Strong plasmon coupling between two gold nanospheres on a gold slab

In this work, the plasmon coupling effect between two gold nanospheres on a gold slab is investigated. At plasmon resonance frequencies, electrons on the surface of the slab are absorbed into spheres and contribute to plasmon oscillation. This effect can help enhance the local electric field and optical coupling force between the two spheres.

In this paper, the plasmon coupling between two gold nanospheres on a gold slab is investigated. Two different resonance modes could be established in this system because of the induced current on the surface of the slab. The exchange current between the two spheres further enhance the local electric field in the structure, resulting in a strong plasmon coupling force between the two spheres.   Figure 1 (a) is one gold sphere in the air. In Figure 1 (b), the gold 3 sphere is placed on a gold slab. Figure 1 (c) shows two identical gold spheres placed on a gold slab. The radius of each of the three structures is R=200 nm. The size of the slab is 2000×2000 nm with a thickness of 150 nm. The gap between the two spheres in Figure 1 (c) is defined as g, which is tuned from 10 to 75 nm in this work. In this work, the sphere of this size are chosen because it is very simple nanostructures and widely found in experiments.

Structure and numerical model
Besides, its resonance is not too strong to overtax the simulation software. In all three configurations, the incident EM wave propagates in the (-y) direction with its polarization in the x direction. Open boundaries are used in all the directions.

Two plasmonic resonance modes
First, one gold sphere (R=200 nm) without a slab is investigated. In the simulations, an EM wave incident to a plane is directed onto the sphere with its polarization along the x direction [ Fig. 1 (a)]. In the program, the local electric field x E near the surface of the sphere is recorded. In principle, if we could determine the local e field and conductivity of material, we could obtain the current density based on equation j σ E = ⋅ . Here, the metal is defined as drude model and its σ could be calculated. E could be obtained directly in the simulations, and then current density could be calculated. Excited by the incident wave, electrons oscillate in the sphere. This produces plasmon resonance. In the simulation, when the incident frequency is swept from 0 to 400 THz, the recorded electric field shows one resonance peak at 182 THz.
At this frequency, the distribution of the induced current density inside the sphere j is obtained from the simulation, as given in Fig. 2

(a). Based on continuity equation of charges
, the distribution of charge density is able to obtained as Then the plasmon mode in nanospheres of this size is a kind of surface modes. For such a surface mode, its eigen frequencies depend on its geometry size. 5 Next, one sphere on gold slab is investigated [ Fig. 1 (b)]. The size of the slab is much larger than the sphere; thus the influence of the boundary of the slab can be ignored. For the local field near the surface of sphere, only one resonance peak is found at 227 THz. The current density at this resonance mode is given in Fig. 2 (d). The results show that when the sphere is placed on the surface of the metal slab, the electrons on the surface of slab are absorbed into the sphere, contributing to the plasmon resonance. As more electrons accumulate on the two sides of sphere [ Fig.2 (e)], the plasmon resonance becomes stronger, and better local field enhancement is produced. The sphere on the slab can still be considered The properties of the two spheres on the slab [ Fig. 1 (c)] are investigated. In the program, the x-component local electric field x E in the gap between the two spheres is recorded. In our simulations, the gap between the two spheres is tuned from 10 to 75 nm. When the incident frequency is swept from 0 to 400 THz, the recorded electric field, x E , is obtained in the simulations. This is shown in Fig. 3 (a). For the structure with a different gap, g, the  Fig. 2 (k)]. Thus, the two spheres cannot be seen as two separate dipoles, they must be seen as one big effective electric dipole [ Fig. 2 (l)]. This low-frequency mode is called the single big dipole mode (SBD).
In the simulations, the local E field enhancement between the two spheres with a different gap is calculated [ Fig. 3 (a)]. The gap size g is tuned from 10 to 75 nm. For the lower resonance mode, the oscillation of electrons is not confined inside one sphere and could flow from one sphere to the other. The whole system could be seen as U-shaped LC circuit.
The low eigenfrequency is actually the LC resonance frequency ω 1 / LC = . The gap between two spheres could be seen as the capacitance of the LC circuit. When gap g is increased, the capacitance C~1/g will decrease. As a result, the resonance frequency will increase. For comparison, the frequency dependence of the E field between the two spheres without a slab is also calculated. Gap values of 10, 20, and 30 nm are chosen. The recorded electric fields are given in Fig. 3 (b). For all the three curves, only one resonance peak was found (denoted as 0 ω ). The E field between the two spheres is also increased when g is  Aside from the comparison between the structure with slab and the structure without slab, the difference between the TSD mode and the SBD mode of the structure with slab [ Fig. 3 (a)] is also compared. When the gap is reduced from 75 to 10 nm, both E fields of the two modes increase. However, the E field of the SBD mode ( ) 1 E ω increases more rapidly than that of the TSD mode ( ) 2 E ω . When g=75 nm, the E field of the SBD mode is nearly equal to that of the TSD mode ( ) ( ) 1 2 E ω E ω . When g is reduced, SBD becomes stronger than TSD. At g=10 nm, the E field of the SBD is approximately twice that of the E field of TSD For the TSD mode, most of the electrons remain confined in each sphere, and the field enhancement mainly comes from the addition of two separated small electric dipoles. For the SBD mode, a strong exchange current between the two spheres allows more electrons join the oscillation. More incident EM wave energy is absorbed into the energy of electrons. This makes the E field enhancement of the SBD mode bigger than that of the TSD mode. Actually, we have already calculated the structure in which spheres do not touch the slab. In the results, the two spheres cannot exchange electrons with each other. The strong local field enhancement disappears and resonance coupling is much weaker. These improves that the touching is very important. 8

Optical force between two spheres on the slab
Previous works [19][20][21][22] have shown that the plasmon coupling between two metal spheres could result in a strong coupling force between the two spheres. Our results show that the two gold spheres on a gold slab present a much stronger plasmon resonance and local field enhancement than the two spheres without the slab. The gold slab is then expected to help increase the optical coupling force between the two spheres.
The optical force between the two spheres on a slab and the optical force between the two spheres without a slab are compared. The calculation of the optical force is based on the Maxwell tensor method. In the simulations, both the electric field ( ) Firstly, we calculate the frequency dependence of the optical force between the two spheres on the slab with the gap g=50 nm and the incident intensity being 1.0 2 mW / μm . The result is given in Fig. 4 (a). The simulation shows that the optical force between the two spheres is an attractive force. At these two plasmon resonance frequencies, the optical force is enhanced as anticipated. For the SBD mode at a low resonance frequency, the optical force is 67 pN, whereas for the TSD mode at a high resonance frequency, the optical force is merely 21 pN. The former is three times that of the latter. The exchange current is formed between the two spheres in the SBD mode. This exchange current causes a much stronger optical force 9 in the SBD mode than that in the TSD mode. For a single sphere in air, the scattering force by a plane wave is about 0.8 pN at resonance frequency, which is smaller than SBD mode by two orders of magnitude. Second, the dependence of the attractive optical force on the gap between the two spheres is calculated. The results are given in Fig. 4 (b). Only optical forces at the two resonance frequencies are provided. For comparison, the results for the two spheres without the slab are given. The changes of optical force for all the three cases are stronger for a smaller gap.
However, both modes of the two spheres on a slab possess stronger forces than the resonance mode of the two spheres without the slab (denoted as the red dotted curve). For a large gap value, the optical force of the SBD mode (denoted as the black dotted curve) is nearly equal to the TSD mode (denoted as the blue dotted curve). However, when the gap decreases, the optical force of the SBD mode increases much faster than the TSD mode. At g=10 nm, the optical force of the SBD mode is four times that of the TSD mode and eighteen times that of the case without the slab. These results prove that the exchange current induced in the slab helps increase the optical force between the two spheres greatly.

10
In summary, this work shows that, given nanospheres placed on a slab, electrons on the surface of slab are able to join in the plasmon oscillation and contribute to the resonance mode. This effect makes the plasmon resonance much stronger. Moreover, it produces much better local field enhancement. Two resonance modes are introduced in this system: the SBD mode at low frequency and the TSD mode at high frequency. Much stronger optical coupling force is produced by the SBD mode than the TSD mode because of the exchange current between the two spheres. Aside from the optical force, the slab could also be used to enhance the Raman scattering and other nonlinear optical effects in nanospheres in the future.