conical nanohole without spacer reveals that the most efficient trapping site (ΔT∼10.8-39.5%) for any refractive index is the site located at the coordinates (x0, z0) ∼ (−92, 124). Furthermore, negative figures of merit observed for the site located at (x0, z0) = (−32, 4) nm, for both nanoholes incite that at this site presence of any material with the refractive index in the given range results in a T < T0.
5. Conclusion
The main objective of this study is to design a plasmonic pump for trapping and controlled transportation of nanoparticles along the depth of vertical conical nanoholes. Results of 3D numerical simulations show that by changing the input source wavelength, one can control the coordinates of plasmonic hotspots along the nanoholes sidewalls vertically, enabling control of the vertical transportation of the nanoparticles as desired. The presence of an oxide spacer in a three-layer Au/SiO2/Au having a conical nanohole through it, while decoupling the LSPs hotspots of the top and bottom Au layer adds a degree of freedom in repositioning the trapping sites along the nanohole sidewall at the expense of slightly reduced trapping efficiency. Another apparent advantage of these designed plasmonic pumps over their counterparts used for vertical nanomanipulation of particles in the absence of ionic current related issues. The simulation results pave the way for slowing down the vertical transportation of the DNA strand.
Funding
Tarbiat Modares University (IG-39703).
Disclosures
The authors declare no conflicts of interest.
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