## Abstract

The discovery of single-molecule sensitivity via surface-enhanced Raman scattering on resonantly excited noble metal nanoparticles has brought an increasing interest in its applications to the molecule detection and identification. Periodic gold bowtie nanostructures have recently been shown to give a large enhancement factor sufficient for single molecule detection. In this work, we simulate the plasmon resonance for periodic gold bowtie nanostructures. The difference between the dipole and the quadrupole resonances is described by examining the magnitude and phase of electric field, the bound surface charge, and the polarization. The gap size dependence of the field enhancement can be interpreted by considering cavity field enhancement. Also, additional enhancement is obtained through the long-range collective photonic effect when the bowtie array periodicity matches the resonance wavelength.

© 2011 OSA

## 1. Introduction

It has been shown that a bowtie-shaped nanoantenna, where two metallic triangular prisms facing tip-to-tip and separated by a small gap, produces a large electromagnetic field confined to the gap region [1–5]. This field enhancement enables the detection of single molecule via surface-enhanced Raman scattering (SERS) [6–11]. While the fabrication of a nanoantenna can be achieved using electron beam lithography and lift-off techniques, it is fruitful to have simulation results to provide guidelines in choosing the geometry of the nanostructure, the wavelength and polarization of the incident light, and the dielectric properties of the constituents of the system to obtain the optimum field enhancement. The purpose of the present study is to perform simulations on a model system to examine the different resonance modes and to provide the essential trends of how the plasmon resonance is influenced by the geometric parameters. Specifically, the effects of the gap size and the periodicity of bowtie arrays on the plasmon resonance of the periodic gold bowtie nanostructures are studied.

## 2. Finite-difference time-domain simulations

A model system of the periodic gold bowtie nanostructures is used for simulations. The side length of the equilateral triangle is 100 nm and the thickness of the Au bowtie is 40 nm with a 0.5 nm thick analyte, *p*-aminothiolphenol (*p*-ATP), on the top surface of the bowtie. At the bottom surface of the bowtie, an 8 nm thick Cr is used to bond the Au bowtie to Si substrate. A schematic drawing of the top view of the bowtie structure on the *x*-*y* plane is shown in Fig. 1(a)
and the side view on the *x*-*z* plane is shown in Fig. 1(b). While a perfectly sharp tip at the apex of the triangle cannot be achieved because of limitations by using the electron beam lithography followed by the vapor deposition of gold, simulations of imperfect bowtie structures have been performed by either truncating the tips of the triangle [12,13] or assuming a curvature at the apex [3]. In the present study, the truncated bowtie shown in Fig. 1(a) is adopted as a simplified structure in simulations such that other parameters (e.g., gap between apexes and inter-bowtie distance) can be changed systematically to elucidate the essential trends of how other parameters affect plasmon resonances. First, the difference between the dipole and the quadrupole resonances is examined. Then, the mechanism of the gap size dependence of the field enhancement is explored. Finally, the effect of the bowtie array periodicity on the field enhancement is studied.

Lumerical FDTD Solutions [14], a commercial electromagnetic software based on the finite-difference time-domain method, is used to perform the simulation. Considering the periodic structure, the unit cell adopted in simulations has a dimension corresponding to the inter-bowtie distance on the *x*-*y* plane. Unless noted otherwise, the unit cell of 400 nm × 300 nm on the *x*-*y* plane is adopted in simulations. The dielectric properties of Au, Cr, and Si used in simulations are taken from Palik’s handbook [15] and the dielectric constant of *p*-ATP is 7 [16]. A plane wave (500 to 1000 nm wavelength) polarized across the junctions between the triangles (i.e., along the *x*-direction) is illuminated from the bowtie side (i.e., in the negative *z*-direction). The mesh sizes in the bowtie region (including analyte, gap, and Cr layer) vary from 0.25 to 1 nm, and an automatic graded mesh is used in the region outside the bowtie.

## 3. Results

The simulation results of both the near-field (local field) and the far-field (reflectance) are recorded. For the local field, *E* intensity (i.e., |*E*|^{2}) is the largest on the plane containing the bowtie surface among all the constant-*z* planes, and it is located at the opposing apexes.

#### 3.1 Maximum E intensity enhancement and minimum reflectance

For an apex width, *w*, of 20 nm, the maximum *E* intensity enhancement (i.e., |*E*|^{2}/|*E*
_{0}|^{2} where *E*
_{0} is the electric field of the illumination wave) and the reflectance are shown in Figs. 2(a)
and 2(b), respectively, as functions of the illumination wavelength, *λ*, at different gap sizes, *d*. There are two peaks in each curve in Fig. 2(a). These two plasmon resonances have also been obtained elsewhere using discrete dipole approximation and the peaks at the longer and the shorter wavelengths have been qualitatively described as dipole and quadrupole resonances, respectively [17,18]. When the two truncated triangles are sufficiently apart, the dipole and the quadrupole resonances occur, respectively, at *λ* = ~640 nm and ~530 nm and red shift occurs for both resonances as *d* decreases. It should be noted that red shift of the dipole peak becomes more evident as *d* becomes smaller. This is in agreement with the plasmon ruler equation that the plasmon wavelength shift for polarization along the interparticle axis decays nearly exponentially with the interparticle gap [19–21]. It is shown in Fig. 2(a) that the quadrupole peak is less sensitive to the gap size and is higher than the dipole peak when *d* is greater than 10 nm. The dipole peak increases as *d* decreases. It has also been reported elsewhere for Ag nanoparticles that the quadrupole resonance is much less sensitive to the interparticle interactions than the dipole resonance [22].

The minimum reflectance corresponds to the maximum scattering and absorption of the system. There are two valleys in each curve in Fig. 2(b) for the reflectance and both valleys show red shift as *d* decreases. The far-field resonance can occur at a different wavelength compared to the near-field resonance because far-field is dictated by the average behavior of the entire bowtie. While the magnitude of local intensity enhancement strongly depends on the gap size as shown in Fig. 2(a), Fig. 2(b) shows that the magnitude of minimum reflectance is relatively insensitive to the gap size.

For the gap size *d* = 10 nm, the dipole and the quadrupole resonances occur, respectively, at *λ* = 655 and 540 nm. The *E* intensity enhancement profiles on the plane containing the bowtie surface are shown in Figs. 3(a)
and 3(b), respectively, for *λ* = 655 and 540 nm. The maximum enhancement is always at the apexes. However, compared to the dipole resonance, the quadrupole resonance has more significant enhancement at the sides. This is in agreement with the results obtained elsewhere [17]. While the *E* intensity enhancement is highly confined to the gap region for dipole resonance, it spreads to the bowtie surface for quadrupole resonance.

#### 3.2 Phase of electric field and bound surface charge

It is instructive to examine the phase of the electric field and the bound surface charge on the bowtie surface to distinct the quadrupole resonance from the dipole resonance. The dominant component of the electric field is *E _{x}* in most regions because of the polarization direction. The field has a harmonic time dependence and the solution of the time-independent component,

*E*, from FDTD simulations is a complex number, such that

_{x}where |*E _{x}*| and

*ϕ*are the magnitude and the phase of

*E*, respectively, and

_{x}While the *E* intensity; i.e., |*E*|^{2}, enhancement profiles are shown in Figs. 3(a) and 3(b), the corresponding phase profiles of the dominate component of the electric field *E _{x}*,

*ϕ*, are shown in Figs. 3(c) and 3(d). Only

*E*is discussed here to avoid the complexity of the three components of the electric field. For

_{x}*λ*= 655 nm (i.e., dipole resonance), the phase is ~–π in both the opposing apex and the end of bowtie regions, and it is ~π in between those two regions. However, it should be noted that –π and π are the same in terms of the trigonometric functions. Hence, the phase is relatively uniform on the bowtie surface for dipole resonance. For

*λ*= 540 nm (i.e., quadrupole resonance), the phase is about zero in the opposing apexes region and progressively shifts toward π/2 as the position moves toward the ends of the bowtie.

The bound surface charge density can be derived from the electric field. In the presence of an electric field, **E**, the polarization of the material, **P**, is

where *ε _{m}* and

*ε*

_{0}are the electric permittivities of the material and vacuum, respectively. This polarization results from the alignment of dipoles with the electric field and is related to the bound surface charge density,

*ρ*, via

_{b}where **n** is the unit vector normal to the surface. Combination of Eqs. (3) and (4) yields

Hence, using the electric field solutions obtained from FDTD simulations, *ρ _{b}* can be obtained from Eq. (5), and the sign of

*ρ*is related to the phase of

_{b}**E**. On the bowtie surface, the permittivity of Au,

*ε*

_{Au}, is used to calculate

*ρ*from Eq. (5).

_{b}Taking the left triangle in Fig. 1(a) as an example, the normal vectors **n** at the opposing apex and the end of bowtie are $\stackrel{\rightharpoonup}{x}$ and –$\stackrel{\rightharpoonup}{x}$, respectively, and *ρ _{b}* becomes

For *λ* = 655 nm, *ϕ* ~–π and *E _{x}* is negative in both the opposing apex and the end of bowtie regions. Also, because Au has small dielectric loss; i.e., Im[

*ε*

_{Au}] is small, and Re[

*ε*

_{Au}] is negative,

*ε*

_{Au}–

*ε*

_{0}is negative. As a result,

*ρ*is positive in the opposing apex region but is negative in the end of bowtie region. For

_{b}*λ*= 540 nm,

*ϕ*approaches 0 and π/2 (i.e.,

*E*is positive and positive but very small), respectively, in the opposing apex and the end of bowtie regions. As a result,

_{x}*ρ*is negative in the opposing apex region but is slightly positive in the end of bowtie region. To satisfy the charge balance condition, some positive charges should exist at the side edge of the bowtie. The bound surface charges on the other sides and the corners of the bowtie can also be calculated by considering

_{b}*E*,

_{x}*E*, and the unit vector normal to the corner. Since the field enhancement for

_{y}*λ*= 655 nm is more confined in the gap region as shown in Fig. 3(a), the bound surface charges are most likely to exist in the opposing apex region and in the end of bowtie region. The field enhancement for

*λ*= 540 nm has more significant enhancement at the sides in Fig. 3(b), and the bound surface charges around the corners need to be considered in describing the physical mechanism. The schematic drawings of the charge distribution for

*λ*= 655 and 540 nm at a specific time are shown, respectively, in Figs. 3(e) and 3(f). Like the field, the charges have a harmonic time-dependence and it is much more complicated than Figs. 3(e) and 3(f); however, the dipole plasmon resonance and quadrupole plasmon resonance can be seen in Figs. 3(e) and 3(f). The polarizations

**P**, which can be calculated from Eq. (3), can also be described by the interactions between positive and negative bound surface charges. For simplicity, only the polarizations in large amplitude at a specific time are plotted as the arrow symbols in Figs. 3(e) and 3(f). For

*λ*= 655 nm, the polarization gives the dipole moments in each triangular prism and most of the polarizations are in the opposing apex regions. This concludes that the dipole plasmon resonance occurs for

*λ*= 655 nm case. For

*λ*= 540 nm, it has not only the similar dipole moments and plasmon resonance as described for

*λ*= 655 nm case, but also the quadrupole plasmon resonance for the opposite polarizations in the two corners of the end of the bowtie regions. This concludes that the dipole plasmon resonance that is mostly in the opposing apex regions and the quadrupole plasmon resonance that is in each triangular prism both occur for

*λ*= 540 nm case.

#### 3.3 Cavity field enhancement

The FDTD simulation results of the maximum *E* intensity enhancement as a function of the gap size, *d*, are shown in Fig. 4(a)
at different apex widths. For *d* > 50 nm (not shown in figure), the maximum |*E*|^{2} enhancement is insensitive to the gap size (i.e., the two prisms are sufficiently far apart to mostly eliminate near-field coupling between prisms). However, for *d* < 50 nm, the maximum |*E*|^{2} enhancement versus *d* relation can be fitted by a power law, such that

where *A* and *m* are fitting parameters with *A* being the intercept of the fitting line with the ordinate and *m* being the slope in the log-log plot. It can be seen in Fig. 4(a) that the maximum |*E*|^{2} enhancement decreases with both the increasing gap size and the increasing apex width.

It is of interest to explore the above power-law dependence. The *E* field intensity in a plasmonic cavity, which can be formed by the junction between two closely spaced metal nanoparticles, is related to the cavity geometry and quality factor, the coupling efficiency, and the absorption and radiation loss [23]. A general equation to describe this field intensity for all plasmonic cavity structures is unattainable and a semi-quantitative approach is used in the present study to describe the cavity field enhancement. For a bowtie cavity, the field intensity is inversely proportional to the effective volume, proportional to the square of the cavity quality factor, and affected by the absorption and radiation loss. For a small gap size, the field is highly confined in the gap region at its resonant wavelength and the absorption loss for the bowtie cavity varies with the gap size. According to [23], the absorption loss for a metallic cavity is inversely proportional to its quality factor. Because the quality factor of a plasmonic cavity increases slightly as the cavity becomes smaller, the effects of the absorption loss and the quality factor are not significant. The radiation loss also varies with the gap size; however, it is neglected for simplicity in analyses because the field is always highly confined at the resonant wavelength. While the effective volume of the bowtie cavity is proportional to the true volume of the bowtie cavity which is the product of the apex width, *w*, the gap size, *d*, and the height of the bowtie, the *E* field intensity should be approximately inverse-proportional to the product of *w* and *d* when the height is fixed. Hence, for a fixed bowtie height, the |*E*|^{2} enhancement should be approximately proportional to (*wd*)^{–1}. This dependence is modified by other factors such as the difference between the effective volume and the true volume of the cavity, the quality factor, and the absorption and radiation loss. The FDTD data shown in Fig. 4(a) for three different apex widths are re-plotted in Fig. 4(b) as a function of *wd*. The three data sets can be fitted by a power law with a slope of –1.08 in a log-log plot which is in good agreement with the predicted value of –1.

#### 3.4 Long-range collective photonic effect

While the gap size dictates the short-range interaction between the two triangular prisms of the bowtie structure, the inter-bowtie distance controls the long-range interaction between neighboring bowties. Using the T-matrix method, the long-range collective photonic effect has been concluded such that additional enhancement can be achieved when the periodicity of the arrays of nanostructures matches the plasmon resonance wavelength of the nanostructures [24,25]. To study this collective photonic effect, truncated bowties with apex width of 10 nm and gap size of 10 nm are considered in FDTD simulations. Because the polarization of the incident light is in the *x*-direction, the inter-bowtie distance controlling this long-range interaction is the one in the *x*-direction. Fixing the (center-to-center) inter-bowtie distance in the *y*-direction at 300 nm, the maximum *E* intensity enhancement is plotted as a function of the inter-bowtie distance in the *x*-direction, *c*, in Fig. 5
. A major peak occurs at *c* = 650 nm that coincides with the resonance wavelength which, in turn, confirms the long-range collective photonic effect. A secondary peak occurs at *c* = 340 nm that is slightly longer than the half resonance wavelength, 325 nm. This secondary peak could result from the long-range interaction between the second neighboring bowties with a modification from the presence of the first neighboring bowties.

## 4. Conclusion

In conclusion, we use FDTD to simulate the plasmon resonance of periodic gold bowtie nanostructures with truncated apexes. The illumination light is polarized along the direction connecting the two opposing apexes of the two triangles of a bowtie. Bowties with different gap sizes are considered to systematically examine how the gap size affects the near-field *E* intensity and the far-field reflectance of periodic gold bowtie nanostructures. Both dipole and quadrupole resonances are evident in the simulation results and both resonances show red shift with decreasing gap size. While the *E* intensity enhancement is highly confined to the gap region for dipole resonance, it has more significant enhancement at the sides for quadrupole resonance. Also, these two resonances are different in the phase of electric field, the bound surface charge distribution on the bowtie surface, and the polarization. Both resonances show red shift with decreasing apex width. We found that the peak at a longer wavelength is caused by the dipole resonance between two triangular prisms, while the other peak at a shorter wavelength has the quadrupole resonance in each triangular prism in addition to the dipole resonance between two triangular prisms. A log-log plot of the field enhancement versus the product of apex width and gap size can be fitted by a straight line with a slope of –1.08 for the periodic bowtie arrays. This is in good agreement with the predicted value of –1 based on plasmon resonance cavity. When the periodicity of the periodic bowtie structures matches the resonance wavelength, a major peak is observed in the maximum *E* intensity enhancement versus the inter-bowtie distance relation. This confirms the long-range collective photonic effect. In addition, when the periodicity is about the half resonance wavelength, a secondary peak is observed. This could be attributed to the long-range interaction between the second neighboring bowties.

## Acknowledgments

This research was jointly supported by National Science Council, Taiwan under contracts no. NSC100-2221-E-002-128 and no. NSC99-2221-E-002-147, and by the Strategic Environmental Research and Development Program of the U.S. Department of Defense.

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