J. Phys. Chem. C 2008, 112, 20233–20240
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Quantitative Analysis of Dipole and Quadrupole Excitation in the Surface Plasmon Resonance of Metal Nanoparticles Fei Zhou, Zhi-Yuan Li,* and Ye Liu Laboratory of Optical Physics, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100190, China
Younan Xia Department of Biomedical Engineering, Washington UniVersity in St. Louis, St. Louis, Missouri 63130 ReceiVed: August 8, 2008; ReVised Manuscript ReceiVed: October 16, 2008
Metal nanoparticles have received increasing attention for their peculiar capability to control local surface plasmon resonance (SPR) when interacting with incident light waves. In this article, we calculate the optical extinction spectra of silver nanocubes with the edge length ranging from 15 to 200 nm by using the discrete dipole approximation method. An increasing number of SPR peaks appear in the optical spectra, and their positions change when the nanocube size increases. We have developed a method to quantitatively separate the contributions of the individual dipole component and quadrupole component of the optical extinction cross sections. This allows us to specify unambiguously the physical origin of each SPR peak in the spectra. We have also extensively analyzed the distribution patterns of electric fields and electric charges within and around the silver nanoparticle. These patterns clearly show the dipole and quadrupole excitation features at the SPR peaks. The near-field analyses are consistent with the far-field extinction spectra analyses. This suggests that the combination of far-field spectra and near-field pattern analysis can greatly help to uncover the intrinsic physics behind light interaction with metal nanoparticles and excitation dynamics of local surface plasmonic waves. I. Introduction In recent years, significant attention has been paid to the study of metal nanoparticles1 for their applications in nanoantennas,2-6 surface-enhanced Raman scattering (SERS),7-10 and medical therapy.11-17 These applications are based on the unique optical properties of metal nanoparticles.18,19 When the incident electromagnetic wave interacts with a nanoparticle, localized surface plasmon resonance (LSPR), which is a collective oscillation of conduction electrons, will be excited. This phenomenon leads to large enhancements of the local electromagnetic field at the nanoparticle surface.20 LSPR of a nanoparticle is very sensitive to its components, shape, size, and dielectric environment. 21 Nanospheres,22 triangular nanoplates,23 nanorods,24,25 and nanocubes26,27 of different sizes have been studied for their optical spectra and surface-enhanced Raman scattering (SERS) properties. Recent studies show that gold nanocages particles with an empty inner space can be synthesized on a template of single-crystal silver nanocube solid particles.11-13,28 By controlling the edge length and wall thickness of gold nanocages, the SPR peak can be conveniently tuned to around 800 nm, which is the wavelength window for several important biomedical technologies to work. For instance, gold nanocages can serve as image contrast agents in optical coherence tomography11,13 and photoacoustic tomography,16 which are two important biomedical diagnosis technologies. In addition, gold nanocages have very large optical absorption cross sections and have shown promising potential for applications in the targeted photothermal destruction of cancer cells.11,12,15 * Corresponding author. E-mail:
[email protected].
Theoretical analysis and numerical simulation have been proven to be very important in designing optimal geometries and structures of metal nanoparticles for a particular application purpose, such as SPR tuning, local field enhancement, or others. Different numerical tools that are developed and used in electromagnetics and optics have been employed. Among them, the discrete dipole approximation (DDA) method27,29-31 is the most popular because it can handle metal nanoparticles of arbitrary geometries, structures, and composites. The approach works in the frequency domain of interest and can account for the dispersive nature of metal materials automatically. The experimental measurement dispersion data of the permittivity of a particular metal can be directly utilized, and this will reduce the numerical inaccuracy when some fitting theoretical models such as the Drude model, Debye model, or Lorentz model is used in the calculations, a situation encountered in the numerical simulation by means of the finite-difference time-domain method.32,33 Previous studies show that when the size of a metal nanoparticle is very much smaller than the incident light wavelength the optical properties of the metal nanoparticle are dominated by dipole resonance excitation so that an SPR peak occurs in the optical extinction spectrum. When the size increases, higher-order plasmonic wave modes such as electric quadrupole or magnetic dipole modes can be excited at the metal nanoparticle, and a series of SPR peaks can appear in the optical spectrum. For a simple spherical metal nanoparticle, the analytical Mie theory can precisely yield all of the relevant optical properties including the near-field distribution, far-field radiation angular spectrum and the optical extinction, scattering, and absorption cross sections.21,34,35 The contribution of dipole,
10.1021/jp807075f CCC: $40.75 2008 American Chemical Society Published on Web 11/26/2008
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quadrupole, and other higher-order wave modes to these optical cross sections can be calculated. The physical origin of different SPR peaks in the optical spectra can be classified, and each peak can be attributed to a particular class of plasmonic excitation mode. However, for a general metal nanoparticle with arbitrary geometrical and physical configurations, the contribution of different plasmonic modes cannot be solved analytically. The physical origin of each SPR peak in the optical spectrum is classified as either dipole or quadrupole excitation mainly through experience or through analysis of the near-field patterns around the metal particle at the wavelength corresponding to the SPR peak. However, the former empirical approach is largely qualitative and sometimes becomes inaccurate or even erroneous, whereas the latter method is not always convenient and becomes ambiguous for very complex electromagnetic field patterns, which might consist of the superposition of several modes. To solve this problem, we have developed a method to identify the SPR peaks quantitatively in the optical spectra on the basis of the conventional DDA method. The remainder of this article is organized as follows. In section II, we briefly introduce the DDA method and show the method that we developed to calculate the contributions of the dipole component and quadrupole component from the DDA simulations. In section III, we take silver nanocubes as an example to study the influence of size on LSPR properties. The modulation on the dipole and quadrupole SPR peak and the corresponding local field enhancement characteristics will be discussed and compared. In section IV, we will conclude our results. II. Theory A. Discrete Dipole Approximation (DDA). DDA is a popular and powerful technique for computing the optical spectrum (scattering, absorption, and extinction) and near-field distribution of metal nanoparticles of arbitrary geometry. In this method, the particle is represented as a cubic array of electric dipoles. Assume the number of dipoles is N. The jth dipole has a polarizability Rj, and the total electric field at the dipole is Ej.Therefore, the polarization Pj in response to the total electromagnetic field can be described by
Pj ) RjEj
(1)
where Ej is the local electric field, which is the sum of the incident field and the retarded radiation fields from the other dipoles,
Ej ) Einc,j -
∑ AjkPk
(2)
k*j
complex conjugate gradient (CCG) algorithm in combination with the fast Fourier transform technique. When the polarization of each dipole is known, the extinction cross-section can be computed using
Cext )
4πk |Einc|2
exp(ikrjk) rjk3
{
k2rjk × (rjk × Pk) +
1 - ikrjk rjk2
Ej ) Einc,j -
∑ AjkPk
(6)
k*j
where the subscript j denotes an observation point outside the metal particle. The local field within the metal particle is directly related to the dipole moment at each discrete cell and is given by eqs 1 and 2. If a more deliberate field distribution is required, then the interpolation technique can be used over the original field pattern at the discrete cells. Besides the local field distribution, it is also helpful to examine the electric charge distribution at the surface of metal particles under the excitation of incident light. The pattern can offer a clear picture of how the geometry of metal particles can influence the congregation and oscillation of conduction electrons within the particle. From the charge distribution pattern, one can recognize the nature of a particular plasmonic wave mode at resonance. Because there are no free charges, the charge density is just the polarized charge density, which can be calculated as
F(r) ) - ∇ · P(r)
(7)
B. Contributions of the Dipole Component and Quadrupole Component. To study the contributions of the dipole component and quadrupole component quantitatively, we first calculate the overall dipole moment and quadrupole moment of the particle from the solution of the discrete dipole distribution within the metal particles in the DDA simulations. The overall dipole moment of the particle can be calculated easily by N
P)
∑ Pj
(8)
j)1
[rjk2Pk -
For a system with discrete charges, the traceless quadrupole moment tensor Q is defined as
}
3rjk(rjk · Pk)] (3) Here, k is the wavenumber, and rjk ) rj - rk. Let Ajj ) that we can obtain
AP ) Einc
(5)
j
The solution of dipole moments at the discrete cells by means of DDA provides information about not only the far-field optical extinction spectrum but also the basis of the local field distribution within and around the metal particle. The local field outside the particle, which yields the near-field pattern, is just the superposition of the radiation field from all of the discrete dipoles and the incident field, namely,
The incident field is assumed to be a plane wave and is in the form of Einc,j ) E0eik · rj where k is the wave vector. A is the dipole-dipole interaction matrix,
AjkPk )
* · Pj) ∑ Im(Einc,j
R-1 j
so
(4)
P and Einc are 3N-dimensional vectors, and A is a 3N × 3N matrix. Equation 4 can be solved iteratively by means of the
Qij )
∑ (3xk,ixk,j - rk2δij)qk
(9)
k
where xk,i is the ith component of the position of the kth charge. Here we have assumed that each dipole is composed of two point charge, qk and -qk. rk′ ) (x′, y′, z′) and rk′′ ) (x′′, y′′, z′′) are the positions of the two charges, respectively. The first item can be derived from
Analysis of Dipole and Quadrupole Excitation
Qi,j ) 3
J. Phys. Chem. C, Vol. 112, No. 51, 2008 20235
∑ [xk,ixk,jqk + xk,ixk,j(-qk)] k
)
3 2
∑ [(xk,i + xk,i)(xk,j - xk,j) + (xk,j + xk,j) × k
(xk,i - xk,i)]qk )
3 2
∑ (xk,iPk,j + xk,jPk,i)
(10)
k
Then the expression of the overall quadrupole moment of the dipole array can be represented as36
1 Qij ) Qij - (Q11 + Q22 + Q33)δij 3
(11)
The extinction cross-section of the dipole component can be calculated from
Cext )
4πk Im(E*inc · P) |Einc|2
(12)
where Einc is the average incident electric field inside the particle. For a quadrupole, the vector scattering amplitude of far-field radiation light from the metal particle is given by
X)-
k4 [nˆ × Q(nˆ)] × nˆ 6|Einc|
(13)
where n is a unit vector in the direction of r and Q(n) is defined as having components QR ) ∑β QRβnβ. Then the extinction cross-section of the quadrupole component is
Cext )
4π 2πk2 Re{(X · eˆx)θ)0} ) Re{Q(eˆz) · E*inc} 2 k 3|Einc|2
(14) where eˆx and eˆz are unit vectors in x and z directions, respectively. Here we have assumed that the incident light propagates along the z-axis direction and the electric field is polarized along the x-axis direction. The extinction efficiency 2 , where reff ) a(3/4π)1/3 is the is defined by Qext ) Cext/πreff normalized radius of the cubic particle with an edge length of a. III. Extinction Spectra Analysis To study the scaling effect, we take silver nanocubes with different sizes ranging from 15 to 200 nm as an example. Figure 1 shows the calculated normalized extinction cross-section (or extinction efficiency) of silver cubes with different sizes. The particle is embedded within an air background with a refractive index of 1.0. The incident light propagates parallel to the (001) axis of the nanocube, and the electric field is polarized parallel to the (110) direction of the nanocube. Figure 1A,B shows the extinction efficiencies of nanocubes with edge lengths of 15-90 and 100-200 nm, respectively. We plot them in different panels in Figure 1 so as to be seen clearly. Two LSPR peaks are observed for smaller particles with sizes below 80 nm, and a third peak appears between the two peaks when the nanocube edge length is longer than 90 nm. To analyze the relation between the edge length and LSPR better, we consider the statistics of the wavelength and extinction efficiency corresponding to the LSPR peaks, and the results are shown in Figure 1C. The solid lines are for the extinction efficiency, and the
Figure 1. (A) Calculated optical extinction spectra for silver nanocubes with the edge length increasing from 15 to 90 nm. (B) Calculated optical extinction spectra for silver nanocubes with the edge length increasing from 100 to 200 nm. All data are calculated by the DDA metho, and. 27 000 dipoles are used. (C) Dependence of the resonance wavelength and extinction efficiency on the size of silver nanocubes for several resonance peaks. The solid lines are for the extinction efficiency, and the dashed lines are for the resonant wavelength.
dashed lines are for the resonant wavelength. From this Figure, we can find that one of the LSPR peak is pinned around 400 nm (peak 1, the black dashed line) and does not shift with the size changes of the nanocubes whereas the other two peaks (red dashed line and green dashed line, labeled peaks 2 and peak 3, respectively) red shift obviously with increasing nanocube size. The solid lines indicate that for each peak the extinction efficiency reaches its maximum at a certain nanocube size. For peaks 1-3, the sizes are 50, 70, and 150 nm, respectively. To understand the intrinsic nature of the plasmon resonance peaks better, we can interpret the spectrum as the sum of separate contribution of the dipole component and quadrupole component. To get the separate contributions, we first calculate
20236 J. Phys. Chem. C, Vol. 112, No. 51, 2008 the overall dipole moment and quadrupole moment using eqs 8 and 11, respectively. Then, substituting the dipole moment and quadrupole moment into eqs 12 and 14, we can evaluate their extinction efficiency. The calculation results are shown in Figure 2. We also plot the total extinction efficiency in Figure 2 for comparison. For small nanocubes such as 20 and 50 nm, the optical spectra in Figure 1A show that there are two LSPR peaks. For the 20 nm particle, the two peaks (1 and 3) are located at wavelength of 393 and 417 nm, respectively, and they have almost the same magnitude. An analysis of the dipole distribution indicates that both LSPR peaks are dipole resonance modes because the total extinction overwhelmingly comes from the dipole contribution whereas the contribution from the quadrupole is negligibly small (Figure 2A). The two peaks seem to result from the splitting of the dipole resonance peak for a small silver particle as a result of the sharp corners of a nanocube. For the 50 nm particle, peaks 1 and 3 now red shift to 400 and 432 nm, respectively. The magnitude of peak 3 is much larger than that of peak 1, and as a result, peak 3 is dominant whereas peak 1 looks like a shoulder of peak 3. As can be seen from Figure 2B, the two peaks are also dipole resonance modes because the dipole contribution is still overwhelmingly larger than the quadrupole contribution, although the quadrupole contribution is larger than at the 20 nm particle. When the nanocube is as large as 90 nm, three LSPR peaks (peaks 1-3) appear in the optical spectra (Figure 2C). They are located at 404, 440, and 480 nm, respectively. An analysis of the dipole and quadrupole contributions shows that the newly appearing peak (peak 2) is a quadrupole peak, despite the fact that the dipole contribution is larger than the quadrupole contribution in quantity. The reason is that there is a peak in the quadrupole contribution curve but no peak exists around that wavelength for the dipole contribution curve, so the peak must correspond to a quadrupole resonance mode. Peak 3 is a pure dipole peak because the quadrupole contribution is nearly zero. This peak shows a significant red shift from the dipole resonance peak for small particles as shown in Figure 2A. Besides, it is greatly broadened. Peak 1 is a hybrid peak containing both dipole and quadrupole contribution because there exists a peak at both the dipole and quadrupole contribution curves at this wavelength. Besides, the dipole peak has a larger magnitude than the quadrupole peak, so peak 1 is dominantly a dipole resonance mode. Among the three peaks, peak 3 (a pure dipole peak) has the largest magnitude, so the overall extinction curve looks like a broad dipole peak centered at 480 nm with two shoulders appearing on the short-wavelength side of this peak. For the 160 nm nanocube, there still exist three peaks located at 404, 500, and 650 nm. The nature of the peak is similar to that of the 90 nm particle. Peak 3 is still a pure dipole resonance peak and peak 2 is a quadrupole resonance peak, whereas peak 1 is a hybrid peak and originates from the mixture of dipole and quadrupole resonance modes. Nevertheless, the quadrupole contribution grows significantly compared with that of the 90 nm nanocube. Peak 2 now becomes the dominant peak in the overall extinction spectrum curve. It is the result of the superposition of a greatly enhanced quadrupole peak over the shoulder of the longwavelength broad dipole peak. The quadrupole contribution exceeds the dipole contribution at this wavelength. At the wavelength of peak 1, the quadrupole contribution approaches the magnitude of the dipole contribution, and as a result, peak 1 is no longer a dominant dipole resonance peak.
Zhou et al.
Figure 2. Analysis of the contributions of the dipole component and quadrupole component as a function of the incident light wavelength at different sizes of nanocubes: (A) 20, (B) 50, (C) 90, and (D) 160 nm. Information pertaining to the maximum and average field enhancement factors is also plotted. The fields are calculated in the enclosed surface surrounding the nanocube with a 2 nm gap from the surface of nanocube.
Analysis of Dipole and Quadrupole Excitation IV. Local Field Analysis As we have mentioned in the Introduction, information about the local field within and around a metal particle can provide very helpful insight into the physical process of light interaction with the particle and the excitation of plasmonic wave modes. In fact, the local field distribution has been widely utilized to discern the physical nature of a particular resonance mode (for example, dipole or quadrupole modes) appearing in the extinction spectrum curve. In this section, we will discuss local field characteristics of silver nanocube particles of different sizes. We will draw a bridge between local field information and farfield extinction spectrum information that has been discussed in section III. Let us first look at the local field enhancement feature of the silver nanocube particle. As is well known, local field enhancement is a very important subject because it is closely related to SERS. A sufficiently high field enhancement factor can make single-molecule detection possible by using metal nanoparticles. We have considered the field distribution around the surface of the silver cubic particle, from which the maximum enhancement factor and average enhancement factor over the whole surface can be obtained. The field enhancement factor at a point r is defined as |E(r)|/|Einc|, namely, the ratio of local field amplitude to incident field amplitude. The dependence of the local field enhancement factor on the excitation wavelength of incident light is displayed in Figure 2A-D for silver nanocubes with sizes of 20, 50, 90, and 160 nm, respectively. In the calculations, we have selected an enclosed cubic surface with a 2 nm gap from the surface of the nanocube. For the sake of comparison, the local field enhancement information is plotted in the same Figures as the optical extinction spectrum curves. From Figure 2A-D, we can find several interesting features. First, the spectrum curve of the maximum field enhancement factor shows clear resonance behavior, and the resonance peaks agree with the resonance peaks in the extinction spectra. This indicates that when resonance plasmonic modes are excited resonance enhancement in both the extinction efficiency and near-field amplitude takes place. Second, the average field enhancement factor is much smaller (about 1 order of magnitude) than the maximum enhancement factor. This means that a significant field enhancement effect takes place only in some small regions. We will find in the latter that these hot regions are located close to the sharp corners and edges of the nanocube particle. In other regions that occupy a large fraction of surface area, the field enhancement effect is weak. Third, the field enhancement factor does not scale proportional to the extinction efficiency. The field enhancement factor at some resonance peaks of extinction is much larger than at other peaks. As a result, these stronger peaks become dominant, and the weak peaks are shadowed and hard to discern in the overall field enhancement spectrum curve. For the 20 nm nanocube, the two extinction peaks are almost the same in strength, but their field enhancement factors have a difference of more than 60%. In the case of the 90 nm nanocube, the maximum field enhancement ratio corresponds to the quadrupole peak, although the dipole peak is 4 times higher than the quadrupole peak in extinction efficiency. The 160 nm nanocube has two very strong peaks in the field enhancement spectrum curve, one located at 506 nm and the other located at 412 nm. The 506 nm field peak agrees well with the 500 nm quadrupole extinction peak. The 412 nm field peak does not exactly overlap with but is close to the 404 nm hybrid resonance extinction peak, so we believe that the 412 nm peak is induced by the hybrid dipole and quadrupole resonances. Although the 404 nm extinction peak
J. Phys. Chem. C, Vol. 112, No. 51, 2008 20237 is more than twice as small in magnitude as the 500 extinction peak, the corresponding 412 nm field peak has almost the same magnitude as the 506 nm peak. From the above analysis, we can find that when the nanocube is small enough that it has no quadrupole resonance mode, the dipole resonance mode causes the maximum field enhancement. When the quadrupole resonance mode appears, the induced field enhancement effect will become dominant, and as a comparison, the field enhancement effect due to dipole resonance is suppressed and is shadowed. As can be seen in Figure 2C,D, the long-wavelength dipole mode (480 nm peak for the 90 nm nanocube and 650 nm peak for the 160 nm nanocube) does not exhibit an apparent resonance feature of field enhancement. The above discussion shows that the field enhancement has a very complicated behavior. It depends on the size of the nanoparticle, the incident light wavelength, and the nature of excited plasmonic modes. Now we turn to look at the near-field pattern for resonance peaks of extinction spectra and see the correlation between nearfield and far-field information. Figures 3 and 4 show the calculated electric field amplitude and charge density distribution of the 20 nm silver nanocube in the (110) face at the 393 and 417 nm resonance peaks, respectively. We choose the (110) section because the incident electric field is parallel to the (110) direction, and then all four corners where polarized electric charges overwhelmingly accumulate will be explicitly shown in the Figure. Figure 3A shows the coordinate system that we use to carry out the simulation. We define the diagonal direction [the (110) direction] as the w axis. Figures 3B and 4A show the local electric field amplitude distribution at wavelengths of 393 and 417 nm, whereas Figures 3C and 4B reveal the corresponding polarized electric charge distribution, respectively. From the Figures, we can clearly see that both modes are dipole modes because charges of different sign are separated on the left and right edges. These charges are strongly accumulated at the four sharp corners of the nanocube. In these situations, the conduction electrons are excited in phase with the incident electromagnetic field. This agrees well with the result we get from the extinction spectra in which both peaks are dipole resonance modes. The main difference between the two modes is that the charge is more concentrated in the corners for the 417 nm resonance mode, but for the 393 nm resonance mode there are some charges that are dispersed away from the corners to the edge. As a result of this difference in charge accumulation, the maximum local electric field enhancement factor at 417 nm is larger than at 393 nm, as has been found in Figure 2A. We should notice that in this Figure there are some data points that are very close to (a distance of 0.333 nm from) the apexes of the nanocube, so the maximum local field enhancement factor here is much larger than the data shown in Figure 2A, which are calculated in an enclosed cubic surface that is 2 nm away from the nanocube surface. When the size of nanocube is large enough to support quadrupole resonance modes, the situation is different. For the 160 nm nanocube, there appear three extinction peaks, and one of them is due to the quadrupole resonance. We calculate the near-field amplitude and polarized electric charge distributions of the 160 nm nanocube at the three resonance peaks, and the results are displayed in Figures 5-7. The coordinate system that we adopt in the simulation is the same as Figure 3A for the 20 nm nanocube. Figures 5A and 6A show the local electric field amplitude distribution at the 650 and 500 nm resonance peaks, respectively, whereas Figures 5B and 6B show the corresponding polarized electric charge distribution. At the 650 nm incidence, the sign of the electric charge on each (001) edge
20238 J. Phys. Chem. C, Vol. 112, No. 51, 2008
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Figure 4. (A) Calculated near-field amplitude and (B) polarized electric charge distribution on the (110) face of a 20 nm silver nanocube at the 417 nm resonance peak.
Figure 3. Calculated near-field amplitude and polarized electric charge distribution in the (110) face of a 20 nm silver nanocube at the 393 resonance peak. (A) Coordinate system for the simulation. The w axis is in the [110] direction, and the origin is at the center of the nanocube. (B) Electric field amplitude distribution. (C) Polarized electric charge distribution. In the near-field images, the rectangles are outlines of nanocubes. The incidence is along the [001] direction, and the electric field is in the [110] direction with an amplitude of E0 ) 1 V/m. The field amplitude is in units of E0.
is uniform whereas the sign of the electric charge on the two (001) edges is opposite to each other, so we can confirm that the 650 nm peak corresponds to a dipole resonance mode that is induced by the (110) polarized incident electric field. However, both the local field and the charge density are stronger on the bottom side of the nanocube (the back surface relative to the incident light) than on the upper surface (the front surface) of the nanocube particle. This might be induced by the retardation effect of the incident light for this large nanoparticle. At the 500 nm incidence, every neighboring corner has an electric charge of opposite sign, so this peak mainly corresponds to a quadrupole resonance mode. The phase retardation effect becomes significant for the 160 nm nanocube, and the quadrupole resonance mode can be excited. Taking a closer look at the charge distribution, we find that the charge density is larger on the bottom side of the nanocube than on the upper side by about 30%; therefore, the excited plasmonic mode at this wavelength is not a pure quadrupole mode. It also involves some fraction of the dipole component. This is consistent with the
Figure 5. (A) Calculated near-field amplitude and (B) polarized charge distribution in the (110) face of a 160 nm silver nanocube at the 650 nm resonance peak.
extinction spectrum analysis. As has been found in Figure 2D, the 500 nm peak is composed of a quadrupole resonance peak superimposed on the shoulder of the broad 650 nm dipole resonance peak, and as a result, both the quadrupole and dipole components are mixed in the total field distribution. Figure 7A,B shows the near-field amplitude and polarized electric charge distribution of a 160 nm silver nanocube at 412 nm incidence. The charge pattern exhibits a complicated accumulation behavior of polarized electric charge. The charges
Analysis of Dipole and Quadrupole Excitation
Figure 6. (A) Calculated near-field amplitude and (B) polarized charge distribution in the (110) face of a 160 nm silver nanocube at the 500 nm resonance peak.
J. Phys. Chem. C, Vol. 112, No. 51, 2008 20239 distribution analysis is again consistent with the optical spectrum analysis as shown in Figure 2D, where the 404 nm resonance peak is found to involve the same order of magnitude contribution from dipole and quadrupole modes. Note that in Figures 3-7 we mainly pay attention to the mode field profile in order to see the physical origin of the plasmonic excitation, so we have adopted the same DDA discrete cells in calculating the optical near field as in calculating the optical spectra, where 30 × 30 × 30 ) 27 000 discrete cells are used for all of the silver cubic particles. In other words, in Figures 3 and 4 the calculation cell size is 0.667 nm for the 20 nm particle whereas in Figures 5-7 the calculation cell size is 5.33 nm for the 160 nm particle. As a result of 8 times denser discrete cells and the existence of points that are 8 times closer to the apex and surface of the particle, the apparent value of the maximum field amplitude, which is located close to the apex of the cubic particle, is stronger in the case of the 20 nm particle than in the case of the 160 nm particle. However, when we adopt 8 times denser calculation points for the 160 nm particle we have found the same order of magnitude of the maximum near-field amplitude as for the 20 nm particle. However, the decay of the field away from the apex of the cubic particle is much faster in the 20 nm particle than in the 160 nm particle, so one can see that the maximum field enhancement factor in Figure 2D (the 160 nm case) is stronger than in Figure 2A (the 20 nm case), where the field in these two images is measured at an enclosed plane that is 2 nm away from the surface of the cubic particle. V. Conclusions
Figure 7. (A) Calculated near-field amplitude and (B) polarized electric charge distribution in the (110) face of a 160 nm silver nanocube at the 412 nm resonance peak.
at the two (001) edges and corners (left and right sides of Figure 7B) have opposite signs, whereas the charges at the two (110) edges and corners (upper and lower sides of Figure 7B) are not strictly opposite in sign. For instance, although the charges in the upper two corners are of a single sign, the charges at the lower two corners are composed of positive and negative charges. For this reason, the charge pattern of Figure 7B can be assumed to be the superposition of dipole and quadrupole modes, and no mode is dominant here. The local field and charge
We have developed an efficient method based on the DDA simulation to discern quantitatively the contribution of dipole and quadrupole plasmonic modes to the optical extinction spectra when light is incident on metal nanoparticles. Information pertaining to the calculated discrete dipole moment distribution within metal nanoparticles allows one to obtain the overall effective dipole and quadrupole moments, from which their optical scattering, absorption, and extinction efficiency can be solved quantitatively. We have used this method to investigate surface plasmon resonance occurring in silver cubic nanoparticles with an edge length ranging from 15 to 200 nm. An increasing number of SPR peaks appear in the optical spectra, and their positions change when the nanocube size increases. We have been able to separate quantitatively the contributions of individual dipole components and quadrupole components to the optical extinction spectra and assign unambiguously each SPR peak to the excitation of either a dipole, or quadrupole, or hybrid resonance mode. For small nanoparticles, only dipole resonance modes can be excited, whereas for larger nanoparticles, quadrupole resonance modes can also be excited because of the onset of the electromagnetic retardation effect. We have also extensively analyzed the distribution patterns of electric near fields and polarized electric charges within and around the silver nanoparticles. These patterns clearly show the dipole and quadrupole excitation feature at the SPR peaks. The near-field analyses are consistent with the far-field extinction spectra analyses. This suggests that the combination of far-field spectra and near-field pattern analysis can greatly help to uncover the intrinsic physics behind light interaction with metal nanoparticles and the excitation dynamics of local surface plasmonic waves. We have found that strong local field enhancement occurs when SPR takes place and that the maximum field enhancement
20240 J. Phys. Chem. C, Vol. 112, No. 51, 2008 is located around the sharp corners of the cubic nanoparticle. For small particles where only dipole modes can be excited, the field enhancement is mainly induced by the excitation of the dipole plasmonic wave. However, when quadrupole modes are excited, the corresponding field enhancement will become dominant over the enhancement induced by the dipole modes. The field enhancement factor does not scale proportionally to the corresponding optical extinction efficiency. At some wavelengths where the optical extinction effect is weak, the field enhancement factor can be very large and comparable to the SPR peak with a large extinction efficiency. These features can help one to design metal nanoparticles with sufficiently large local field enhancement effects at a particular wavelength window, and this will find useful application in SERS, singlemolecule detection, nonlinear optics, and other areas. Acknowledgment. We acknowledge financial support from the National Natural Science Foundation of China under grant nos. 10525419, 60736041, and 10874238 and the National Key Basic Research Special Foundation of China under grant no. 2004CB719804. Note Added after ASAP Publication. This article was published ASAP on November 26, 2008. Corrections have been made to eq 6. The correct version was published on December 5, 2008. References and Notes (1) de Abajo, F. J. G. ReV. Mod. Phys. 2007, 79, 1267. (2) Fromm, D. P.; Sundaramurthy, A.; Schuck, P. J.; Kino, G.; Moerner, W. E. Nano Lett. 2004, 4, 957. (3) Kuhn, S.; Hakanson, U.; Rogobete, L.; Sandoghdar, V. Phys. ReV. Lett. 2006, 97, 017402. (4) Merlein, J.; Kahl, M.; Zuschlag, A.; Sell, A.; Halm, A.; Boneberg, J.; Leiderer, P.; Leitenstorfer, A.; Bratschitsch, R. Nat. Photon. 2008, 2, 230. (5) Yang, J.; Zhang, J.; Wu, X.; Gong, Q. Opt. Express 2007, 15, 16852. (6) Muhlschlegel, P.; Eisler, H. J.; Martin, O. J. F.; Hecht, B.; Pohl, D. W. Science 2005, 308, 1607. (7) McLellan, J. M.; Li, Z. Y.; Siekkinen, A. R.; Xia, Y. N. Nano Lett. 2007, 7, 1013. (8) Nie, S. M.; Emery, S. R. Science 1997, 275, 1102. (9) Doering, W. E.; Nie, S. M. J. Phys. Chem. B 2002, 106, 311.
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