1. Introduction

Carbon nanotubes (CNTs) are high-aspect ratio hollow pipe-like chiral structures made of carbon atoms that exhibit conductivity peaks in infrared, visible, through ultraviolet (UV) frequency range (hundreds of gigahertz to 1000 terahertz (THz) in frequency, 1000 $\mu m$ to 300 $nm$ in wavelength) [1–4]. Experimental analyses manifest that the conductivity peaks for CNTs below 100 THz occur due to their length-dependent localized surface plasmon resonances (LSPR), and beyond 100 THz the interband absorption phenomena dominates [1,2,4]. Advancement in sophisticated fabrication and nano-manipulation techniques made it possible to realize customized CNT configurations with high precision and scalability [5–8]. Concurrently, several efforts have been given to model the realistic shapes and complex spatial distribution of CNT networks accurately with full-wave electromagnetic solvers [9–11]. Many theoretical analyses have studied the electromagnetic properties of CNT networks commonly found in nanocomposite environments. These efforts can be broadly classified into two different approaches: (1) the dilute limit effective medium approximation (EMA) studies, such as the Waterman-Truell [12], and Maxwell-Garnett (MG) approximations [13,14], and (2) full-wave electromagnetic analysis, such as using the method of moment (MoM), finite element method (FEM), or finite difference time domain method (FDTD) [9–11,15–19]. Although the EMA approach provides simplified relations to evaluate the frequency dependent lossy dielectric properties of CNT composites, it suffers from multiple inconsistencies when compared to the experimental data [1,20]. The EMA approach assumes low filler density and homogeneous environment, and thus it ignores the strong electromagnetic interactions among the CNTs and the interfaces of the embedding medium when in close proximity. The results from the EMA can therefore be significantly different from the full-wave simulation results [11]. Recently with full-wave MoM modeling we have shown that the near electromagnetic field coupling between two CNTs in proximity create dual plasmonic resonances that are unique and depend on the configurational properties of the two CNT assembly, otherwise named as the CNT dimer [21]. This dual plasmon resonance phenomena can not be explained by the EMA models. Melnikov et al. has also pointed out the inconsistencies in the EMA approach while modeling the charge transport phenomena between two crossing CNTs in sub-nanometric vicinity [22]. The electromagnetic response of CNT dimers was experimentally measured by Wang et al. [4]. New hybridized plasmonic modes were generated due to inter-tube coupling effects [4]. These experimental measurements provide motivation for developing accurate computational techniques that account for the inter-tube coupling effects and how they vary with the CNTs’ environment.

Recent computational and experimental studies also show evidences of significant impact on the electromagnetic response of embedded CNTs due to the supporting dielectric medium. In an experimental study Long et al. showed that formation of a concentrated layer of CNTs near the composite surface enhance the surface conductivity of the CNT composite [23]. They used controlled UV exposure to selectively etch away the epoxy layer from the composite surface to reduce the distance between CNT rich micro-domains and the dielectric interface. Zhang et al. used FDTD simulation to show that adding ultrathin dielectric layer on top or below CNT based meta-material film can significantly red-shift the THz plasmon peaks and weaken the resonances [19]. In addition to the embedded condition, the response of CNTs are also found to vary when they are placed above a dielectric substrate as reported by Blancon et al. [24]. While measuring the extinction cross-section of an individual CNT supported on a silicon substrate, they observed noticeable broadening and red-shifts in CNT resonances, and modified relative peak amplitude as compared to the freely suspended CNT case [24].

The above studies indicate that the finite-thickness supporting substrate, which typically has a relative permittivity ($\varepsilon _r > 1$) and a loss tangent ($\tan (\delta ) > 0$), significantly change the embedded CNT response. In this present paper we are particularly interested in studying the effect of finite thickness dielectric slab on the electromagnetic response of embedded two CNT network (CNT dimers) and asses its potential in sensing application. Previous computational studies related to two CNT networks have considered a homogeneous or free-space environment [21,22]. The effect of a finite thickness embedding medium and its interaction with the embedded two CNT network has not yet been explored. In our previous work we have demonstrated the sensing modalities of CNT dimer in free-space environment using full-wave MoM simulations [21]. We explicitly demonstrated the near field coupling phenomena by introducing symmetry and asymmetry in CNT dimer assembly [21]. With respect to an incident plane wave excitation, the assembly of two CNT appears as a symmetric CNT dimer if both the CNTs are identical in every aspect, such as shape, size, alignment and chirality, otherwise it forms an asymmetric CNT dimer. It was observed that a length asymmetric CNT dimer illuminated by a plane wave excitation in free-space environment exhibits two fundamental plasmonic resonances: a bonding resonance (BR) and an antibonding resonance (ABR) [21]. The ABR occurs when the currents in the two CNTs are out of phase, whereas the BR occurs when the currents in the two CNTs are in phase. The BR typically appears at a higher frequency than the ABR. This dual plasmonic resonance phenomena has also been reported previously for other metallic nano-dimers involving structural asymmetries [25,26]. In a recent study Gerislioglu et al. report that modifying the dielectric environment of gold dimers results in a wide variety of additional resonances [27].

In this paper our goal is to investigate how the BR and ABR of CNT dimer assembly respond to the change in its dielectric environment, especially when we transform homogeneous medium into finite thickness dielectric slab. We use our in-house full-wave MoM multilayer solver to account the effect of the finite thickness embedding medium and its interaction with the embedded CNT network [28,29]. The in-house solver accurately solve the multiple reflections, transmissions, and absorption in the finite-thickness substrate and their effects on the embedded CNTs [28,29]. Using this solver in our present study we demonstrate how the electromagnetic response of a CNT dimer varies with the distance between the CNTs and the interface of the dielectric substrate, and create new plasmonic resonance modalities that are not observed when the CNTs are placed in a homogeneous dielectric environment.

2. Computational model

Frequency domain full wave electromagnetic modeling of plasmonic structures requires highly dense mesh elements to accurately capture the nanoscale details especially when operating at THz or higher frequency range. Carbon based plasmonic materials, such as two dimensional graphene sheet, or three dimensional high aspect ratio CNTs face high volume discretization issues and unrealistic simulation time when solved using conventional MoM or FEM solvers even when simulated in the free space environment. Recently alternative novel numerical approaches are proposed to minimize these discretization problems for plasmonic structures. For example, Shapoval et al. proposed ’meshless’ algorithm based on the singular integral equations and Nystrom-type discretizations for simulating finite and infinite graphene strips in free space [30]. Hassan et al. proposed one dimensional equivalent arbitrary thin wire (ATW) model for high aspect ratio CNTs for free space MoM analysis and maintaining high solution accuracy [9,21]. In addition to that, when high aspect ratio CNTs are embedded in layered medium their numerical modeling becomes a multiscale problem that increases the computational complexity by manifold. The conventional MoM and FEM models require thousands to millions of surface/volume discretization elements to accurately characterize the embedding dielectric layer as well as the embedded nanoscale tubes, and demand high computational time and resources as reported previously [11,17]. Addressing these issues, we have recently developed an alternative MoM solver for studying electromagnetic scattering response of CNT composites that drastically reduce the discretization requirements and boost the computational speed by several hundred order over commercial MoM or FEM solvers while maintaining same level of accuracy [29]. The in-house solver uses the multilayer Green’s function technique to avoid the explicit discretization of the embedding layer interfaces [31]. So far, we implemented the $G_{xx}$ and $G_{zz}$ component of the multilayer Green’s function which allows us to simulate horizontal (parallel to the x-axis) and vertical (parallel to the z-axis) CNTs [29]. To model arbitrarily-shaped CNTs will require the implementation of all nine components of the multilayer Green’s function which will be implemented in future work. The embedded high-aspect ratio CNTs are modeled as arbitrary thin wires (ATW) that need only one-dimensional discretization [9,21]. The accuracy and computational efficiency of this method are validated rigorously at multiple stages of development against commercial solvers [21,28,29]. In this article, we have used this in-house solver to study the CNT dimer embedded in a lossy dielectric layer.

To establish a proof of concept, presently we investigate parallel CNT dimer embedded in a lossy dielectric slab as shown in Fig. 1. The computational model considers a generic lossy three-layer structure where the CNTs are placed in layer 2 on the xz-plane aligned in the x-direction, with a vertical separation of $\Delta z = |z_1-z_2|$, and center to center lateral separation of $\Delta x$ as shown in Fig. 1(a) and (b). The layered structure is illuminated by a plane wave ($\bar {E}^{i}$) incident normally on the top dielectric interface ($z=d$). The strength of the illuminating plane wave is set as $|\bar {E}^{i}|$ = 1 V/m for all computational experiments performed in this article. Thus, the incident power density on the top dielectric interface is $\frac {|\bar {E}^{i}|^2}{\eta _0}=2.65 \times 10^{-4}\;\textrm {mW}/\textrm {cm}^{2}$, where $\eta _0=377\; \Omega$ is the free space wave impedance.

 

Fig. 1. (a) Three-dimensional schematic view of a lossy three-layer medium with embedded CNTs in layer 2, aligned in x-direction and distributed on the XZ-plane. The structure is illuminated by a x-polarized plane wave normally incident on the top dielectric interface (z = d) with $|\bar {E}^{i}|$ = 1 V/m. (b) Two-dimensional schematic view of the CNT distribution on the XZ-plane in layer 2 with all design parameters indicated.

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The configuration in Fig. 1 includes almost twenty different design parameters such as the dielectric properties of the three layers (permittivity, permeability, conductivity), thickness of layer 2, polarization and angle of incidence of the exciting electric field, CNT properties (vertical and horizontal locations, lengths, diameters, and complex impedance profile of CNTs). In addition to the length asymmetry, other structural variations can be included in the CNT dimer configuration, such as the shape anisotropy, misalignment that will lead to different scattering response. Thus, a large number of simulations is needed to quantify the effect of each parameter. However, the effect of many of these parameters were quantified in previous studies [21]. Therefore, in this paper we will only focus on the new resonance behavior exhibited by CNTs as they are positioned at different heights inside a dielectric slab with a finite thickness.

The CNTs have been assigned frequency dependent complex impedances derived from their Drude-like surface conductivity model similar to the previous works [21,29]. This conductivity model agrees well with the quantum mechanical dynamic conductivity model for CNTs from the microwave through 100 THz frequency range, provided the CNTs are single-walled, metallic and have small radius (< 2 nm) [32]. In this present study, the CNTs are considered as single-walled with (9,9) armchair chirality that has a radius of $a= {0.61}\;{nm}$ [15]. The portion of the incident field tangential to the CNT, that is the x-component of the incident field, excites currents on the CNT’s wall according to its conductivity. Other components of the incident field are normal to the CNT and therefore do not interact with it. This current satisfies the boundary condition that relates the total tangential electric field to the induced axial current on a CNT ($\bar {I}_{cnt}$) according to:

3. Computational results and discussion

3.1 Resonance of a single CNT embedded in a finite thickness dielectric slab

We start our analysis by investigating the effect of the height of a single CNT inside a finite thickness dielectric slab. Figure 2(a) shows a schematic diagram of a horizontal CNT of length $L$ embedded in a lossy dielectric slab at a height $z_0$, and is illuminated by a plane wave polarized along the CNT length. The slab has a thickness $d={50}\;\textrm {nm}$, relative permittivity $\varepsilon _{r2}=10$, relative permeability $\mu _{r2}=1$, and conductivity $\sigma _{2}=1$ S/m. We assume that the electrical properties of the slab do not vary with its thickness. The slab is backed by air above/below the top/bottom interfaces. The present choice of slab properties are close to that of the silicon, one of the most commonly used substrate. However, the in-house solver can support a wide range of material properties ($\varepsilon _{ri}$ = 1 to 20, $\mu _{ri}$ = 1 to 20, $\sigma _i$ = 1 S/m to 20 S/m, for i=1,2,3), including 1 GHz to 10 THz operating frequency, nm to mm thick substrate, nm to cm range of lateral separation between source and observation point, to cover most CNT composite applications [29].

 

Fig. 2. (a) Schematic diagram of a horizontal CNT embedded in a finite thickness lossy dielectric slab at a particular height ($z_0$). The structure is illuminated by a x-polarized plane wave normally incident on the top dielectric interface (z = d) with $|\bar {E}^{i}|$ = 1 V/m. (b) Variation of CNT resonance frequency with changing CNT height inside the dielectric slab ($f_r$), and comparison against CNT resonance in homogeneous medium ($f_h$) with same relative permittivity ($\varepsilon _{r}=10$).

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The absorbed power ($P_{abs}$) spectrum is calculated for two different CNT lengths ($L= {96}\;\textrm {nm}$ and $L= {86.4}\;{nm}$) and with varying height of the CNT ($z_0 = 5, 15, 25, 35, 45$ nm) inside the slab. The CNT resonance frequencies inside the dielectric slab ($f_r$) are plotted in Fig. 2(b) showing a nearly symmetric ‘U’ shape variation. The resonance frequency ($f_r$) for both CNT lengths reaches its minimum value when the CNT is placed at the middle of the dielectric slab ($z_0=25$ nm), and increases when approaching either of the top or bottom interface of the slab. This indicates that the embedded CNT experiences varying dielectric loading inside the slab which is maximum at the middle of the slab and minimum near the slab-air interfaces. If we place the same CNT in free-space it will resonate at a much higher frequency ($f_{air}$) as reported in [21]. The free-space resonance is $f_{air} =$ 23 THz for a 96 nm long CNT and $f_{air} =$ 25 THz for an 86.4 nm long CNT [21]. Free-space can be considered as a dielectric slab with thickness $d = 0$, means no dielectric loading. Instead of free-space, if we consider the same CNT embedded in an unbounded homogeneous dielectric ($d \rightarrow \infty$), it will resonate at a frequency such that $f_h < f_{air}$ if $\varepsilon _r > 1$ and $\mu _r > 1$. A CNT embedded in a dielectric slab with a finite thickness ($0 < d < \infty$) will resonate at an intermediate frequency $f_r$ that has to be in between $f_{air}$ and $f_h$ such that $f_h < f_r < f_{air}$. To confirm that, we calculate the CNT resonances while floating in a lossless homogeneous unbounded medium with the same relative permittivity as the dielectric slab ($\varepsilon _{r}=10$). In homogeneous medium, the 96 nm long CNT resonates at $f_h = 7.12$ THz as depicted by the orange horizontal dashed line (Fig. 2(b)), and the 86.4 nm long CNT resonates at $f_h = 7.79$ THz as depicted by the blue horizontal dashed line (Fig. 2(b)). The CNT resonance inside the dielectric slab ($f_r$) varies with the height of the CNT inside the slab. $f_r$ increases as the CNT moves towards the slab interface and comes closer to the free-space. Since the resonance frequency is typically inversely proportional to the size of the scatterer, we defined the effective electrical length ($L'$) of a single CNT inside a finite thickness dielectric slab by comparing the resonance of the same CNT inside the slab at different heights ($z_0$) with the resonance frequency of the same CNT in a homogeneous medium with permittivity same as the dielectric slab, as given below,

The effective electrical length ($L'$) of a CNT at a particular height $z_0$ inside the slab can be thought of as the length of an otherwise identical CNT in a homogeneous medium, with the same dielectric constant as the slab, which has the same resonance frequency $f_r$. Since $f_r$ is always greater than $f_h$, the effective electrical length ($L'$) will always be less than the physical length of CNT ($L$). The CNT experiences maximum dielectric loading and longest effective electrical length at the middle of the slab. As the CNT moves closer to either of the slab interfaces the dielectric loading reduces, the effective CNT length decreases and thus the CNT resonates at higher frequency. If the slab thickness ($d$) becomes very large in comparison to the CNT length ($L$), then $f_r$ asymptotically approaches $f_h$. On the other hand, if the thickness of the slab drops to zero, the CNT is effectively placed in free-space and the 96 nm and 86.4 nm length CNTs will resonate at 23 THz and 25 THz respectively [21]. Therefore, embedding a CNT in a dielectric slab provides an intermediate resonance frequency to the case when the CNT is floating in an infinite homogeneous dielectric medium and the case when the CNT is floating in free-space.

3.2 Symmetric CNT dimer embedded in a dielectric slab vs in a homogeneous medium

In this section we explain how a finite thickness slab can change the plasmonic resonance behavior of a length symmetric CNT dimer compared to its homogeneous medium response. We also show how the symmetry conditions are defined for embedded CNT dimer. We particularly focus on the longitudinal plasmon excitation (incident electric field being parallel to the CNT length) both in side-by-side and end-to-end dimer arrangements, inspired by the work of Jain et al. who studied plasmon coupling in gold nanorod assemblies in homogeneous medium [25].

Fig. 3(a) shows two CNTs at two different heights ($z_0 = 45$ nm, $33$ nm) embedded in a dielectric slab. Both the CNTs have equal length of $L1=L2=96$ nm and are placed in a non-collinear parallel arrangement where the incident electric field is aligned to the CNT lengths. Thus, a length-symmetric side-by-side CNT dimer is formed with a 12 nm dimer gap. All other parameters remain the same as in Fig. 2(a). For ease of comparison, the choice of CNT length, dimer gap, and arrangement are intentionally kept similar to that of our previous work ([21], Fig. 1), where we found that the non-collinear parallel CNT dimer with identical CNTs in free-space condition generate single plasmonic resonance which exhibits blue-shifts in frequency with decreasing dimer gap.

 

Fig. 3. (a) Schematic view of a symmetric CNT dimer embedded in a lossy dielectric slab. (b) Absorption power spectrum comparison of the symmetric CNT dimer embedded in the lossy dielectric slab (SLAB) Vs in a lossless homogeneous medium (HM) ($\varepsilon _r=10$). (c) Real and (d) imaginary part of the axial current ($I_{cnt}$) at anti-bonding resonance (ABR) and at bonding resonance (BR) flowing on the symmetric CNT dimer embedded in the dielectric slab.

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Presently, we compute the $P_{abs}$ for the symmetric CNT dimer of Fig. 3(a), i.e. when embedded in the dielectric slab as well as the case when the CNT dimer is placed in a lossless homogeneous space with $\varepsilon _r=10$. Both $P_{abs}$ are plotted in Fig. 3(b). For the homogeneous case (blue curve) we see a single resonance at 7.9 THz denoted by $BR_h$ ($h$ stands for homogeneous medium) created by the in-phase currents flowing on both the CNTs typical of the bonding resonance. The anti-bonding resonance disappears as the equal magnitude out-of-phase currents cancel each other [21].

On the contrary, the same symmetric CNT dimer when embedded in the finite thickness dielectric slab produces two distinct resonances at 6.37 THz and at 8.86 THz (Fig. 3(b) orange curve). The position of the ABR and BR depends on all design parameters related to the embedded CNT dimer configuration, such as the CNT properties (shape, size, distribution, orientation, and complex impedance profile), dielectric properties of all three layers, thickness of layer 2, polarization and angle of incidence of the exciting electric field etc [21]. We will discuss the effects of some of these parameters on the CNT dimer resonances in the following subsections. In this section we are more interested in exploring the reason behind the resonance splitting phenomena observed in length symmetric CNT dimer. We plot the real and imaginary parts of the currents at these two frequencies flowing on the CNT dimer in Fig. 3(c) and 3(d), respectively. At 6.37 THz, the current is flowing out-of-phase on both CNTs (purple curve) creating an anti-bonding resonance (ABR). At 8.86 THz, the current is flowing in-phase on both CNTs (green curve) creating a bonding resonance (BR). It is interesting to observe that, although both the CNTs have equal physical length, the induced currents are unequal whether it is ABR or BR. At ABR, CNT 2 has a higher magnitude current than CNT 1. As a result, the out-of-phase current on both CNTs do not cancel each other completely and the ABR emerges at 6.37 THz. At BR, CNT 1 has a higher magnitude of current than CNT 2. This inequality in induced current is caused by the height dependent dielectric loading on the CNTs inside the slab. Being closer to the top interface (next to air), CNT 1 at $z_0=45$ nm experiences low dielectric loading and shorter effective electrical length than CNT 2 at $z_0=33$ nm. Thus, one significant contribution of this paper is to show that the differences in the relative height inside a slab is another mechanism that can break the symmetry in dimers, with identical CNTs, generating dual plasmon resonances. It is also clear from Fig. 3(c)-(d) that the net CNT current magnitude at bonding resonance is larger than that of the anti-bonding resonance. According to Eq. (2), absorbed power increases with increasing magnitude of CNT current ($|\bar {I}_{cnt}|$), which explains why BR has a much higher peak power than the ABR in Fig. 3(b). The width of the resonances depends on the relaxation time ($\tau$) definition used in the $Z_{cnt}$ formulation [21]. Typical value of relaxation time for CNTs fall in the range $\tau =$ 0.3 ps to 3 ps [15]. The CNT resonance width increases with decreasing value of $\tau$. In our present study we used $\tau =$ 3 ps similar to [21]. The BR for the embedded CNT dimer inside the dielectric slab is $\thicksim 12\ \%$ blue-shifted as compared to the homogeneous case (BR$_h$). Due to the reflection at the top interface and losses in the dielectric slab ($\sigma _2=1$ S/m), the peak absorbed power in the CNT dimer also decreases significantly ($\thicksim 19$ dB) as compared to the lossless homogeneous case.

To confirm our hypothesis, we study a different kind of symmetric embedded CNT dimer as shown in Fig. 4(a), where the two CNTs are collinear as well as being parallel, commonly known as end-to-end dimer. All other parameters are same as in Fig. 3(a). Both the CNTs in Fig. 4(a) are placed at the same height $z_0=$ 45 nm with an end to end gap of 12 nm. We compute the $P_{abs}$ for this end to end symmetric CNT dimer configuration when embedded in the dielectric slab as well as the case when the CNT dimer is placed in a lossless homogeneous infinite medium with $\varepsilon _r=10$, and both are plotted in Fig. 4(b). For the homogeneous case (blue curve) we only see a bonding resonance at 6.97 THz ($BR_h$) that follows a similar explanation as before in Fig. 3(b). In contrast to Fig. 3(b), the embedded CNT dimer of Fig. 4(b) (orange curve) exhibits only BR at 7.75 THz and no ABR. Since both the CNTs are placed at the same height inside the slab they experience the same dielectric loading and as a result have the same effective electrical length. Thus, the equal magnitude out-of-phase current on both CNTs cancel each other completely and the ABR disappears. Figure 4(c)-(d) show the real and imaginary components of the excited BR current of the embedded CNT dimer. The equal amplitude of BR current on both CNTs proves that they have equal effective electrical length when placed at the same height inside the slab. If we simulate the configuration in Fig. 4(a) at higher frequencies, additional resonances will appear [10]. However, none of these resonances will be ABR since the two CNTs have the same effective electrical length, and therefore, their currents will be in phase [10]. Thus, using two identical CNTs it is possible to excite both ABR and BR (Fig. 3), or only BR (Fig. 4), just by varying the relative locations of embedded CNTs inside the slab.

 

Fig. 4. (a) Schematic view of a symmetric end to end CNT dimer embedded in a lossy dielectric slab. (b) Absorption power spectrum comparison of the symmetric end to end CNT dimer embedded in the lossy dielectric slab (SLAB) Vs in a lossless homogeneous medium (HM) ($\varepsilon _r=10$). (c) Real and (d) imaginary part of the axial current ($I_{cnt}$) at the BR flowing on the symmetric CNT dimer embedded in the dielectric slab.

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3.3 Effect of lateral and vertical separation in embedded CNT dimer

We introduce lateral separation ($\Delta x$) in the non-collinear dimer configuration as shown in Fig. 5(a). The dimer assembly is similar to that of Fig. 3(a) except that the top CNT is now shifted by $\Delta x$ which breaks the symmetry. All other parameters remain same as in Fig. 3(a). Figure 5(b) compares the $P_{abs}$ for $\Delta x =$ 0, 10 nm, and 200 nm. In addition to that $P_{abs}$ for the individual CNTs of the dimer, i.e. single 96 nm long CNT at two different heights $z_0 =$ 33 nm and 45 nm, are also simulated and included in Fig. 5(b). For $\Delta x = 0$, there is no lateral shift and the CNTs experience maximum electromagnetic coupling. When the lateral separation is increased to $\Delta x = 10$ nm, the CNT-CNT coupling reduces, the ABR shifts to higher frequency, the BR shifts to lower frequency. For $\Delta x = 200$ nm the CNTs are far apart from each other. The electromagnetic coupling between the CNTs becomes insignificant and they behave as isolated CNTs. The ABR or BR phenomena do not prevail, rather the ABR converges to the CNT2 resonance with higher effective electrical length (CNT2 at height $z_0 =$ 33 nm), and the BR converges to the CNT1 resonance with lower effective electrical length (CNT1 at height $z_0 =$ 45 nm). The $P_{abs}$ for $\Delta x = 200$ nm becomes the sum of the individual $P_{abs}$ of CNT1 at 45 nm height, and CNT2 at 33 nm height as evident from Fig. 5(b).

 

Fig. 5. (a) Embedded CNT dimer with variable lateral separation ($\Delta x$). (b) Variation in absorption power spectrum with different values of lateral separation ($\Delta x$) and comparison against individual CNT responses.

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Next, we study the effect of vertical separation between the CNTs in an embedded dimer assembly. As shown in Fig. 6(a), we chose a length symmetric CNT dimer, where the top CNT is positioned at a fixed height $z_1 =$ 45 nm and the bottom CNT is free to move along the z-axis. Thus, the center-to-center separation between two CNTs ($\Delta z$) or commonly known as the dimer gap is varied. All other parameters remain the same as in Fig. 3(a). Figure 6(b) compares the $P_{abs}$ for three different values of $\Delta z =$ 6 nm, 12 nm, and 30 nm. As $\Delta z$ increases, CNT2 approaches the center of the dielectric slab and its effective electrical length increases. At the same time, it moves away from CNT1 reducing the electromagnetic coupling between the two CNTs. These combined effects cause blue-shifts in the ABR and red-shifts in the BR. That is, the separation between the ABR and BR decreases as the center to center separation between two CNTs ($\Delta z$) increases similar to what was reported for CNT dimers in free-space [21].

 

Fig. 6. (a) Embedded length symmetric CNT dimer with variable dimer gap ($\Delta z$). (b) Variation in absorption power spectrum with different values of dimer gap ($\Delta z$).

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If the same dimer is placed in homogeneous medium, and the CNTs are moved sufficiently apart (lateral and/or vertical) then the ABR and the BR will merge to the single resonance of an isolated 96 nm long CNT floating in homogeneous medium.

3.4 Length asymmetric CNT dimer in a dielectric slab

In this section, we perform a comparative study between symmetric and asymmetric CNT dimers embedded in a dielectric slab with a finite thickness. Figure 7 (top) shows three such embedded CNT dimer configurations: Case 1, Case 2, and Case 3. Case 1 is the same symmetric dimer configuration with 96 nm long CNTs used in Fig. 3(a). Case 2 and Case 3 differ from Case 1 only by the CNTs’ lengths. In Case 2, the top CNT is $L1=96$ nm long but the bottom CNT has a shorter physical length of $L2=86.4$ nm. Case 3 is the opposite of Case 2, where the top CNT has a shorter physical length of $L1=86.4$ nm and the bottom CNT is $L2=96$ nm long. In each case, the two CNTs are placed at two different heights ($z_0=$45 nm, 33 nm) inside the slab such that the dimer separation is 12 nm. Figure 7 (bottom) compares the computed $P_{abs}$ for all three cases. Both the ABR and BR show significant shifts in resonance frequency and peak power value among these three cases which is caused by the differences in dielectric loading on the embedded CNTs depending on their relative locations. To better explain the variation in $P_{abs}$ we calculate the effective electrical lengths of the embedded CNTs used in Cases 1-3 for different heights using the relation (3) and are listed in Table 1. The equivalent CNT dimers with computed effective electrical lengths are shown in Fig. 7 in a dashed box just below the actual configurations. In all three cases, the effective electrical length of embedded CNT ($L'$) is smaller than their actual physical length ($L$). The symmetric dimer in Case 1, which has a zero difference in physical length ($\Delta L=L_1-L_2=0$), while embedded in a dielectric slab behaves as a length asymmetric CNT dimer with a difference in effective electrical length $\Delta L'=L'_1-L'_2=6$ nm. In Case 2, the asymmetric CNT dimer has a difference in physical length of $\Delta L=10$ nm, but the difference in effective electrical length is only $\Delta L'=3$ nm. However, when we swapped the position of CNTs from Case 2 to Case 3, we find a large difference in effective electrical length of $\Delta L'=14.5$ nm. The power absorbed by a CNT ($P_{abs}$) at any frequency increases with the square of magnitude of the resulting current on the CNT dimer. At the ABR, the current increases with the difference in effective electrical lengths ($\Delta L')$ [21]. In Case 1, ($\Delta L'=6$ nm), the peak $P_{abs}$ at the ABR is higher than that of Case 2 ($\Delta L'=3$ nm) and lower than that of Case 3 ($\Delta L'=14.5$ nm). On the contrary, the peak $P_{abs}$ at the BR increases with the sum of effective electrical lengths of the dimer ($\sum L'=L_1'+L_2'$). For Case 1, $\sum L'=177.6$ nm, for Case 2 and Case 3 they are close to $\sum L'=169$ nm. If the asymmetric CNT dimers of Case 2 and Case 3 are placed in a lossless homogeneous space with $\varepsilon _r=10$, an identical $P_{abs}$ will be achieved as expected. In spite of the swapped CNT position, no differences in resonance frequency or in peak power level will be seen between Case 2 and Case 3 in the infinite homogeneous medium. Hence, it is clear that the shift in resonance and power level found in Fig. 7 is due to the presence of the finite thickness dielectric slab.

 

Fig. 7. (top) Case 1: symmetric embedded dimer; Case 2 and Case 3: asymmetric embedded dimer with swapped CNT position. (middle) Effective electrical lengths of the CNT dimers. (bottom) Absorption power comparison of the above three cases of embedded CNT dimers.

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Table 1. Effective electrical length for embedded CNT

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3.5 Canceling the ABR of CNT dimer embedded in a dielectric slab

In Fig. 7, we showed that the asymmetric CNT dimer of Case 2 ($\Delta L'=3$ nm) had the minimum $P_{abs}$ at the ABR among all three cases. The ABR vanishes when the two CNTs in the dimer are approximately symmetric. However, the contribution of this section is to highlight that the symmetry for CNTs embedded in a dielectric slab with a finite thickness has to be in the effective length and not the physical length. To realize an equivalent symmetric CNT dimer embedded in a dielectric slab we need $\Delta L' \rightarrow 0$. In Case 2, $\Delta L'$ can be reduced either by decreasing $L'_1$ or by increasing $L'_2$. This can be achieved either by tuning the CNT physical length or by varying CNT height. First, we chose to tune the length of CNT2 ($L_2$) keeping all other parametric values constant. The value of $L_2$ was increased from 86.4 nm, in fine steps until $L_2=90$ nm. It was found that with increasing value of $L_2$ the peak power at ABR decreases initially as shown in Fig. 8 (bottom). At $L_2=88$ nm the ABR disappears from the $P_{abs}$ spectrum (green curve). Further increments of $L_2$ beyond 88 nm re-excite the ABR with increasing peak power. For $L_2=88$ nm, the resultant difference in effective electrical length is found to be only $\Delta L' \approx 1$ nm as shown in Fig. 8 (top), which agrees with our hypothesis. It is important to emphasize that the ABR disappears at $L_2=88$ nm only when the two CNTs are perfectly aligned. If the CNTs are misaligned, we estimate that the ABR will re-appear again at a rate of 2-3 dBW/(degree of misalignment) [21].

 

Fig. 8. Finding a special case of embedded asymmetric CNT dimer which eliminates its ABR from the absorption power spectrum. (top) Initial configuration with $L_2=86.4$ nm shows both ABR and BR. Tuned configuration with $L_2=88$ nm shows only BR and no ABR. (middle) Effective electrical lengths of CNTs in embedded condition. For the Initial configuration (left) $\Delta L'\approx 3$ nm and for the tuned configuration (right) $\Delta L'\approx 1$ nm. (bottom) Absorption power spectrum for different values of $L_2=86.4$ nm to 90 nm.

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Next we chose to vary CNT height to manipulate ABR peak power. The symmetric CNT dimer of Fig. 9(a) has two 96 nm long CNTs separated by a fixed dimer-gap of $\Delta z= 12$ nm. The dimer can be translated up-down inside layer 2. The z-position of the central axis of the dimer is indicated by $zm$. Three different dimer locations are studied and the corresponding absorption spectrum is presented in Fig. 9(b). When $zm = 39$ nm, this is the same configuration as of Case 1 in Fig. 7, where CNT1 and CNT2 are located at 5 nm and 17 nm below the top interface respectively. Both the ABR and the BR get excited. Similarly, when $zm = 11$ nm, CNT1 and CNT2 are located at 17 nm and 5 nm above the bottom interface respectively. Due to the configuration’s symmetry both CNTs experiences similar dielectric loading as of the previous configuration and produces similar $P_{abs}$. However, when $zm = d/2 =$ 25 nm, i.e. the dimer axis is at the center of the slab, CNT1 is 19 nm below the top interface, and CNT2 is 19 nm above the bottom interface. Thus, both CNTs experience similar dielectric loading and thus exhibit similar effective lengths. The ABR disappears as the $\Delta L'=L'_1-L'_2$ approaches zero. Being located near middle of the slab, both the CNTs experience high dielectric loading and the BR moves to lower frequency.

 

Fig. 9. Tunning the height of embedded symmetric CNT dimer to eliminate its ABR from the absorption power spectrum. (a) Schematic of a length symmetric CNT dimer with a fixed dimer gap ($\Delta z= 12$ nm) and varying dimer height ($zm$). (b) Absorption power spectrum for different values of $zm$. The ABR vanishes for $zm=25$ nm, when both the CNTs exhibit same effective electrical length.

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In summary, in this work, we studied several CNT dimer configurations embedded in a dielectric slab and, in many cases, a unique resonance behavior was exhibited. In the future, we will investigate different techniques to experimentally fabricate some of the CNT dimer configurations studied in this work. Multiple experimental techniques have been recently proposed for CNT fabrication [21]. One of the most promising techniques, is Atomic Force Microscopy (AFM) manipulation that can control the shape and location of a CNT with a precision on the order of fractions of a nanometer, and therefore, would be a promising technique for fabricating the CNT dimer configurations studied here [21].

4. Conclusions and future work

This study shows that carbon nanotubes (CNTs) embedded in a finite thickness dielectric slab exhibit new plasmon resonance phenomena that cannot be explained by the homogeneous medium model. The electromagnetic scattering response and the resonance change with the relative distance between the embedded CNTs and the interfaces of the dielectric slab. As the CNTs move towards the slab interface, their effective electrical length decreases causing a blue-shift in their resonance frequency. By optimizing the CNTs’ lengths, distribution, and their proximity to the slab interfaces, it is possible to control their resonance frequencies and absorption power. Therefore, physically symmetric CNT dimers can behave like asymmetric CNTs when embedded in a dielectric slab. The shift in the bonding resonance (BR) and the anti-bonding resonance (ABR), and the variation in peak power level under the influence of the embedding layer can be very useful in developing CNT dimer based nano-sensing applications, such as,

In practical applications, the fabricated CNTs are mostly skewed and misaligned. Thus, any potential sensitivity issues arising from non-uniformity of embedded CNTs should be investigated in details while developing CNT dimer based sensors. Thus, our future work will extend the present analysis to more complex embedded CNT distribution, including the sensitivity towards diverse orientations, alignments, and inter-tube proximity.