## Abstract

This paper describes a comparative study of finite-difference time-domain (FDTD) and analytical evaluations of electromagnetic fields in the vicinity of dimers of metallic nanospheres of plasmonics-active metals. The results of these two computational methods, to determine electromagnetic field enhancement in the region often referred to as “hot spots” between the two nanospheres forming the dimer, were compared and a strong correlation observed for gold dimers. The analytical evaluation involved the use of the spherical-harmonic addition theorem to relate the multipole expansion coefficients between the two nanospheres. In these evaluations, the spacing between two nanospheres forming the dimer was varied to obtain the effect of nanoparticle spacing on the electromagnetic fields in the regions between the nanostructures. Gold and silver were the metals investigated in our work as they exhibit substantial plasmon resonance properties in the ultraviolet, visible, and near-infrared spectral regimes. The results indicate excellent correlation between the two computational methods, especially for gold nanosphere dimers with only a 5-10% difference between the two methods. The effect of varying the diameters of the nanospheres forming the dimer, on the electromagnetic field enhancement, was also studied.

©2009 Optical Society of America

## 1. Introduction

Surface-enhanced Raman scattering (SERS) [1-6] has been employed as an
important spectroscopic tool for sensitive and specific detection of chemical, biological, and
medical analytes and is of great interest to many fields such as biology, medicine, biomedical
engineering, environmental and industrial monitoring, as well as defense and security [7-11]. The
advantage of employing SERS as a sensing platform is that SERS spectra exhibit narrow spectral
features characteristic of the vibrational states of the detected analyte species, which allows
specific detection of these species in the presence of other components in complex mixtures.
Raman scattering can be described as an inelastic light scattering process in which a target
sample on which light is incident absorbs one photon and emits another photon at the same time,
the second photon being either at a lower frequency (i.e. Stokes scattering) or at a higher
frequency (i.e. Anti-Stokes scattering) than the incident light frequency. While Raman
scattering cross-sections are extremely small - typically between 10^{-30} to
10^{-25} cm^{2} per molecule - thereby limiting its ability to detect the
analyte species, Surface-enhanced Raman scattering (SERS) increases the Raman scattering
cross-section substantially enabling the application of this process for extremely sensitive and
specific detection of the analytes [1-3]. There are two main sources of electromagnetic enhancement
(EM) of SERS [7-11] – The first source of the enhancement is due to excitation of surface
plasmons (SPs) - collective oscillations of conduction band electrons on the surface of
nanoparticles and films - which are characterized by intense electromagnetic fields confined to
the surface [12-17]. SP excitation leads to significant enhancement in the localized
electromagnetic fields in the vicinity of the metallic nanostructures which in turn leads to an
increase in the Raman emission intensity which is proportional to the square of the applied
field at the molecule. Further SERS enhancement originates from SP related enhancement of the
Raman signals that emanate from the molecules. The relationship between the electromagnetic
enhancement (EME) of the SERS signal intensity and the localized electric field in the vicinity
of metallic nanostructures has been described in the past literature by the product of the
average local field intensity (E_{loc}) at the surface of the SERS active surface [1]:

where the frequency of the incident radiation is ω_{i}, E_{inc} is
the incident electric field strength, ωS is the Stokes frequency of the emitted Raman
signal. When the Stokes frequency of the emitted Raman signal is spectrally close to the
frequency of the incident radiation, the value of the electromagnetic enhancement is
proportional to the fourth power of the localized field intensity (E_{loc}) at the
surface of the SERS active surface. It is therefore important to theoretically estimate the EM
field enhancement in the nano-scale gaps between the metallic nanostructures - such as dimers of
nanoparticles - forming the SERS substrates so that substrates with extremely high enhancement
factors can be designed and fabricated. In the SERS signals obtained from the substrates, the
electric field enhancement is averaged over the surface area of the metallic nanostructures on
which there is adsorption of the molecules generating the SERS signals [18]. On the other hand, in single molecule SERS, the maximum value of the SERS
enhancement factor is considered important as compared to the averaged SERS signal. The maximum
enhancement factor of SERS can be greater than the value of SERS signals, averaged over the
surface area of the SERS substrate on which the molecules are adsorbed, by many orders of
magnitude. Single molecule SERS has been described previously in literature and generally
employs a confocal microscope in conjunction with a Raman spectrometer to detect SERS signals
from localized regions containing high EM fields [19-20]. It is believed that the
anomalously strong Raman signal originates from “hot spots”, i.e., regions
where clusters of several closely-spaced nanoparticles are concentrated in a small volume. The
high-intensity SERS then originates from the mutual enhancement of surface plasmon local
electric fields of several nanoparticles that determine the dipole moment of a molecule trapped
in a gap between metal surfaces. In the study described in this paper, we focus on accurately
determining the maximum value of EM field enhancement midway between two gold nanospheres
forming dimer structures, i.e. in the middle of the “hot spot” between the
nanospheres, which would be useful for single molecule detection employing SERS. Along with
developing SERS substrates containing an assembly of metallic nanoparticles and nanostructures
on a solid surface, SERS signals are measured from metallic nanoparticles and dimers of metallic
nanoparticles in solution [21]. Moreover SERS studies on
living cells, incubated with Raman dye-labeled metallic nanoparticles, have also been recently
reported and show promise in fields such as imaging and spectroscopy [22-23].

In order to determine analytical solutions of electromagnetic (EM) fields around individual metallic nanoparticles having spherical geometries, Mie theory [18], which is usually employed, can describe a red-shift in the Mie resonance peaks, along with appearance of higher order resonances, with an increase in the nanoparticles size. Closed-form analytical calculations employing Mie theory become impossible when EM fields are calculated for nanostructures with complex non-spherical geometries or for an array of nanoparticles and nanostructures. Previously, several authors have employed multipole expansion based methods to calculate electromagnetic fields around multiple spherical nanoparticles [24-25]. The multipole expansion method for two or more spheres is sometimes called the “superposition method” [26], which refers to the idea that the total scattered field can be computed as a superposition of the fields scattered from the individual spheres. Each individual field is represented as a multipole expansion with respect to a coordinate system centered on that sphere, and the coefficients in each expansion are related by translational formulas based on the spherical harmonic addition theorem [26]. Since the method is semi-analytical and convergence of the multipole expansion can be verified, the method is capable of giving very accurate results, as noted in [26]. A number of authors have used the superposition method for EM calculations involving clusters of spheres [24-25, 27-30]. Much of this work has employed full-wave analyses that require vector spherical harmonics. Li et al. [24] employed multipole spectral expansion (MSE) method to determine electromagnetic fields in between a chain of metallic nanospheres and determined that the chain of metallic nanospheres having varying diameters and gaps between neighboring spheres leads to focusing of electromagnetic energy to the ends of the chain of nanospheres. Pelligrini et al. [25] employed generalized Mie theory using multipole expansion to calculate far-field properties of strongly interacting metallic spherical nanoparticles having diameters between 10 nm and 60 nm, with the gaps between the nanoparticles lying between 0.2 nm and 8 nm. They determined the extinction cross-sections of an assembly of metallic nanoparticles with varying diameters including dimers, chains of the metallic nanoparticles, as well as core-satellite structures with large clusters surrounded by small clusters of nanoparticles by employing multipole expansion of the incident and scattered fields in vector spherical wave functions around each sphere and applying independent solution of boundary conditions at the surface of each particle together with vector translation theorems. In analytical modeling of metallic nanoparticles substantially smaller than the wavelength of light, one does not need to employ the generalized Mie theory as was described by Pelligrini et al. [25] and multipole expansions with the quasistatic approximation is sufficient and considerably simplifies the computational calculations. The multipole expansion method described in our paper enables calculation of electromagnetic fields in the near field of the metallic nanoparticles as compared to far-field calculations described by Pelligrini et al. [25]. In our own work, we employ two assumptions that greatly simplify the analysis: (1) axial symmetry (the E-field polarization direction is aligned along the axis of the dimer), and (2) the quasi-static approximation, which should hold when the particles are much smaller than the incident wavelength. The latter assumption avoids the need for vector wave functions. Another benefit of the quasi-static approximation is that the only wavelength dependence in the calculation enters through the wavelength dependence of the dielectric function. In the quasi-static approximation, we solve Laplace’s equation rather than the Helmholtz equation and, as a consequence, the multipole expansion functions are wavelength independent. This procedure results in calculation times significantly shorter than what full-wave algorithms require, particularly if calculations are required over a range of wavelengths. Hence in this paper, we first employ a multipole expansion based semi-analytical method to accurately determine (high accuracy results from the method being semi-analytical and the verification of convergence of the multipole expansion) electromagnetic fields in between the two nanospheres forming dimer structures and subsequently compare the results of numerical simulations with those obtained from the multipole method.

In order to determine electromagnetic fields around multiple nanostructures or nanostructures of more complex geometries, it becomes necessary to employ numerical methods such as the discrete dipole approximation [31-32], the finite-element integration technique [33], and the finite difference time domain method [34-36]. The finite-difference time-domain method (FDTD) has been employed by several authors for determining electromagnetic fields around individual metallic nanoparticles of different geometries such as nanospheres [37], nanostars [34], and metallic torus [38], as well as two dimensional periodic and Fibonacci arrays of gold nanopillars [39]. In previous calculations of EM fields in the vicinity of nanoparticle dimers using the FDTD method, near field and far field response of dimers of metallic nanoshells was described by Oubre et al. [35]. They employed FDTD modeling to determine the extinction cross-sections as well as electromagnetic field distributions in the vicinity of metallic nanoshell dimers. Previously, Nordlander et al. [36] compared FDTD calculations with those carried out by employing the plasmon hybridization method [40] to determine plasmon energy and excitation cross sections of nanoparticle dimers. In their calculations, the Drude model was employed to define the dielectric constant of gold as compared to an extended Debye model, which was employed in our FDTD calculations. Moreover, our paper provides a comparison between near field EM calculations carried out using the FDTD method and results obtained using semi-analytical multipole expansion. Our paper also discusses calculations of near-field EM fields for spacings between the metallic nanospheres forming the dimer much smaller than the diameter of the nanospheres (the minimum spacing between the nanospheres forming the dimer in our work was taken to be 1/8 of the diameter of the nanosphere). Sweatlock et al. [33] have previously employed three dimensional finite-element integration techniques to determine electromagnetic fields in the region between chains of closely coupled silver nanospheres 10 nm in diameter and a 0-4 nm spacing variation between the nanoparticles. The finite-element integration technique used by Sweatlock et al. [33] employs discretization of the integral form of the Maxwell’s equations as compared with the discretization of the differential form as used in the FDTD method. In the finite-element integration technique employed by Sweatlock et al. [33], a Drude model was used to define the dielectric constant of gold, whereas an extended Debye model was employed in our FDTD calculations. The Debye model is known to be more accurate for metals such as silver and gold in UV and visible spectral regions [37]. Norton and Vo-Dinh [41] previously described a multipole-based method for calculating the optical response of a linear chain of metallic nanospheres or nanospheroids. In this paper, we employ both the FDTD and the multipole expansion [42] based semi-analytical methods to quantify the EM field enhancement in the region between the two nanospheres forming the dimer structures when coherent light in the ultraviolet, visible, and infrared regions is incident on these structures, and compare the results obtained by these two processes. The objective of this paper is to bring about the differences between FDTD simulations and analytical calculations and as a specific example or case, we selected dimers to bring about the differences in calculations (of EM fields in the near-field of the nanoparticles forming the dimers) made by these two processes. One of the objectives of the comparison of the FDTD and analytical results is to bring about the differences in the results that occur in the different nanoparticle size regimes as well as different spacings between the nanoparticles. The specific case of Au and Ag dimers was selected to explain the differences between FDTD and analytical processes as it describes a very important problem in the fields of SERS and biomedical engineering. These calculations could be employed for designing SERS substrates or for intra cellular SERS measurements - SERS based cellular imaging or spectroscopy using uptake of Raman dye-labeled dimers. Increase in the localized electric and magnetic fields in the vicinity of the metallic nanoparticles is dependent on the shape and size of the nanoparticles and on the nano-scale gaps between the nanoparticles. The effect of nano-scale spacing between metallic nanoparticles forming a dimer, and of the diameters of the nanoparticles, on the electromagnetic field enhancement was evaluated. Values of electromagnetic enhancement of SERS, as a function of the nano-scale spacing between metallic nanoparticles forming a dimer and of the diameters of the nanoparticles, were also estimated in this study. This should help guide the design and fabrication of SERS substrates based on one or two dimensional arrays of metallic nanoparticles forming dimers or particle chains and having high SERS enhancement factors.

## 2. Finite-difference time-domain (FDTD) simulations

Finite-Difference Time-Domain algorithms [43] analyze structures by solving the differential form of coupled Maxwell’s equations. They can be used for the analysis (electric and magnetic fields distribution determination, energy and power distributions, etc.) of metallic nanoparticles, nanorods, nano-apertures and other nanostructures on planar substrates, in liquid media, in or on the surface of waveguides, etc. The FDTD method involves discretization of the Maxwell’s equations in both the time and the space domains in order to find the E and H fields at different positions and at different time-steps. An FDTD algorithm, called the Yee’s algorithm’ is employed to determine fields at different points in space at different times. In an FDTD lattice or grid, every E component is surrounded by four H components and every H component is surrounded by four E components.

Our group has previously employed FDTD for modeling electromagnetic fields around metallic nanostructures of different geometries [16]. In the work described in this paper, FDTD is employed for calculating electromagnetic fields around metallic nanoparticle dimers taking into account the effects of the size of the nanoparticles and the spacing between the nanoparticles forming the dimers, as well as the material of the nanostructure. FDTD analysis described in these calculations incorporates the effects of dispersion relations, i. e. the effects of wavelength dependence of the dielectric constants of the metallic structures. In this work, we employed an FDTD software called FullWAVE 6.0 by R-Soft to carry out FDTD analysis for our plasmonic nanostructures. This software enables FDTD analysis of the metallic media to include Debye or Lorentz models of dispersion relations of the dielectric constants of the metals. In the simulations, we use the following dispersion model (an extended Debye model) for determining the dielectric constant for gold and silver:

where *Δ _{εk}, a_{k}, b_{k}* and

*c*are constants that provide the best fit for various metals (gold and silver) when compared to optical constant data for these metals given by Palik et al. [44]. The following values of Δε

_{k}_{k}, a

_{k}, b

_{k}, and c

_{k}for gold were employed in our work - Δ ε

_{1}: 1589.516, Δ ε

_{2}: 50.19525, Δ ε

_{3}: 20.91469, Δ ε

_{4}: 148.4943, Δ ε

_{5}: 1256.973, Δ ε

_{6}: 9169; a

_{1}: 1, a

_{2}: 1, a

_{3}: 1, a

_{4}: 1, a

_{5}: 1, a

_{6}: 1; b

_{1}: 0.268419, b

_{2}: 1.220548, b

_{3}: 1.747258, b

_{4}: 4.406129, b

_{5}: 12.63, b

_{6}: 11.21284; c

_{1}: 0, c

_{2}: 4.417455, c

_{3}: 17.66982, c

_{4}: 226.0978, c

_{5}: 475.1387, c

_{6}: 4550.765. The following values of Δε

_{k}, a

_{k}, b

_{k}, and c

_{k}for silver were employed in our work - Δ ε

_{1}: 1759.471, Δ ε

_{2}: 135.344, Δ ε

_{3}: 258.1946, Δ ε

_{4}: 22.90436, Δ ε

_{5}: 1749.06, Δ ε

_{6}: 11756.18; a

_{1}: 1, a

_{2}: 1, a

_{3}: 1, a

_{4}: 1, a

_{5}: 1, a

_{6}: 1; b

_{1}: 0.243097, b

_{2}: 19.68071, b

_{3}: 2.289161, b

_{4}: 0.329194, b

_{5}: 4.639097, b

_{6}: 12.25; c

_{1}: 0, c

_{2}: 17.07876, c

_{3}: 515.022, c

_{4}: 1718.357, c

_{5}: 2116.092, c

_{6}: 10559.42.

3-D solutions were obtained for electromagnetic fields around the gold and silver nanosphere dimers. Electromagnetic fields (i. e. E and H fields in the x, y, and z directions) in the vicinity of the metallic nanosphere dimers were calculated assuming plane wave illumination, with wavelengths varying between 300 nm and 850 nm. The magnitude of the incident electric fields was taken to be unity and the enhancement of electromagnetic fields, around the dimers, evaluated. A schematic of the metallic nanosphere dimers that were evaluated is shown in Fig. 1. The time steps (Δt) employed in these simulations were selected to be small enough such that the Courant stability criterion [43] described by Eq. (3) was satisfied for the different grid sizes (Δx, Δy, and Δz) employed, where c is the speed of light:

For FDTD calculations involving the nanosphere dimers, the grid sizes in x, y, and z directions (Δx, Δy, and Δz) were selected to be such that the value of the E field intensities around the nanospheres became independent of the grid sizes. This process was carried out by initially selecting values of grid sizes in x, y, and z directions (Δx, Δy, and Δz) to be of the order of minimum spacing between the nanoparticles (for example 2.5 nm grid size for a 2.5 nm spacing) forming the dimer and subsequently reducing the values of the grid spacings until further reduction in the grid spacing had no further influence on the calculated electromagnetic fields around the metallic nanospheres forming the dimers. This usually required at least 3 or 4 FDTD grids in the region between the nanospheres. The FDTD software employed in this work allows selection of different grid sizes for the edge of the metallic nanostructures and the bulk media. In our calculations, 0.5 nm grid spacing on the edges of the nanosphere was employed for all the spacings s (2.5 nm, 5 nm, 10 nm, and 20 nm) between the nanospheres forming the dimers where as the bulk grid size selected were 1 nm, 2 nm, 2 nm, and 3.5 nm, respectively for the nanosphere spacings s being 2.5 nm, 5 nm, 10 nm, and 20 nm, respectively. The FDTD calculations were carried out for TE polarization of the electric field, i.e. the incident E-field being polarized along the axis of the two spheres.

## 3. Analytical evaluations

In this section, we summarize the solution based on a multipole representation of the
scattered fields. Assume that the incident electric field is directed along the line joining the
centers of the two spheres (the z-axis) labeled 1 and 2 as shown in Fig. 1. Let *R _{i}* and

*ε*denote the radius and dielectric constant of the

_{i}(ω)*i*-th sphere and let

*L*denote the distance between their centers. We define spherical coordinate systems centered on each sphere; thus, for example, a point

**r**in the i-th coordinate system may be written

**r**=(

*r*) or

_{i}, θ_{i}**r**=(

*r*), where u

_{i}, u_{i}_{i}=cos

*θ*. From geometry, we have the relations

_{i}*r*=√(

_{1}*r*

^{2}

_{2}+

*L*+

^{2}*2 r*) and

_{2}u_{2}L*u*=(

_{1}*r*+

_{2}u_{2}*L*)/

*r*and, conversely,

_{1}*r*=√(

_{2}*r*+

_{1}^{2}*L*-

^{2}*2 r*) and

_{1}u_{1}L*u*=(

_{2}*r*-

_{1}u_{1}*L*)/

*r*.

_{2}The electric field is expressed as the gradient of a potential in the quasi-static
approximation. We denote by *ψ ^{(i)}_{p}(r_{i},
u_{i}), ψ^{(i)}_{s}(r_{i}, u_{i})*,
and

*ψ*the potentials associated with the primary (incident) field, the scattered field and the field internal to the i-th sphere defined with respect to the spherical coordinate system

^{(i)}_{in}(r_{i}, u_{i})*(r*centered at this sphere. The potentials obey Laplace’s equation subject to boundary conditions and are represented as expansions in Legendre polynomials as follows. For the

_{i}, u_{i})*i*-th sphere, these fields are given by the following expressions:

where *P _{n}* is the Legendre polynomial of degree n. The sums begin
with

*n*=1 when there is no net charge on the spheres. The total potential at a point,

**r**, external to the two spheres is given by the sum of the potentials associated with the incident field and the fields scattered from each sphere:

The boundary conditions on the surface of each sphere requires continuity of the potentials
and continuity of the dielectric function multiplied by the radial derivative of the potentials
at each point on the surface. Applying these boundary conditions to both spheres, multiplying
the resulting equations by *P _{m} (u_{i})*, integrating with
respect to ui between -1 and 1 and using the orthogonality of the Legendre functions, we obtain
the following linear system of equations for the coefficients

where γ_{i}=ε_{i}ε_{0}

and *c _{m}*=

*(2*. There are two sets of these equations: one with

_{m}+1)/2*(i=1, j=2)*and another with

*(i=2, j=1)*. The matrix elements

*Q*and

^{(ij)}_{mn}*R*can be evaluated analytically by making use of the addition theorem [42] as:

^{(ij)}_{mn}Substituting this into *Q ^{(ij)}_{mn}* and

*R*and again using the orthogonality of the Legendre functions gives the following relationship:

^{(ij)}_{mn}and the relationship *R ^{(ij)}_{mn}*=

*m*

*Q*. The system of equations defined by (8) and (9) is then truncated to

_{mn}^{(ij)}*N*terms and upon computing the matrix elements

*Q*and

^{(ij)}_{mn}*R*can be solved for the coefficients

^{(ij)}_{mn}*a*and

^{(i)}_{m}*b*. These, in turn, are substituted into Eq. (5) and the gradient computed to obtain the scattered electric field. For larger gap sizes,

^{(i)}_{m}*N*=20 was found to be sufficient. It was found that a higher value of N was required for convergence when the interparticle distance between the metallic nanospheres forming the dimers was reduced. This could be attributed to the higher localized electromagnetic fields for smaller gap between the nanospheres forming the dimer which require a larger number of multipole terms N for correct representation of the fields. At the smallest gap used here (2.5% of the sphere diameter),

*N*=40 provided excellent convergence. In the numerical evaluation of Eq. (13), we carried out calculations to compute the value of EM fields (electric field intensities) using 40 harmonics, 60 harmonics, as well as 80 harmonics and saw that the results were almost the same for use of more than 40 harmonics, i.e. there was convergence of the results up to one part in 10

^{5}.

## 4. Results and discussions

It is important to let the FDTD simulation reach stability, i. e. for the amplitude of the
square of the electric field intensity to reach a constant value as a function of time, so that
reliable values of the electromagnetic field enhancements in the vicinity of the nanostructures
could be obtained. All FDTD simulations described in this paper were allowed to run for
sufficiently long time so that the simulation stability was achieved before E and H fields in
the near field of the dimer nanostructures were determined. Figure 2 shows intensity of the E-field as a function of the incident field, polarized
along the axis of two 70 nm gold spheres forming a dimer, plotted against the normalized
simulation time *cT*. In this simulation, the spacing between the two nanospheres
was 5 nm and wavelength of the incident light for this plot was 600 nm. It can be seen in Fig. 2 that the E-field intensity stabilizes after around
5000 FDTD simulation steps (when *cT* ~5 µm) where c is the speed of
light and *T* is the time during the simulation) out of a total simulation time
of 12000 time steps employed in this simulation (*cT*=12 µm is the end
of the simulation). Figure 3(a) shows the spatial
distribution of the E-field as a function of the incident field, when the incident E-field is
polarized along the axis of two 70 nm gold spheres forming a dimer with 5 nm spacing between the
nanospheres. In the simulation, an EM field (a plane wave with a 600 nm wavelength) was incident
towards the nanosphere dimer along the Z direction and the spatial distribution shown in Fig. 3(a) is the X-Z plane. The E-field was determined at a
simulation time *cT* ~5.28 µm. Figure
3(b) shows a horizontal cut of the E-field made along ~ Z=0 displaying the spatial
distribution of the E-field enhancement along the X axis, i. e. the axis connecting the two
spheres forming the dimers. It can be observed that there is high concentration of the E-Field
in between the two nanospheres where there is maximum enhancement of the electric field of ~30.

The effect of the spacing s, between two adjacent metallic nanospheres forming a dimer, on the
magnitude of the electric field (as a ratio of the incident electric field), was evaluated by
employing both the FDTD and multipole expansion methods. Figure
4 shows FDTD calculations and near-field analytical multipole expansion calculations of
the spectral distribution of the magnitude of the electric field enhancement *E*,
midway between two gold nanospheres forming a dimer, evaluated for different spacings between
the nanospheres. It can be observed from FDTD calculations shown in Fig. 4 that the maximum enhancement of the electric field, occurring between
wavelengths 546-560 nm, increases substantially as the spacing between the adjacent
gold nanospheres is decreased. The value of the electric field enhancement for a 20 nm spacing
between the gold nanospheres is ~1.8, whereas a 2.5 nm spacing gives an enhancement of is ~15.6.
One also observes a slight red-shift in the plasmon resonance from ~546 nm for a 20 nm spacing
between the gold nanospheres and ~560 nm for a 2.5 nm spacing between the nanospheres. It can be
observed from the multipole expansion calculations shown in Fig. 4 that the maximum enhancement of the electric field, occurring between
wavelengths 532-545 nm, increases substantially as the spacing between the gold
nanospheres is decreased. The value of the electric field enhancement for the 20 nm spacing
between the gold nanospheres is ~1.7 whereas for 2.5 nm spacing this increases to ~14.8. Similar
to the FDTD calculations, one also observes a slight red-shift in the plasmon resonance from
~532 nm for the 20 nm spacing between the gold nanospheres and ~545 nm for the 2.5 nm spacing
between the nanospheres. One can observe from Fig. 4
that the spectra obtained by the FDTD calculations as well as by the multipole expansion method
are similar in shape. Moreover, the highest E-field enhancement factor (for the least spacing
between the gold nanospheres, i. e. 2.5 nm) is ~15.6 for the FDTD calculations whereas it is
~14.8 for the multipole expansion calculations – a difference of only ~5% between the E-field
enhancement values. Figure 5 shows FDTD and near-field
analytical multipole expansion calculations of the spectral distribution of electric field
enhancement midway between the two silver nanospheres forming a dimer for different nanosphere
spacings. It can be observed from the FDTD calculations shown in Fig. 5 that the maximum enhancement of the electric field increases
substantially as the spacing between the adjacent silver nanospheres, is decreased. The value of
the electric field enhancement for the 20 nm spacing between the nanospheres is ~3.75, whereas
for the 2.5 nm spacing is ~50. The FDTD calculations also show a very small red-shift in the
plasmon resonance from ~402 nm for the 20 nm spacing between the gold nanospheres and ~405 nm
for the 2.5 nm spacing between the nanospheres. The multipole expansion calculations for the
silver nanosphere dimers shown in Fig. 4 also showed as
substantial increase in the value of the electric field enhancement, from 3.3 to 62, as the
spacing between the gold nanospheres forming the dimer is decreased from 20 nm to 2.5 nm.
Similar to FDTD calculations for gold and silver dimers and multipole expansion results for the
gold dimers discussed above, one also observes a slight red-shift in the plasmon resonance from
~376 nm for the 20 nm spacing between the silver nanospheres and ~397 nm for the 2.5 nm spacing
between the nanospheres. On comparing the FDTD and analytical multipole expansion calculations
shown in Fig. 5, one observed that the difference
between the spectra obtained by the FDTD calculations and the multipole expansion method is
greater for the case of silver dimers as compared to gold dimers. Moreover, the highest E-field
enhancement factor (for the least spacing between the gold nanospheres, i. e. 2.5 nm) is ~62 for
FDTD calculations whereas it is ~50 for the multipole expansion calculations – a difference of
~20% between the E-field enhancement values. One can also note a higher value of maximum
electric field enhancement factor midway between the silver nanospheres (~50 employing FDTD
method and ~62 employing multipole expansion method) as compared to gold dimers (~15.6 employing
FDTD method and ~14.8 employing multipole expansion method) as higher plasmon resonances are
excited in silver as compared to gold due to a larger magnitude of the ratio of the real to the
imaginary part of the dielectric constant of silver as compared to gold [15]. We selected 2.5 nm as the minimum spacing between the nanospheres forming
dimer structures in our FDTD calculations as the computational time involved in these
calculations increases dramatically for gaps between the dimer nanosphere that are smaller than
2.5 nm. This happens because one has to employ even smaller FDTD grid sizes for smaller gaps
between the dimer nanospheres such that the grid sizes do not have any effect on the EM fields
in the vicinity of the dimer nanospheres. In our FDTD computations, we employed ~4 FDTD grids in
the region between the nanospheres. This implies that for a very small spacing - for example 1
nm - between the nanospheres forming a dimer, the maximum grid size selected would be 0.25 nm
which dramatically increases the FDTD computational time. It is also known that it is extremely
difficult to develop metallic nanostructures containing gaps, between neighboring
nanostructures, that are smaller than 5 nm by employing standard nanofabrication processes such
as electron beam lithography, focused ion beam milling, deep UV lithography, as well as
nanosphere lithography [14, 45]. Dimers and closed packed arrangements of metallic nanoparticles have been
developed using wet chemistry by employing linker molecules such as alkanethiols [45] to create nanoscale separations between the metallic
nanoparticles that are as small as ~2-3 nm [45]. Although different lengths of linker molecules such as aryl dithiol molecules
[46] could be employed to achieve spacings between
metallic nanoparticles that are even smaller than 2 nm, it is difficult to reliably insert probe
molecules - i. e. molecules to which target analyte molecules to be detected can attach - inside
gaps between dimer nanospheres that are smaller than 2-3 nm. The minimum spacing
between the dimer nanospheres that was selected in our FDTD calculations, i. e. 2.5 nm, happens
to fit with the experimentally practical spacing for use in SERS. As is evident from Eq. (1), the value of electromagnetic enhancement (EM)
of SERS is approximately equal to the fourth power of the electric field enhancement value. The
values of the SERS EM enhancement factor were estimated midway between the two metallic
nanospheres forming a dimer. Figure 6 shows the
magnitude of the SERS EM Enhancement Factor plotted on a logarithmic scale as a function of
wavelength of the incident field i. e. log_{10} (E^{4}), for different spacings
between the nanospheres dimers. Evaluations were carried out for gold and silver nanosphere
dimers using both FDTD simulations and analytical calculations using the multipole expansion
method. It can be observed from the FDTD and analytical calculations for gold nanosphere dimers
that the maximum SERS EM enhancement (for 2.5-nm spacing in between the nanospheres) is
~10^{5} whereas the maximum enhancement factor in between silver dimers is
~10^{7}. The higher values of SERS enhancement factor in between silver dimer
nanospheres illustrates that silver might by a better choice than gold for developing SERS
substrates based on an assembly of metallic nanoparticle chains or dimers. However, silver is
susceptible to oxidation when exposed to chemical and biological analytes being detected using
SERS, whereas gold is chemically inert.

Figure 7 shows the effect of varying the diameter
*D* of two gold nanospheres forming a dimer on the magnitude of the electric
field enhancement, evaluated by employing FDTD simulations as well as analytical calculations
using the multipole expansion method. The spacing s between the two nanospheres was 5 nm in
these calculations. One can observe from the FDTD simulation results shown in Fig. 7 that there is an increase in the electric field
enhancement factor, from 6.2 to 36, as the diameter of the nanospheres forming the dimers is
increased from 20 nm to 70 nm. One can observe from the multipole expansion method results shown
in Fig. 7 that there is an increase in the electric
field enhancement factor, from 5.8 to ~30, as the diameter of the nanospheres forming the dimers
is increased from 20 nm to 70 nm. We observe that there is not as much variation between the two
methods (FDTD and multipole expansion method) for the gold nanoparticle size being 20 nm (5%
variation between the results of the two processes) whereas the difference is larger when the
gold nanoparticle size is 70 nm (20% variation between the results of the two processes). While
the multipole expansion method using the quasi-static approximation is known to be more accurate
than the FDTD method for nanoparticle diameter D≪λ - more specifically
extremely accurate for D<λ/20 [1]
as is the case for the 20 nm nanoparticles forming the dimers - the EM field enhancement
calculations using the FDTD computations are more accurate than the multipole expansion method
for larger diameters of the nanoparticles (the FDTD method accounts for the retardation
effects). From the FDTD simulation results shown in Fig.
7, one can also observe large shifts in the wavelength at which a maxima in E-field
enhancement occurs, from 550 nm to 600 nm when the diameter of the nanospheres is increased from
20 nm to 70 nm. The multipole expansion results shown in Fig.
7 show smaller shifts (as compared with the FDTD results) in the wavelength at which a
maxima in E-field enhancement occurs, from 533 nm to 547 nm when the diameter of the nanospheres
is increased from 20 nm to 70 nm. These smaller shifts (shown in the multipole expansion
results) could be explained by the fact that the multipole calculations do not take the
retardation effects into account (the quasi-static approximation is employed), which are
accounted for in the FDTD calculations. Figure 8 shows
the magnitude of the SERS EM Enhancement Factor plotted on a logarithmic scale as a function of
wavelength of the incident field i. e. log_{10} (E^{4}), for different diameters
of nanospheres forming the dimers. Evaluations were carried out for gold nanosphere dimers, with
a 5 nm spacing in between the nanospheres, using both FDTD simulations and analytical
calculations using the multipole expansion method. It can be observed from the FDTD and
analytical calculations for gold nanosphere dimers that the maximum SERS EM enhancement for the
20 nm diameter nanospheres is ~10^{3}, whereas the maximum enhancement factor is
~10^{6} for the 70 nm diameter nanospheres. The dependence of the SERS enhancement
factor, and the wavelengths at which the enhancements occur, not only on the gaps between the
metallic nanospheres forming the dimers but also on the diameters of the nanospheres allows us
to carefully design the SERS substrates to enhance the SERS EM enhancement factors and also to
tune the wavelengths at which the maximum SERS enhancement occurs with the wavelength of the
incident light employed in SERS measurements. One has to note that although the maximum SERS
enhancement factor is greater for the dimers with the 70 nm diameter nanospheres as compared
than the 20 nm diameter nanospheres, the number of regions with the smallest gaps or SERS
“hotspots” per unit area are higher for an assembly of dimers with 20 nm
diameter nanospheres. In the case of SERS based cellular imaging or

spectroscopy, uptake of Raman dye-labeled dimers of metallic nanospheres (connected by linker molecules) is expected to be easier for dimers with smaller diameter nanospheres.

## 5. Conclusion

In this paper, a comparison of the finite-difference time-domain (FDTD) and analytical calculations was carried out, and as a specific case, we carried out evaluations of electromagnetic fields at hot spots in nanosphere dimers of plasmonics-active metals. The objective of the comparison of the FDTD and analytical results was to bring about the differences in the results that occur in the different nanoparticle size regimes as well as different spacings between the nanoparticles and a comparison between these two processes has not been described in previous literature. The specific case of Au and Ag dimers was selected to describe the differences between FDTD and analytical processes as it is exemplifies a very important problem in the fields of SERS and biomedical engineering. More specifically these calculations could be employed for designing either SERS substrates or for intra cellular SERS measurements. The enhancement of the electric field in the region between the two nanospheres forming the dimer, for both gold and silver nanospheres, was determined and an excellent correlation between these two computational methods was observed - especially for gold nanosphere dimers with only a 5-10% difference between the two methods. In the case of silver nanosphere dimers the difference between the FDTD and analytical calculations was ~20% for the closest spacing. The effect of spacing between two nanospheres forming the dimer on the electric field enhancement and therefore the EM enhancement of SERS was determined and a substantial increase in the enhancement factor observed when the spacing between the nanospheres is 2.5 nm. The effect of the diameter - of the nanospheres forming the dimer - on the electric field enhancement and EM enhancement of SERS was also studied and it was observed that the multipole expansion method employed for the analytical calculations, which employs the quasi-static approximation, departs from the FDTD results for the larger diameters. This is expected since the FDTD method provides a complete solution of the Maxwell’s equations to determine the EM fields and includes the retardation effects.

## Acknowledgments

This work was sponsored by the U. S. Army Research Office (Grant No. W911NF-04-D-0001-0008) and the National Institutes of Health (Grant No. R01 EB006201).

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