Formation and Evolution Mechanism of Plasmon Resonance from Single Ring-Shaped Nanotube to Dimer and Arrays

The formation and evolutionmechanisms of plasmon resonance from single Ring-shaped nanotube to dimer and arrays are studied; an attempt has beenmade to bridge the gap between single-tube, dimer, and array. Results show that resonantmodes can be divided into three types: quadrupole, hexapole, and octupole resonance from visible to near infrared region, and each mode maintains relatively stable resonant characteristics, but the optical transmission properties including redshift and blueshift of the modes and band gap are highly tunable by adjusting the number of nanotube and intertube spacing values. The field-interference mechanism has been suggested to explain the physical origin.


Introduction
Surface plasmon (SP) stems from the coupling between metallic nanostructures and light and great interest has been aroused for its rich physical phenomena, mechanisms, and potential application [1][2][3][4].
The studies on single metallic nanostructures include nanospheres [5], nanoprisms [6], nanorods [7], nanocubes [8], and shells [9].The results show that the plasmonic properties depend strongly on the size and shape of the nanostructure and material properties of the surrounding medium.
Coupled metallic nanoparticle pairs are referred to as plasmonic "dimer" [10].The effects of symmetry breaking on plasmon resonance of dimer have been analyzed in many works [11,12], and the "hot spots" may be formed by the pair of closely spaced nanoparticles and caused by the strong coupling interactions between the near fields of two closely spaced nanoparticles [11,13].
Single nanostructure, dimer, and arrays have been highly studied, respectively.And the above researched nanostructures such as circular-shell-core nanoparticle or nanotube are the common structure; the metallic Ring-shaped nanotube and its array have not been discussed comprehensively; and the existence of length  of the Ring-shaped nanotube (shape of nanotube) influences greatly the transmission spectrum and plasmon resonance.In addition, we want to research the formation and evolution mechanisms of plasmon resonance between single particle, dimer, and array.
Based on the above analysis, in this paper, we propose single Ring-shaped nanotube, dimer, and arrays and discuss its optical transmission properties by using Finite-Difference Time-Domain (FDTD) method.First, we make a simple comparison between the Circular-shaped and Ring-shaped nanotube arrays.Then, from the transmission spectra of the simulation results, we can see the formation and evolution process of the regularly resonant transmission modes and band gap.A field-interference mechanism has been suggested and investigated.In addition, the transmission properties have been further characterized with various intertube spacing between adjacent nanotubes.The present study has led to an improved understanding of the formation and evolution of plasmon resonant modes in nanostructures, and the proposed models and results may have great potential applications in subwavelength near-field optics.

Model and Method
Figure 1 shows schematically two-dimensional Ringshaped nanotube with infinite length along the  direction.Figure 1(a) is the single-tube, in the whole paper, the outer radius, inner radius, and the length of the Ring-shaped nanotube are fixed  1 = 50 nm,  2 = 40 nm, and  = 100 nm, respectively.A nanodimer composed by two nanotube pairs is shown in Figure 1(b), and the intertube spacing between adjacent nanotubes can be changed by varying the parameter .Further, the array of ten nanotubes is depicted in Figure 1(c).In addition, the metal is chosen to be Au.The relative permittivities of the gold are approximated by the Drude model, which is defined as () = 1 −  2  /( 2 +   ), where   = 1.37 × 10 16 s −1 is the bulk plasmon frequency and   = 4.08 × 10 13 s −1 represents the electron relaxation time [29].The outside and inner-core outside dielectric medium of Ring-shaped nanotube are defined as air; the dielectric constant is  = 1.
We use two-dimensional FDTD [30] method to simulate the structure with a lattice of   ×   = 1800 nm × 300 nm for single-tube, dimer, and array, and infinity along the  direction, and the spatial mesh and time steps are set as Δ  = Δ  = 1 nm and Δ = Δ/2, where  is the velocity of light in vacuum.The left and right surfaces of the calculated region along the  direction are truncated by perfectly matched layer (PML) absorbing boundary conditions, and due to the periodicity of the structure along  direction, the top and bottom boundaries along  direction are treated by periodic boundary conditions.A Gaussian single pulse of light with a wide frequency profile is sent along the  direction with TM polarization.

The Differences between the Circular-Shaped and Ring-Shaped Nanotube Arrays
Let us make a simple comparison between the Circularshaped and Ring-shaped nanotube arrays.We fix the outer radius  1 = 50 nm, inner radius  2 = 40 nm, and separation distance of tubes  = 20 nm both for Circular-shaped and Ring-shaped nanotube arrays and set length  = 100 nm of the Ring-shaped nanotube.
The transmission spectrum of the band structure can be formed in both the Circular-shaped and the Ring-shaped nanotube arrays.From Figure 2(a), we can see that only one forbidden band has been formed in range from 428 nm to 577 nm, and it shows that the resonant peaks on the left side are dense and irregular, while the resonant peaks on the right side are regular.While two forbidden bands form intransmission spectrum of the Ring-shaped nanotube array, which is showed in Figure 2(b), the right forbidden band is very wide ranging from 680 nm to 1000 nm, the left forbidden band is narrow ranging from 370 nm to 500 nm, and five regular resonant peaks appear between the two forbidden bands.From the results, we can find that the existence of length  of the Ring-shaped nanotube influence greatly the transmission spectrum and plasmon resonance; the details of resonant properties of Ring-shaped nanotube are studied in next sections.This is why we choose this structure.

Numerical Results and Discussion: Formation and Evolution Mechanisms of Plasmon Resonance
First of all, we fix the outer radius  1 = 50 nm, inner radius  2 = 40 nm, and separation distance of tubes  = 20 nm and discuss the effect of the tube number  on optical transmission properties and then study the formation and evolution mechanisms of plasmon resonance from single Ring-shaped nanotube to dimer and array.The transmission spectrum with different numbers  = 1, 2, 10 are shown in Figures 3(a)-3(c), namely, the single-tube, dimer, and array, respectively.We find some features by comparing Figures 3(a)-3(c).First, the resonant modes can be grouped into three types.Type 1 is quadrupole resonance including the peaks of A1, A2, and A3, and the Type 2 peaks B1, B2, and B3 belong to hexapole resonance; the Type 3, C1, C2, and C3, is octupole resonance; we will discuss the multipole resonant properties by simulating the electric field distributions in next chapter.Second, from Figure 3, we can find that the properties of the three types of modes exhibit certain regularity with  increasing.When only the single-tube system is considered, the higher-order modes are excited in the visible and near infrared region, just like the resonant peaks B1 and C1, while the quadrupole resonance peak A1 stays away from this area and distributes in infrared region (not shown here).Then, we take into account the dimer system; due to the coupling effect between nanotube pairs, quadrupole resonance peak A2 gradually forms in near infrared region; we also consider that quadrupole resonance peak A2 blueshifts into near infrared region.At the same time, the higher-order modes B1 and C1, the dip between B1 and C1, and the dip on the right B1 redshift.
It is more important that the transmission band begins to form; forbidden band can clearly be seen in the transmission spectra from Figure 3(b).As shown in Figure 3(c), due to the coupling effect between each nanotube, regular quadrupole resonance peak A3 and hexapole resonance peak B3 establish; for example, six hexapole resonance peaks B3 appear.In addition, two wide forbidden bands have been formed, the right forbidden band ranges from 680 nm to 1000 nm, the left band gap is smaller than the right, and it ranges from 380 nm to 500 nm.The formation and evolution process is similar to the forming process of band lines in photonic crystal, which can be used to design an optical filter.
In order to analyse the basic physical mechanisms for above phenomena, we plot the electric field distributions    and the corresponding peaks have been labeled in Figure 3, respectively.We consider that red colored areas represent positive and blue means negative.
First, we can find that positive and negative electric field   spread alternately near the edges of the nanotube from the left column of Figures 4(a) and 4(b).Figure 4(a) shows that the left sign of   is the same of the right, while they are opposite on the top and bottom for single nanotube, and electric field also exists in the middle and inner of the tube; its sign is opposite to the left and right.We consider that it is a hexapole field distribution for the plasmon modes in the nanotubes.It is the reason that we regard this transmission peaks as the peaks of Type 2 hexapole resonance in Figure 2. In addition, Figure 4(b) depicts the octupole resonance at the Type 3 C1 in Figure 3, but the signs of the top and bottom, left and right, and outside and inner change alternately.Besides, we can see analogous hexapole and octupole resonant patterns in the dimer and array, which are shown in Figures 4(d) and 4(e) for B2 and C2 and Figures 4(i) and 4(k) for B3 and C3.The only difference is that some of the nanotubes form a group; this phenomenon is particularly evident in Figure 4(i) B3, where each of the two tubes forms a group, and five groups appear totally.
Second, Figure 4(c) for A2 and Figure 4(f) for A3 show the quadrupole resonance; the electric field   distributes nearly outside of the tube, and we could barely see the electric field inner of the tube.The signs of the surface charges at two sides of the gap are opposite; the electric fields between two nanotubes are coupled together closely and intensified, which is just in analogy with a simple capacity device, so the field distributions are mostly concentrated at every other gap between adjacent nanotubes [24][25][26].The results indicate that the oscillation of charge density exists between the nanotubes.In other words, the surface plasmon wave really exists and propagates along the surface.Due to the nanotube with different lengths that may be used as chain waveguides, the understanding of the plasmonic behaviors in coupled nanotubes of arrays is especially important [24].
Last, we draw the spatial distributions of the   at wavelengths of band gap D3 and E3, which are depicted in Figures 4(h) and 4(j); electric field only emerges on left several tubes.No excitation of resonant mode exists owing to the forbidden band gap, and incident light cannot transmit.
Successively, we investigate the effect of different spacing intertube on transmission spectrum and electric field distributions.
Firstly, the transmission spectra varying intertube spacing between adjacent nanotubes for dimer is shown in Figure 5(a).The transmission spectra blueshift obviously with increasing of the intertube spacing; the peak intensities of A  5 are 1134 nm, 1035 nm, and 989 nm, respectively.During transmission peak A that is the quadrupole resonant peak, the field distributions of   only appear outside the nanotube.When separation distance is  = 20 nm, similar to that of nanotube array [24][25][26], the opposing and uniform surface charges are aligned alternatively between the gap, so the field distributions are concentrated between adjacent nanotubes.As  = 40 nm and 60 nm, the uniform surface charges are lessened due to less coupling between nanotubes.When the intertube distance gets farther apart, the uniform surface charges reduce further and the intensity of electrical field attenuates sharply.This phenomenon exists obviously in Figures 6(a)-6(c) for dimer.For the case of array, from Figures 6(d)-6(f), we can see that some of the nanotubes form a group when we increase the intertube spacing, this phenomenon is particularly evident in array, and five groups appear totally when  = 40 nm, and four groups form as  = 60 nm.Due to the group formation, the electric fields   couple strongly with inner of each group, so we do not see that the intensity of electrical field inner of the group attenuates, but electrical field between the group attenuates evidently.

Conclusion
In conclusion, we proposed single Ring-shaped nanotube, dimer, and arrays and discussed its optical transmission properties; an attempt has been made to bridge the gap between single particle dimer, and array.Results show that the resonant modes can be divided into three types: quadrupole, hexapole, and octupole resonance from the visible to near infrared region, and each mode maintains relatively stable resonant characteristics, but the resonant transmission sensitively depends on the number of nanotubes; magnitude modification, redshift and blueshift of the resonance modes, and band gap are observed.From the transmission spectra and electric field distributions of the single nanotube to dimer and arrays, we can see the formation and evolution process of the regularly resonant transmission modes and band gap.In addition, the transmission properties of the dimer and array have been further characterized with various intertube spacing between adjacent nanotubes.The presented study not only sharpens our understanding of the formation and evolution of plasmon resonant modes in nanostructures but also provides a profound comprehension into the plasmonic interference.And we hope that the proposed models and results can be used to design structures with optimized nanophotonic devices.

Figure 4 :
Figure 4: The field distributions of   at the transmission peaks and band gap (A, B, C, D, and E) are labeled in Figure 2, left column for single-tube, middle column for dimer, and right column for array.