JOURNAL OF CHEMICAL PHYSICS
VOLUME 120, NUMBER 1
1 JANUARY 2004
Electromagnetic fields around silver nanoparticles and dimers
Encai Hao and George C. Schatza)
Department of Chemistry and Institute for Nanotechnology, Northwestern University, Evanston,
Illinois 60208-3113
共Received 31 March 2003; accepted 3 October 2003兲
We use the discrete dipole approximation to investigate the electromagnetic fields induced by
optical excitation of localized surface plasmon resonances of silver nanoparticles, including
monomers and dimers, with emphasis on what size, shape, and arrangement leads to the largest local
electric field 共E-field兲 enhancement near the particle surfaces. The results are used to determine what
conditions are most favorable for producing enhancements large enough to observe single molecule
surface enhanced Raman spectroscopy. Most of the calculations refer to triangular prisms, which
exhibit distinct dipole and quadrupole resonances that can easily be controlled by varying particle
size. In addition, for the dimer calculations we study the influence of dimer separation and
orientation, especially for dimers that are separated by a few nanometers. We find that the largest
兩 E兩 2 values for dimers are about a factor of 10 larger than those for all the monomers examined. For
all particles and particle orientations, the plasmon resonances which lead to the largest E-fields are
those with the longest wavelength dipolar excitation. The spacing of the particles in the dimer plays
a crucial role, and we find that the spacing needed to achieve a given 兩 E兩 2 is proportional to
nanoparticle size for particles below 100 nm in size. Particle shape and curvature are of lesser
importance, with a head to tail configuration of two triangles giving enhanced fields comparable to
head to head, or rounded head to tail. The largest 兩 E兩 2 values we have calculated for spacings of 2
nm or more is ⬃105 . © 2004 American Institute of Physics. 关DOI: 10.1063/1.1629280兴
I. INTRODUCTION
describe the electrodynamics of nanoparticles with arbitrary
shapes, degree of aggregation, and complex external dielectric environments.
Over the last 15 years, new theories have been introduced for performing classical electrodynamics studies of
light scattering from particles of arbitrary shape, including
the discrete dipole approximation 共DDA兲,21–23 finite difference time domain methods,24 T-matrix methods, the multiple
multipole method,25 and the modified long wavelength approximation 共MLWA兲.21,26 Among them, the DDA is a particularly powerful method for isolated nanoparticles or small
aggregates that are in a complex surrounding environment.
In the DDA, the object of interest is represented as a cubic
array of N polarizable elements.23,27 The response of this
array to an applied electromagnetic field is described by selfconsistently determining the induced dipole moment in each
element. This information can be used to determine far-field
properties like extinction efficiencies and scattering cross
sections, and also near-field properties such as the E-field
close to the particle that determines the electromagnetic enhancements in SERS.
Although SERS has been the subject of extensive studies, and a literature consisting of hundreds of papers, a complete understanding of the mechanism of the enhancement is
still missing. On the basis of many earlier studies,28 –34 an
important contribution to the SERS enhancement comes
from the electromagnetic 共EM兲 enhancement mechanism, in
which plasmon excitation in the particle creates an enhanced
E-field near the particle, which in turn leads to enhanced
Raman excitation and emission. Two types of enhancements
are of interest: The average of 兩 E兩 2 over the particle surface,
Although it was discovered nearly a century ago, Mie
theory 共the classical electromagnetic theory of spherical
particles兲1–3 continues to play a crucial role in describing the
optical properties of metal nanoparticles, especially silver
and gold particles. One of the important uses of Mie theory
is in describing the size dependence of localized surface
plasmon resonances; indeed one finds that Mie theory easily
describes red shifts and broadening of the dipole plasmon
resonance as particle size is increased, as well as the appearance of quadrupole and higher resonance contributions. Recently, new synthetic methods including wet-chemistry
methods4 – 6 and lithographic techniques7 have provided significant challenges to electrodynamics theory through the
production of high yields of nanoparticles with well-defined
nonspherical shapes. When spherical metal particles are
transformed into nanoscale rods4,8,9 disks,10,11 or triangular
prisms,6 the surface plasmon resonances are strongly affected, typically red-shifting and even splitting into distinctive dipole and quadrupole plasmon modes. In addition,
nanoparticle aggregates with controllable dimensions, such
as DNA-linked gold nanoparticle assemblies12–14 and molecularly bridged metal nanoparticle arrays,15 are currently of
great interest due to their unique optical and electrical properties, with possible applications in surface enhanced Raman
spectroscopy 共SERS兲,16 –18 electronics,15 and biosensors.19,20
All of these factors motivate the need for a theory that can
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2004/120(1)/357/10/$22.00
357
© 2004 American Institute of Physics
Downloaded 27 Sep 2004 to 128.210.142.204. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
358
J. Chem. Phys., Vol. 120, No. 1, 1 January 2004
which is relevant to conventional SERS measurements, and
the peak value of 兩 E兩 2 , which is important in single molecule
SERS. In conventional SERS, the Raman enhancement factor is around 106 , 28,35,36 which means that 兩 E兩 2 would have to
be 103 if the electromagnetic mechanism were to explain the
full enhancement factor, while in single molecule SERS, the
enhancement factor is around 1012, 16,17,37– 40 so 兩 E兩 2 would
have to be 106 if all of the enhancement factor were electromagnetic. For conventional SERS, there have been a variety
of estimates of electromagnetic enhancement factors,26 and
these typically give average 兩 E兩 2 values of a few hundred,
which suggests that most but not all of the enhancement is
electromagnetic. However, the situation for single molecule
SERS is much less clear, as the properties which determine
the peak 兩 E兩 2 are incompletely known. Peak 兩 E兩 2 values are
relatively modest for isolated spheres 共⬃100兲, however, they
are significantly higher (⬎103 ) for spheroids and nanoprisms, due in part to red-shifted plasmon excitation 共which
gives the metal a more free-electronlike response兲 and to
sharp points that produce lightening-rod effects. In many of
the theoretical studies,32,33,37,41– 44 it was recognized that the
fields between two spheres are strongly enhanced, indeed
兩 E兩 2 enhancements greater than 105 were estimated in some
of the early literature32,33,37 mostly based on quasistatic approximations. Very recently, Lu and co-workers45 have obtained similarly large enhancements in finite element electrodynamics studies of atomic force microscopy 共AFM兲 tips
interacting with spherical silver nanoparticles. These large
enhancement estimates have been used in the interpretation
of single molecule SERS.37,46,47 For example, Brus and coworkers found that the active nanoparticles for single molecule SERS are compact aggregates consisting of a minimum of two individual particles.17,38 Also, Xu and Käll
et al.33 demonstrated that single molecule SERS associated
with hemoglobin between two colloidal particles may be
subject to 兩 E兩 2 enhancements as large as 105 . However it has
never been clarified how the field between two nanoparticles
depends on the shape, size, and orientation of the nanoparticles, and in particular what is the highest possible SERS
enhancement based on electromagnetic effects. In addition,
for monomers or dimers that support multiple plasmon resonances, it is not clear which of the resonances 共dipole, quadrupole, etc.兲 give the largest SERS intensities. This is one
reason why there has been confusion as to whether single
molecule SERS can occur for molecules on isolated particles
or not.17,39,40
In this paper, we report DDA calculations for Ag nanoparticles and dimers of Ag nanoparticles in which size,
shape, orientation and spacing of the nanoparticles are varied
over a significant range. Although there have been earlier
studies of fields associated with metal particles in two
dimensions,48,49 here we consider three-dimensional compact
particles that are relevant to recent experiments. We especially consider spherical and triangular shapes, as the Ag
dimer resonances for these shapes exhibit distinct dipole and
quadrupole plasmon resonance features which aid in interpreting the spectra. Our analysis will emphasize peak E-field
values, although for completeness we show the complete
near-field for many particle shapes and arrangements. Al-
E. Hao and G. C. Schatz
though individual particles will be examined, we find that
dimers give maximum E-fields that are a factor of 10 larger
than any compact monomer structure, so most of our analysis will be on dimers. E-field enhancement factors as high as
105 are found, and with surprising insensitivity to local radius of curvature, but with strong dependence on particle
spacing. We show that the dimer of Ag triangular prisms has
great potential in single molecule SERS studies because of
the much stronger induced E-fields and unique plasmon resonances at visible and near infrared 共IR兲 wavelengths.
Our calculations are based on local dielectric constants
from measurements,50 which means that we are not including
for nonlocal or size dependent dielectric response. The nonlocal description has occasionally been considered,51,52 usually for spheres or coupled spheres, with dielectric response
modeled using the hydrodynamic or Lindhard–Mermin models. Significant effects are found for small particles 共diameter
less than 5 nm兲 typically leading to broader plasmon resonances and smaller SERS enhancements. Size dependent dielectric constants arise because of scattering of the conduction electrons off the particle surfaces, which is an effect
which could be important whenever the particle is smaller
than the electron mean free path. Both classical and quantum
theories of this effect have been developed, as recently summarized by Coronado and Schatz,53 and for silver particles
this again leads to a reduction in SERS enhancements, particularly for particle diameters smaller than 5 nm. Since the
particle sizes that we consider in this paper are all much
larger than 5 nm, we expect that the nonlocal and size dependent dielectric response will not have a major effect.
However, a possibly important exception to this can arise
when we consider two large particles that nearly touch. In
this case, the strong inhomogeneous fields at the junction
could lead to high wavevector excitation in the dielectric
response, which will enhance nonlocal contributions. In addition, the use of continuum electrodynamics becomes questionable when the particles are so close together that
electron–electron interactions, and electron transfer between
particles can occur. Because of this limitation, we will restrict our studies of dimer structures to particles that are with
one exception separated by at least 2 nm. In this way we are
restricting our modeling to structures that we feel can be
described realistically by classical electromagnetic theory.
II. CALCULATIONS
The DDA method has been described in detail
elsewhere.21–23,27 All calculations here refer to water as the
external dielectric medium. The silver dielectric constant in
all calculations has been taken from Palik50 but smoothed as
described by Jensen et al.27 All the DDA calculations use a
cubic grid with a grid spacing of 1 nm unless otherwise
noted. This grid spacing leads to high-quality simulations of
extinction spectra which are converged with respect to grid
spacing. However, convergence for the E-field is more difficult to obtain, as demonstrated in Fig. 1 where we plot 兩 E兩 2
versus distance from the center of a 20 nm diameter sphere,
for a direction through along the polarization vector. Included in the figure are results for grid spacings of 1, 0.5, and
0.25 nm. Note that in the DDA results, the field is not cal-
Downloaded 27 Sep 2004 to 128.210.142.204. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
J. Chem. Phys., Vol. 120, No. 1, 1 January 2004
FIG. 1. The E-field enhancement factor vs the distance away from the center
of a 20 nm Ag sphere. The data were taken along the polarization axis. The
grid spacings are 1, 0.5, and 0.25 nm.
culated directly on the particle surface; instead the closest
point is one grid point removed from the surface. The figure
shows a factor of 2 increase in the peak field between 1 and
0.5 nm, and a smaller increase in going to 0.25 nm. In addition, we find from Mie theory calculations that 兩 E兩 2 at the
surface should be about 240, which is to be compared with a
value of 170 for the first grid point of the 0.25 nm grid
spacing, and an extrapolated value of 210 at the sphere surface. Thus we conclude that convergence of the E-field is
approximately achieved for a spacing of 0.25 nm. Unfortunately it would not be possible to do many calculations of
interest with this spacing 共using particle dimensions representative of what are actually made兲. However, in several
tests both with monomers and dimers, we found that the
change in 兩 E兩 2 in going from 0.25 or 0.50 nm spacing to 1
nm is consistently less than a factor of 3 so we assume that
this is a bound on the accuracy of our results. For example,
for the sphere dimer that we discuss later, we found that the
peak value of 兩 E兩 2 is 11 000 for a 1 nm spacing and 28 000
for 0.5 nm spacing, and for the dimer of triangular prisms,
we found a peak 兩 E兩 2 of 53 000 for 1 nm spacing and 59 000
for 0.5 nm spacing. Since a 1 nm spacing makes it possible
to study a wide range of particles that are relevant to the
experiments, we have used this in all the calculations reported below 共unless otherwise noted兲. Obviously this means
that there will be a factor of 2 to 3 uncertainty in the magnitudes of 兩 E兩 2 , but this should be adequate to determine key
properties of interest such as the influence of particle shape
and arrangement on peak field where we are comparing similar structures that have the same grid size.
As mentioned previously, we have chosen to do calculations mostly with spheres and triangles. The sphere calculations all refer to a radius of 36 nm, and we have studied
dimer separations of 2 nm and larger. The triangle calculations mostly refer to triangles that are 60 nm in edge length
and 12 nm thick. We have also done calculations with 30⫻6
and 120⫻24 triangles in order to study the size dependence
of the results. In addition, we have studied ‘‘snipped’’ triangles in which a small region at the end of each tip of the
triangle is removed in order to mimic the effect of rounding
Electromagnetic fields around silver nanoparticles and dimers
359
FIG. 2. The calculated extinction spectra of a spheroid, a rod, and a triangular prism 共with the same effective radius 共15 nm兲. The aspect ratios of
spheroid and rod are 3.4:1 and 2.8:1, respectively. The prism has a 60 nm
edge dimension with thickness of 12 nm.
of the tips that occurs in the experiments. Thus a 2 nm snip
refers to a particle in which 2 nm along the 60 nm edges has
been removed from each tip. Two other shapes we have studied are: Oblate spheroids, with an aspect ratio of 3.4:1, and
an effective radius 共radius that makes the actual volume
match 4/3 ␲ r3 ) of 15 nm, and a cylindrical rod with an aspect ratio of 2.8:1 and the same effective radius.
In the triangle dimer calculations, we have considered
two different configurations: Head to head 共i.e., tip to tip兲
and head to tail 共i.e., tip to side兲, in order to determine how
the plasmon resonance and E-fields vary with particle orientation. In all dimer studies, we have done calculations with
the polarization vector taken in three possible directions,
typically the parallel and two perpendicular directions to the
interparticle axis. The direction of the k-vector is generally of
lesser importance, but for calculations where the polarization
is parallel to the interparticle axis, the k-vector is taken to be
perpendicular to the plane of the prism, while for perpendicular polarization, we take the k-vector parallel to the interparticle axis.
III. RESULTS AND DISCUSSION
A. Monomers
The electrodynamics of nonspherical metal nanoparticles
has long been of interest, especially to estimates of the SERS
enhancement factor.23,28 However, the absence of analytical
theory results for particles other than spheroids has proven to
be a serious hindrance, and in fact there are very few accurate studies of particles other than spheres or spheroids. In
this section we have studied three particle shapes: Rod, oblate spheroid, and triangular prism. Figure 2 presents the
extinction spectra of these three particles, where in all cases
the dimensions have been chosen so that the longest wavelength resonance is near 700 nm. The silver rod and spheroid
show two important plasmon resonances, one associated
with long-axis excitation near 700 nm, and another one with
short axis excitation at 360 nm. The extinction spectrum of
the Ag triangular prism 共with side length⫽60 nm, snip⫽0
nm, and thickness⫽12 nm兲 shows a strong in-plane dipole
Downloaded 27 Sep 2004 to 128.210.142.204. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
360
J. Chem. Phys., Vol. 120, No. 1, 1 January 2004
E. Hao and G. C. Schatz
FIG. 3. 共Color兲 E-field enhancement contours external to monomers with different shapes. 共a兲 and 共b兲 are the E-field enhancement contours external to a
triangular prism polarized along the two different primary symmetry axes, 共c兲 and 共d兲 are the E-fields enhancement contours for a rod and spheroid polarized
along their long axes. The arrows show where is the maximum of E-filed.
plasmon resonance at 700 nm, an in-plane quadrupole resonance at 434 nm, and an out-of-plane quadrupole resonance
at 340 nm. 共See Ref. 6 for details of the assignment of these
features.兲 Figure 3 presents E-fields for the nonspherical particles in Fig. 2, all for the longest wavelength plasmon resonance near 700 nm. 共The shorter wavelength resonances
have smaller E-fields so we do not present the results.兲 The
E-field contours show that the maximum enhancement for
the dipole resonances occurs at the particle tips. The largest
fields ( 兩 E兩 2 ) for the dipole resonance are 3500 共times the
applied field兲 for the triangular prism, 4500 for the rod, and
4700 for the spheroid at the peak resonance wavelength near
700 nm. For the spheroid and rod, the long wavelength resonance red shifts with increasing aspect ratio, and at the resonance wavelengths, the E-field enhancements increase with
increasing aspect ratio while keeping the same effective radius. For the triangular prism, the extinction spectrum is sensitive to its edge length, thickness and snipping. The long
wavelength dipole resonance shifts to red when the edge
length is increased or the thickness decreased, while snipping
moves the plasmon resonance to blue. Although the E-fields
around nonsymmetrical particles depend on their sizes and
shapes, the largest fields ( 兩 E兩 2 ) for the long wavelength resonances are several thousand times the applied field for all the
particle shapes considered.
For spherical Ag particles with radii less than 20 nm,
earlier studies using DDA showed that the maximum E-field
enhancement is less than about 200 near the plasmon
resonance.23 The plasmon wavelength associated with this
resonance is at about 410 nm. Because the E-field enhancement is wavelength dependent, and the dielectric constant is
closer to being free-electronlike at longer wavelengths, it is
interesting to examine the sphere E-field enhancement for
spheres that are sufficiently large that they have a plasmon
resonance near 700 nm 共where the other particles that we
considered above have their resonances兲 rather than at 410
nm. For a sphere with radius of 90 nm, the plasmon resonance occurs at 700 nm, and for this case the E-field enhancement is ⬃25. This value is much smaller than for the
other particles, but this is to be expected as the particle is
much larger and, therefore, would have more radiative damping.
Downloaded 27 Sep 2004 to 128.210.142.204. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
J. Chem. Phys., Vol. 120, No. 1, 1 January 2004
FIG. 4. The calculated extinction spectra of a dimer of Ag nanoparticles 共36
nm in diameter兲 separated by 2 nm. Solid lines are corresponding to polarization along the two primary symmetry axes, while dashed line represents
the overall extinction spectrum.
Electromagnetic fields around silver nanoparticles and dimers
361
approach. Further details of particle interaction effects on
extinction spectra, including surprising blue shifts for two
dimensional particle arrays with separations larger than 100
nm are described by Zhao et al.54 so we omit further consideration of this point. Figure 5 presents contours of the electric field 兩 E兩 2 around the Ag dimer 共taking a cut that passes
through the center of the spheres兲 for 430 and 520 nm. Here
we see that the maximum enhancement for both dipole and
quadrupole resonances occurs at the midpoint between the
two spheres. The largest fields ( 兩 E兩 2 ) for the 共TR兲 quadrupole and dipole resonances are 3500 times the applied field
at 430 nm and 11 000 times the applied field at 520 nm,
respectively. These enhancements are much larger than are
found for isolated Ag spherical nanoparticles where factor of
102 peak enhancements are found for the same sphere size
and grid spacing. We have examined the peak enhancements
at other wavelengths, and we find that peak enhancement is
associated with peak extinction for all the structures that we
have considered.
C. Dimer of triangular prisms
B. Dimer of spheres
One of the most interesting questions raised by recent
single-molecule SERS experiments concerns the possibility
that single-molecule SERS arises from a ‘‘hot’’ site16,17,37,38
between two particles, or perhaps at the junction between
two or more particles. This possibility was suggested by
early theoretical simulations.32 To study this possibility, we
have performed DDA calculations for structures which consist of two particles that nearly touch 共a minimal separation
of 2 nm was considered, as the use of a local dielectric function description becomes questionable for shorter distances兲.
Touching particles were also considered, but slightly smaller
fields were found, so we omit this case from our analysis.
Figure 4 presents the extinction spectrum of a dimer of
36 nm silver spheres that are separated by 2 nm. Included are
three spectra: One for polarization parallel to the interparticle
axis that shows a strong peak at 520 nm and a shoulder peak
at 430 nm, one for perpendicular polarization that shows a
peak at 410 nm, and one that corresponds to an average over
the three possible polarizations. To assign the resonances, we
examined the induced polarization in each DDA element at
the different wavelengths. The behavior is similar to that
reported by Jensen et al.27 in which the 520 nm polarizations
are all parallel to the applied polarization direction while
those at 430 nm are about half parallel and half antiparallel.
If we were to expand the induced polarization in vector
spherical harmonics about the midpoint of the spheres, the
520 nm polarizations would be predominantly dipolar in
character while those at 430 nm would be quadrupolar, so we
will use the terms ‘‘dipole plasmon’’ and ‘‘quadrupole plasmon’’ to label these features. Importantly, the behavior of the
induced polarizations depends strongly on interparticle distance, with dipoles in the interfacial region between the
spheres being strongly influenced by interparticle coupling
when the particles are close together. As a result, the extinction spectra of dimers are very sensitive to separation distance, with the dipole resonance red shifting, and the quadrupole resonance becoming more intense as the particles
It has already been noted above that isolated nonspherical particles show typically larger 兩 E兩 2 than for isolated
spheres because of their ability to a support plasmon resonance at long wavelengths while keeping the effective radii
small. It is therefore of interest to see if dimers of nonspherical particles also yield higher 兩 E兩 2 . One issue of significance
to this study is where the longest wavelength plasmon resonance occurs. Here we have chosen only to consider dimers
that have dipole plasmon wavelengths shorter than 1000 nm,
which means that the particle separations are typically a few
nanometers or more.
Figure 6 shows the extinction spectra of a pair of triangular prisms that are positioned on a common plane, with a
common perpendicular bisector, and with corresponding tips
directed toward each other 共‘‘head-to-head’’兲 and separated
by 2 nm. The extinction spectra and the E-field enhancement
contours obtained for polarization along the x and z axes, i.e.,
perpendicular to the plane of the triangles, and perpendicular
to the interparticle axis, are very similar to what is found for
uncoupled triangles.6,23 However, when the polarization is
along the inter-particle axis 共y polarized兲 there is a significant
red shifting of the dipole plasmon resonance 共from 653 to
932 nm兲, and an additional weak peak at 550 nm appears,
while the in-plane quadrupole peak at 434 nm remains. To
assign the resonances, Fig. 7 plots the induced polarizations
for the 550 and 932 nm resonances. This behavior is analogous to the sphere dimer results described earlier.27 For the
932 nm resonance, the induced polarization plot shows that
the vectors inside the prisms are aligned along the y axis with
the largest induced polarization occurring where the tips
come together. In contrast, at 550 nm we see the same induced polarizations, but the polarizations at the interface between the prisms are directed oppositely on the two particles.
This indicates that the 932 nm peak is the dipole plasmon
resonance, while the 550 nm peak is the quadrupole plasmon
resonance.27
Figure 8 shows E-field contours for the head to head
configuration of the prisms at wavelengths that correspond to
Downloaded 27 Sep 2004 to 128.210.142.204. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
362
J. Chem. Phys., Vol. 120, No. 1, 1 January 2004
E. Hao and G. C. Schatz
FIG. 5. 共Color兲 E-field enhancement contours external to a dimer of Ag nanoparticles separated by 2 nm, for a plane that is along the inter-particle axis and
that passes midway throng the two particles. In the 3D plots, the axis perpendicular to the selected plane represents the amount of E-field enhancement around
the dimer. Left: 430 nm. Right: 520 nm.
the dipole and quadrupole plasmon resonances in Fig. 7. For
both dipole and quadrupole resonances, we see large enhancements at the tips and the interface, but the threedimensional 共3D兲 plots show that the maximum enhancement occurs at the interface between the two prisms. The
largest fields ( 兩 E兩 2 ) for dipole and quadrupole resonances are
53 000 and 5700 times the applied field, respectively. For the
in-plane quadrupole resonance at 434 nm, the maximum en-
FIG. 6. The calculated extinction spectra of a dimer of triangular Ag nanoprisms along three primary symmetry axes with a position of ‘‘tip-to-tip’’
and with a spacing of 2 nm. The prism has a 60 nm edge dimension with
snip of 2 nm and with thickness of 12 nm. 36 480 dipoles are used in the
calculation.
hancement is at the sides of the two prisms and the field is
only 16 times the applied field, very similar to the result for
the individual prism.
Figure 9 represents the extinction spectra for a pair of
FIG. 7. Polarization vectors for quadrupole 共upper兲 and dipole 共bottom兲
resonances at 932 and 550 nm, respectively, for a ‘‘tip-to-tip’’ prism dimer
described in Fig. 6.
Downloaded 27 Sep 2004 to 128.210.142.204. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
J. Chem. Phys., Vol. 120, No. 1, 1 January 2004
Electromagnetic fields around silver nanoparticles and dimers
363
FIG. 8. 共Color兲 E-field enhancement contours external to a ‘‘tip-to-tip’’ prism dimer described in Fig. 6, for a plane that is along the inter-particle axis and that
passes midway throng the two particles. In the 3D plots, the axis perpendicular to the selected plane represents the amount of E-field enhancement around the
dimer. Left: 932 nm. Right: 550 nm.
triangular prisms with a ‘‘head-to-tail’’ orientation and separated also by 2 nm. Again, the polarizations along the x axis
and z axis are very consistent with results for the individual
prisms. However, with the polarization along the interparticle axis 共y axis兲, we see four peaks located at 852, 653,
550, and 434 nm. Examination of the induced polarizations
FIG. 9. The calculated extinction spectra of a dimer of triangular Ag nanoprisms along three primary symmetry axes with a position of ‘‘tip-to-side’’
and with a spacing of 2 nm. The prism has a 60 nm edge dimension with
snip of 2 nm and with thickness of 12 nm. 36 480 dipoles are used in the
calculation.
indicates that the 852 and 653 nm peaks are dipolar, while
the 550 and 434 nm peaks are quadrupolar.
Figure 10 shows E-field enhancement contours at 653
and 852 nm with the induced polarization along the interparticle axis. For both dipole and quadrupole resonances, we
see that the maximum enhancement occurs at the interface
between two prisms. The largest fields ( 兩 E兩 2 ) for dipole resonances at 852 and 650 nm are 57 000 and 3 700 times the
applied field, respectively. The fields ( 兩 E兩 2 ) for quadrupole
resonances at 550 and 434 nm are 570 and 16 times the
applied field, respectively. From the induced polarization
and, therefore, the E-field enhancement for the ‘‘head-tohead’’ dimer in Figs. 6 – 8, we see that the head to head and
head to tail arrangements of the triangular prisms give almost
identical peak enhancements. However, other properties of
the two dimers are quite different. For example, the head-tohead dimer shows much stronger E-field enhancement for
the quadrupole resonances at 550 nm.
Figure 11 shows the orientation averaged extinction
spectra of head-to-tail dimers with different spacings. The
bottom panel refers to 60⫻12 nm triangles with a 2 nm snip,
while the top results are for the analogous dimer of half the
size 共with side length⫽30 nm, thickness⫽6 nm, and snip⫽1
nm兲 and half the grid spacing 共0.5 nm instead of 1 nm兲. The
results in Fig. 11 show four peaks. Among them, the strongest feature is the dipole peak at 653 nm 共or 628 nm for
Downloaded 27 Sep 2004 to 128.210.142.204. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
364
J. Chem. Phys., Vol. 120, No. 1, 1 January 2004
E. Hao and G. C. Schatz
FIG. 10. 共Color兲 E-field enhancement contours external to a ‘‘tip-to-side’’ prism dimer described in Fig. 9, for a plane that is along the inter-particle axis and
that passes midway throng the two particles. In the 3D plots, the axis perpendicular to the selected plane represents the amount of E-field enhancement around
the dimer. Left: 852 nm. Right: 653 nm.
half-sized prisms兲. In addition, there is a weaker quadrupole
peak at 434 nm and a sharp out-of-plane quadrupole peak at
340 nm that does not change with spacing of the triangles.
However, the dipole peak along the interparticle axis is very
sensitive to separation distance especially when the two
prisms are closely spaced. Thus there is a more than a 100
nm red-shift for the 60 nm particle as spacing decreases from
4 to 2 nm 共from 2 to 1 nm for half-size prisms兲.
The E-field enhancement for the prism dimer in Fig. 11
is also very sensitive to spacing. Figure 12 shows the maximum enhancement for the head to tail configuration as a
function of spacing. The figure shows an enhancement that
increases by nearly an order of magnitude for a 2 nm change
in spacing. The results for the 30 and 60 nm prisms are very
similar in appearance, but with the 30 nm result shifted to
smaller distances by an amount which scales with the relative sizes of the nanoparticles. This indicates that larger
E-field enhancements can be obtained for dimers composed
of larger particles. We have furthered tested this conclusion
by examining 120⫻24 nm prisms separated by 4 nm, and we
find results that are somewhat smaller than the 60⫻12 nm
particles separated by 2 nm. This indicates that the simple
scaling of enhancements with particle size, which is an expected result from the quasistatic approximation, breaks
down when the particles are large enough that radiative
FIG. 11. The orientation-averaged extinction efficiency for ‘‘tip-to-side’’
prism dimers with three choices of spacings as noted in the legend. Upper:
the prism has a 30 nm edge dimension with thickness of 6 nm and snip of 1
nm, which is referring to a half-size prism. Bottom: The prism has a 60 nm
edge dimension with snip of 2 nm and with thickness of 12 nm. 36 480
dipoles are used in both calculations.
Downloaded 27 Sep 2004 to 128.210.142.204. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
J. Chem. Phys., Vol. 120, No. 1, 1 January 2004
FIG. 12. The E-field enhancement in the junction of ‘‘tip-to-side’’ prim
dimers 共described in Fig. 11兲 as function of the dimer spacing. Black square
represents the ‘‘half-size’’ prism dimer.
damping effects are significant. A more detailed analysis of
the dimer junction region 共not presented兲 indicates that the
maximum enhancement is located in the middle of the gap
between the 60 nm prisms for very close spacing 共2 nm兲, but
close to one or the other particles for larger separations 共6
nm兲.
For individual triangular prisms, the in-plane dipole
resonance is very sensitive to snipping.6,23 Figure 13 shows
the orientation averaged extinction spectra for dimers of
prisms with different snips. It is interesting that only the
in-plane z axis dipole resonance is sensitive to snipping, with
the 5 nm snipped prism giving a peak that is blue-shifted by
50 nm in comparison to the perfect prism. This is not surprising because the dipole peak is mostly associated with z
polarized excitation. The y axis dipole resonances around
850 nm are weakly sensitive to snipping, but intensities increase if snipped. The maximum E-field enhancement for the
y axis dipole resonances is also not sensitive to snipping,
which means that it is not necessary to make a perfect triangular prism to generate a large E-field near a dimer. For
SERS studies, the dimer of the snipped prisms could be a
FIG. 13. The orientation-averaged extinction efficiency for ‘‘tip-to-side’’
prism dimers separated by 2 nm. The prism has a 60 nm edge dimension
with thickness of 12 nm and snips of 0, 2, and 5 nm. For snip⫽2, 36 480
dipoles are used in the calculation.
Electromagnetic fields around silver nanoparticles and dimers
365
better choice than unsnipped prisms because the contact area
at the interface is larger for that case, while the enhancement
is the same. Note that the lack of dependence of the E-field
enhancement to snipping occurs for 850 nm excitation. We
have also examined electrostatic fields for the same dimers,
and we find that this field is larger for the unsnipped particles
than for the snipped particles. Thus it seems that the ‘‘lightening rod’’ effect, which is a concept that comes from electrostatics is less relevant for the plasmon resonant response
of dimers. A similar result was noted by Kottmann et al.48 in
their studies of isolated triangular wire structures, but here
we find that this effect is even more important for dimers
than it was for isolated wires.
We have also studied tail to tail configurations of the
dimers, and we find that this leads to a somewhat smaller
peak enhancement ( 兩 E兩 2 ⫽14 000) than the head to head and
head to tail configurations, however, the contact area associated with the peak intensity is much larger than for the other
cases.
IV. CONCLUSION
In this paper we have used classical electrodynamics to
study the optical properties of isolated silver nanoparticles
and pairs of these nanoparticles that are separated by a few
nanometers. These particles exhibit distinct dipole, quadrupole and higher multipole plasmon resonances, and excitation of these resonances creates an E-field external to the
particles that is important in determining normal and single
molecule SERS intensities. Because of this, we have attempted to systematically determine which particles and particle dimers lead to the highest E-fields. For individual particles we find that the largest 兩 E兩 2 values are very similar for
triangular prisms, oblate spheroids or cylindrical rods, with
兩 E兩 2 always being less than 104 . For a dimer of nanoparticles, we find 兩 E兩 2 values of closer to 105 for structures
where the particle separation is 2 nm. The enhancement is a
strong function of separation distance, and it scales with particle size such that larger particles give the same enhancements for larger separations. Unfortunately the largest enhancements occur for separations that are too small for
classical electrodynamics to be valid, and for which the DDA
approach is hard to converge. Other structural properties of
the dimer, such as local curvature, are found to play a less
important role in determining E-field enhancements, although larger enhancements are generally found for dimers
that have longer wavelength dipole plasmon resonances. We
find that this lack of sensitivity to particle structure is more
noticeable for the field at the plasmon resonance wavelength
than it is for the electrostatic field. This suggests that quasistatic calculations may be less accurate for determining peak
E-field enhancements than for more averaged properties such
as the extinction spectrum.
The lack of sensitivity of peak E-field on subtle details
of the dimer structure combined with the large SERS enhancement factor we find ( 兩 E兩 4 ⫽1010 for 2 nm separation兲
indicate that many dimer structures should be capable of producing single molecule SERS. In addition, the present results
provide support for earlier conclusions17 that single molecule
SERS is associated with compact aggregates consisting of a
Downloaded 27 Sep 2004 to 128.210.142.204. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
366
J. Chem. Phys., Vol. 120, No. 1, 1 January 2004
minimum of two individual particles. However, the present
studies suggest that not all of the single molecule SERS enhancement factor of 1012 can be ascribed to purely electromagnetic effects.
ACKNOWLEDGMENTS
We thank Richard P. Van Duyne, K. Lance Kelly, and
Louis Brus for many stimulating conversations. We gratefully acknowledge support from the National Science Foundation through the Materials Research Center 共MRSEC兲.
Support was also provided by the Air Force Office of Scientific Research MURI program 共F49620-02-1-0381兲.
G. Mie, Ann. Phys. 共Leipzig兲 25, 377 共1908兲.
C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by
Small Particles 共Wiley Interscience, New York, 1983兲.
3
U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters;
Springer Series in Materials Science 25 共Springer, Berlin, 1995兲.
4
Y. Y. Yu, S. S. Chang, C. L. Lee, and C. R. C. Wang, J. Phys. Chem. B
101, 6661 共1997兲.
5
J. M. Petroski, Z. L. Wang, T. C. Green, and M. A. El-Sayed, J. Phys.
Chem. B 102, 3316 共1998兲.
6
R. Jin, Y. Cao, C. A. Mirkin, K. L. Kelly, G. C. Schatz, and J. G. Zhang,
Science 294, 1901 共2001兲.
7
C. L. Haynes and R. P. Van Duyne, J. Phys. Chem. B 105, 5599 共2001兲.
8
K. Franklin, S. J. Hee, and Y. Peidong, J. Am. Chem. Soc. 124, 14316
共2002兲.
9
N. R. Jana, L. Gearheart, and C. J. Murphy, Chem. Commun. 共Cambridge兲
2001, 617 共2001兲.
10
S. Chen, Z. Fan, and D. L. Carroll, J. Phys. Chem. B 106, 10777 共2002兲.
11
E. Hao, K. L. Kelly, J. T. Hupp, and G. C. Schatz, J. Am. Chem. Soc. 124,
15182 共2002兲.
12
C. A. Mirkin, R. L. Letsinger, R. C. Mucic, and J. J. Storhoff, Nature
共London兲 382, 607 共1996兲.
13
J. J. Storhoff, A. A. Lazarides, R. C. Mucic, C. A. Mirkin, R. L. Letsinger,
and G. C. Schatz, J. Am. Chem. Soc. 122, 4640 共2000兲.
14
C. J. Loweth, W. B. Caldwell, X. Peng, A. P. Alivisatos, and P. G. Schultz,
Angew. Chem., Int. Ed. 38, 1808 共1999兲.
15
W. P. McConnell, J. P. Novak, L. C. Brousseau III, R. R. Fuierer, R. C.
Tenent, and D. L. Feldheim, J. Phys. Chem. B 104, 8925 共2000兲.
16
S. Nie and S. R. Emory, Science 275, 1102 共1997兲.
17
A. M. Michaels, M. Nirmal, and L. E. Brus, J. Am. Chem. Soc. 121, 9932
共1999兲.
18
B. Nikoobakht, J. Wang, and M. A. El-Sayed, Chem. Phys. Lett. 366, 17
共2002兲.
19
C. L. Haynes and R. P. Van Duyne, J. Am. Chem. Soc. 124, 10596 共2002兲.
20
Y. Cao, R. Jin, and C. A. Mirkin, Science 297, 1536 共2002兲.
21
W. H. Yang, G. C. Schatz, and R. P. Van Duyne, J. Phys. Chem. 103, 869
共1995兲.
22
K. L. Kelly, A. A. Lazarides, and G. C. Schatz, Comput. Sci. Eng. 3, 67
共2001兲.
1
2
E. Hao and G. C. Schatz
23
K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, J. Phys. Chem. B
107, 668 共2003兲.
24
R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, Phys. Rev. Lett. 75,
4772 共1995兲.
25
E. Moreno, D. Erni, C. Hafner, and R. Vahldieck, J. Opt. Soc. Am. A 19,
101 共2002兲.
26
E. J. Zeman and G. C. Schatz, J. Phys. Chem. 91, 634 共1987兲.
27
T. Jensen, K. L. Kelly, A. Lazarides, and G. C. Schatz, J. Cluster Sci. 10,
295 共1999兲.
28
G. C. Schatz, Acc. Chem. Res. 17, 370 共1984兲.
29
M. Moskovits, Rev. Mod. Phys. 57, 783 共1985兲.
30
H. Metiu and P. Das, Annu. Rev. Phys. Chem. 35, 507 共1984兲.
31
M. Kerker, Acc. Chem. Res. 17, 271 共1984兲.
32
H. Metiu, Prog. Surf. Sci. 17, 153 共1984兲.
33
H. Xu, J. Aizpurua, M. Käll, and P. Apell, Phys. Rev. E 62, 4318 共2000兲.
34
G. C. Schatz and R. P. Van Duyne, in Handbook of Vibrational Spectroscopy, edited by J. M. Chalmers and P. R. Griffiths 共Wiley, New York,
2002兲, Vol. 1, pp. 759–774.
35
A. Campion, Chem. Soc. Rev. 4, 241 共1998兲.
36
A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, J. Phys.: Condens.
Matter 4, 1143 共1992兲.
37
H. Xu, E. J. Bjerneld, M. Käll, and L. Borjesson, Phys. Rev. Lett. 83, 4357
共1999兲.
38
A. M. Michaels, J. Jiang, and L. E. Brus, J. Phys. Chem. B 104, 11965
共2000兲.
39
S. R. Emory and S. Nie, J. Phys. Chem. B 102, 493 共1998兲.
40
K. Kneipp, H. Kneipp, G. Deinum, I. Itzkan, R. R. Dasari, and M. S. Feld,
Appl. Spectrosc. 52, 175 共1998兲; K. Kneipp, H. Kneipp, I. Itzkan, R. R.
Dasari, and M. S. Feld, Chem. Rev. 共Washington, D.C.兲 99, 2957 共1999兲.
41
M. Inoue and K. Ohtaka, J. Phys. Soc. Jpn. 52, 3853 共1983兲.
42
H. Xu and M. Käll, Phys. Rev. Lett. 89, 246802 共2002兲.
43
M. Quinten, Appl. Phys. B: Lasers Opt. 73, 245 共2001兲.
44
P. K. Aravind, A. Nitzan, and H. Metiu, Surf. Sci. 110, 189 共1981兲; P. K.
Aravind, R. Rendell, and H. Metiu, Chem. Phys. Lett. 85, 396 共1982兲; P.
K. Aravind and H. Metiu, J. Phys. Chem. 86, 5076 共1982兲; P. K. Aravind
and H. Metiu, Surf. Sci. 124, 506 共1983兲.
45
M. Micic, N. Klymyshyn, Y. D. Suh, and H. P. Lu, J. Phys. Chem. B 107,
1574 共2003兲.
46
S. Corni and J. Tomasi, J. Chem. Phys. 116, 1156 共2002兲.
47
W. E. Doering and S. Nie, J. Phys. Chem. B 106, 311 共2002兲.
48
J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, Phys. Rev. B
64, 235402 共2001兲; J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S.
Schultz, Chem. Phys. Lett. 341, 1 共2001兲.
49
M. Futamata, Y. Maruyama, and M. Ishikawa, J. Phys. Chem. B 107, 7607
共2003兲.
50
E. D. Palik, Handbook of Optical Constants of Solids 共Academic, New
York, 1985兲.
51
R. Fuchs and F. Claro, Phys. Rev. B 35, 3722 共1987兲.
52
P. T. Leung and W. S. Tse, Solid State Commun. 95, 39 共1995兲.
53
E. A. Coronado and G. C. Schatz, J. Chem. Phys. 119, 3926 共2003兲.
54
L. L. Zhao, K. L. Kelly, and G. C. Schatz, J. Phys. Chem. B 107, 7343
共2003兲.
Downloaded 27 Sep 2004 to 128.210.142.204. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp