Optical Properties of Metallodielectric Nanostructures Calculated

17740

J. Phys. Chem. B 2004, 108, 17740-17747

Optical Properties of Metallodielectric Nanostructures Calculated Using the Finite Difference Time Domain Method Chris Oubre† and Peter Nordlander* Department of Physics and Department of Electrical and Computer Engineering, Rice Quantum Institute, M.S. 61, Rice UniVersity, Houston, Texas 77251-1892 ReceiVed: June 20, 2004; In Final Form: September 2, 2004

The optical properties of metallodielectric nanostructures, variations of a core-shell geometry, are investigated using the finite difference time domain method. This method provides a convenient, systematic, and general approach for calculating the optical response of a nanostructure of arbitrary symmetry and geometry to an incident light wave. Properties such as the optical absorption and scattering cross sections as well as the local electromagnetic fields and induced charge densities at the surfaces of the nanostructures can be obtained by this method. Issues of convergence with grid size and other simulation parameters are discussed in detail. The method is applied to uniform single nanoshells, nanoshells with surface defects, and nanoshells with shape distortions from a spherical geometry. The results show that, while defects can significantly affect local surface field enhancements, far field properties such as optical absorption and scattering spectra can be remarkably insensitive to defects and distortions.

I. Introduction The optical properties of metallic nanostructures are important from both a scientific and a technological perspective. In recent years, the physical and chemical properties of metallic nanoshells have received particular attention. Nanoshells are metallodielectric nanoparticles typically consisting of a small dielectric core, usually silica, surrounded by a thin layer of a coinage metal, such as silver or gold.1,2 Nanoshells possess several attractive features which make them interesting as nanoscale optical components. They possess a plasmon-derived tunable optical resonance controlled by the dimensions of the core and shell layers, spanning much of the visible and infrared regions of the optical spectrum.3-6 Additionally, nanoshells and other nanoscale metallic structures have been shown to greatly enhance local electromagnetic fields in certain regions near their surfaces at specific wavelengths of light, controlled by nanostructure geometry.7-10 This “subwavelength lens” functionality provides a tool for manipulating light below the diffraction limit. A broad range of applications, including Raman spectroscopy,7 biological sensing,11 environmental sensors,12 whole blood immunoassays,13 drug delivery materials,14 in vivo optical contrast agents,15 photothermal cancer therapy,15-18 and optically actuated materials,19 have been demonstrated with this type of nanoparticle. While the near and far field optical properties of nanoshells can be calculated analytically, any deviations from this spherical symmetry typically make this approach impractical for analysis. To examine the optical properties of metallodielectric nanoparticles of arbitrary shape and symmetry, a numerical approach is ultimately required. In this paper, we present a finite difference time domain study of the optical properties of metallodielectric nanoparticles. This approach will be described in detail in * Corresponding author. E-mail: [email protected]. Visit http:// cmt.rice.edu. † E-mail: [email protected].

sections II and III. In section IV, the method will be applied to several different nanoshell-related nanoparticles. For a single nanoshell, where an exact solution can be obtained analytically, we show how computational parameters, such as mesh size, influence the accuracy of the results. We then study metallodielectric nanoparticles whose straightforward deviations from spherical symmetry demand a numerical analysis to reveal their properties: distorted nanoshells and nanoshells with surface defects. II. The Finite Difference Time Domain Method A. General Overview. The finite difference time domain (FDTD) method is an explicit time marching algorithm used to solve Maxwell’s curl equations on a discretized spatial grid. It can be used for studying both the near and far field electromagnetic responses for heterogeneous materials of arbitrary geometry. The approach has successfully been applied to many nanosystems in the past.8,20-24 The method has several features which make it a highly desirable algorithm.25 It is fully explicit; that is, it uses no linear algebra. This reduces the computational overhead needed to solve a particular problem. Typical methods which rely on linear algebra are generally limited to fewer that 106 electromagnetic unknowns, while FDTD simulations have used as many as 109 unknowns. The sources of errors in the method are well-known and bounded. Modeling a new system is reduced to grid generation, instead of deriving geometry-specific equations. Thus, previously uninvestigated and potentially complex geometries do not require new formalisms. The time marching aspect of the method allows one to make direct observations of both near and far field values of the electromagnetic fields and at any time during the simulation. With these “snap shots”, the time evolution of the electromagnetic fields can be measured and visualized. This opportunity provides new insight into the dynamics of the system under study. Additionally, the method is easily parallelizable due to the local nature of finite differ-

10.1021/jp0473164 CCC: $27.50 © 2004 American Chemical Society Published on Web 10/23/2004

Optical Properties of Metallodielectric Nanostructures

J. Phys. Chem. B, Vol. 108, No. 46, 2004 17741

Figure 1. A typical nanoparticle experiment. A nanostructure is illuminated with a plane wave. Surface plasmons are induced on the nanoparticles, creating local electromagnetic fields. The cross sections are obtained be recording the electromagnetic radiation in the far field.

ences. This localized aspect enables the use of simple parallel methods, such as domain decomposition, on either shared or distributed memory systems. Since the FDTD method uses the curl form of Maxwell’s equations, the method is fully retarded. In Figure 1, we schematically show the type of phenomenon that will be modeled. An incoming light pulse or wave illuminates a target containing one or several nanoparticles or structures. The electromagnetic fields excite plasmons or polaritons in the individual nanoparticles. The plasmons on the individual nanoparticles interact resulting in a complicated and time dependent electromagnetic field. Since the dimensions of the composite target may be of the same order of magnitude as the wavelength of the light, retardation effects must be included in the formalism.26 B. Numerical Methodology. In the following two sections, we will closely follow the methodology of Sullivan27 and Taflove.25 The electric flux dependent form of Maxwell’s equations is given by

∂D B )∇×H B ∂t

(1)

D(ω) ) (ω) E(ω)

(2)

∂H B 1 )- ∇×B E ∂t µ0

(3)

These equations are solved on a discrete grid replacing all derivatives with finite differences. Following the Yee algorithm,28 both D B and H B are offset in both space and time. This allows one to use central differences, making the algorithm second-order accurate. For example, the discrete one-dimensional form of eq 1 is given by

Dn+1/2(k) - Dn-1/2(k) Hn(k + 1/2) - Hn(k - 1/2) ) (4) ∆t ∆x where k is the spatial coordinate and the n superscript denotes time. It is then straightforward to solve for Dn+1/2(k) in terms of Dn-1/2(k), Hn(k + 1/2), and Hn(k - 1/2) which are already stored in memory. Care must be exercised when choosing a value for ∆x. Two main points must be taken into account. First, the grid must be fine enough to describe the smallest spatial features, whether it is the particle geometry or particle separation. Generally, 1020 grid points in each dimension is adequate. Also, the grid must be fine enough to support the wavelengths of interest.

Again, it is recommended to have 20 points per wavelength for the smallest wavelength to be investigated. The time step (∆t) must be chosen with even greater consideration. After the mesh size (∆x) is selected, ∆t must be sufficiently small not to violate causality. In one time step, information can travel from one grid cell to the next. The time step must be small enough that this information is not traveling faster than the speed of light in that medium. This is known as the “Courant condition”25,29 and is given by

∆x ∆t ) SC c0xd

(5)

where d is the dimensionality and SC lies between 0 and 1 and is called the Courant number. The choice for the Courant number is empirical. Due to discretization errors, a maximum of 1.0 cannot be used. Typical values used in our simulations were in the range between 0.5 and 0.9. As stated above, the determination of the correct time step is more important than the proper mesh size. If the mesh size is slightly too large, an error will be introduced into the simulation; however, if the time step is slightly too large, then there will be an exponential growth in the error and the simulation will be unstable and diverge. The dielectric function (r b, ω) is used to simulate the materials in the grid. A value for is specified at every grid point. To simulate gold and silver nanoshells, we implemented a Drude dielectric function.

(ω) ) ∞ -

ω2p ω2 + iδω

(6)

This Drude dielectric function was fitted to a particular frequency range of the Johnson and Christy bulk dielectric data30 for the metal under study. For gold, the numbers we used were ∞ ) 9.5, ωp ) 8.9488 eV, and δ ) 0.069 09 eV. For silver, we used ∞ ) 5.0, ωp ) 9.5 eV, and δ ) 0.0987 eV. A comparison of our fitted Drude model to the bulk values for silver is shown in Figure 2. Corrections to the Drude dielectric function arising from the finite electron mean free path in a nanoparticle can be taken into account by increasing the factor δ in eq 6.31 Such an extra damping does not influence the energies of the plasmons but shortens their lifetimes and results in a broadening of the plasmon resonances in the far field spectra. Such a damping also slightly reduces the electric field enhancements at the surfaces of the nanoparticles. Recent experimental investigations of scattering spectra of individual nanoshells indicate that, for these systems, no extra broadening

17742 J. Phys. Chem. B, Vol. 108, No. 46, 2004

Oubre and Nordlander

Figure 2. Comparison of the Johnson and Christy bulk dielectric data for silver30 (the solid line is the real part, and the dashed line is the imaginary part) and our Drude fit (the triangles are the real part, and the diamonds are the imaginary part).

needs to be included in eq 6. For this reason, we abstain from making such a correction. The use of the electric flux density form of Maxwell’s equations provides two major benefits: a simpler time domain implementation of dielectric functions and programming modularity. The D and H field update equations are independent of the choice of dielectric function. Thus, if an object to be modeled requires a different dielectric function, only the E update equations need to be modified. Dielectric functions such as eq 6 can easily be transformed from the frequency domain to the sampled time domain via Z transforms.32,33 The sampled time domain for of eq 6 is given by

E(n) )

1 [D(n) + S(n)] ∞

(7)

S(n) ≡ [1 + exp(δ∆t)]S(n - 1) ω2p∆t [1 - exp(δ∆t)]E(n) (8) exp(δ∆t)S(n - 2) + δ From eq 8, we see that S depends on S at the two previous time steps. This requires two new sets of arrays to store these values. S also depends on E at the current time step, while E depends on S at the previous time step; thus, first E is updated and then S is updated with the new value of E. III. Specific Implementation Details Our FDTD implementation is separated into three spatially distinct regions: the absorbing boundary, the scattered field region, and the total field region (see Figure 3). To approximate an infinite grid, an absorbing boundary is used to prevent reflections of scattered waves back into the simulation space. The particular type of absorbing boundary that we used is the uniaxial perfectly matched layer (UPML).25,34 The UPML is a lossy medium with ultralow reflectance over a broad spectrum and large angles of incidence. An electromagnetic pulse is inserted and subtracted along the inner square in Figure 3. For example, the pulse shown in Figure 3 is added to the grid along the left side of the total field/scattered field boundary (TFSF boundary) and subtracted out on the right. This approach allows for the monitoring of both near and far fields. Subtracting out the pulse at the TFSF boundary also reduces the load on the

Figure 3. A typical calculation using the FDTD method. The simulation space is separated into three regions, the absorbing boundary (shown in gray), the scattered field region (in between the absorbing boundary and the inner red rectangle), and the total field region (inside of the inner red rectangle). A target particle is illuminated with an electromagnetic pulse, and the electric and magnetic fields as a function of time are recorded for postprocessing spectral analysis.

UPML, thus reducing unwanted reflections. The pulse used in our calculations is a sine modulated Gaussian.

Einc ) e-(n∆t-Nττ) /τ2 sin(ωc(n∆t - Nττ))

(9)

ωmax - ωmin 4 , τ) 2 ωmax - ωmin

(10)

2

with

ωc )

and Nτ ) 4. The quantities ωmax and ωmin are the maximum and minimum frequencies to be investigated. Since a short pulse is a superposition of plane waves with a range of frequencies, the target nanostructure can be probed at several frequencies within a single calculation. To perform our spectral analysis, we keep running discrete Fourier transforms at predefined locations of interest. The decision of which wavelengths to monitor has to be made before the simulation begins. Several quantities can be extracted from a simulation. The time evolution of the electric and magnetic fields and charge distribution can be analyzed. During the simulation, these quantities can be Fourier transformed and spectrally resolved. Scattering and absorbing cross sections are computed as a function of wavelength. Our code also calculates local field enhancements which are of paramount importance in surface enhanced raman spectroscopy (SERS).35 As stated above, the FDTD method is easily parallelized using domain decomposition. We have implemented this procedure using message passing interface (MPI).36 The computational grid is broken into several smaller grids, with each subgrid being updated by a different central processing unit (CPU). To complete the update, subgrids only need to exchange boundary values with adjacent subgrids. With a Beowulf computer, domain decomposition allows one to greatly increase the size of the simulated regions. We have run simulations as large as 80 GB across 100 processors. Our code has been developed in C++, making use of the object-oriented tools. This choice of language allows for the use of dynamic memory allocation and code modularity.


Figure 4. Extinction cross section as a function of wavelength for a (39, 48) nm silver nanoshell with a silica core. The solid line is the result from the FDTD calculation, and the dashed line is from Mie theory. Both calculations used the fitted Drude dielectric function (eq 6). The mesh size is 0.375 nm.

Polymorphism and inheritance increase the functionality and flexibility while simplifying maintenance. In any simulation, the vast majority of run time will be spent in one “for loop”. We have optimized and serialized our main loop to reduce the overhead from these complex constructs. On the basis of a “fixed job size” analysis using Amdahl’s law,37 94% of the code executes in parallel. The calculations in this paper were performed on either a 24 node 2.8 GHz dual Xeon cluster running Red Hat Linux 7.2 or a 128 node 1.0 GHz dual Itainium cluster running Red Hat Linux Advanced Workstation release 2.1AW. In both cases, each node had 4 GB of ram.

J. Phys. Chem. B, Vol. 108, No. 46, 2004 17743

Figure 5. Normalized error as a function of mesh size for FDTD calculations of the dipole peak location (solid line) and height (dotted line) in the extinction cross section of a (39, 48) nm silver nanoshell with a silica core. Results were compared to the classical Mie theory prediction. The 0.375 nm grid calculation was calculated using a larger Courant number of SC ) 0.8 instead of 0.5 and a smaller UPML separation.

IV. Results In our FDTD calculations, typical values of the mesh size ranged from 0.75 to 3.0 nm. The time step was of the order of 1 × 10-19 to 1 × 10-18 s. Through convergence testing, we found that a Courant number of SC ) 0.8, a UPML with a thickness of 10 grid cells, and a UPML of grade 4.0 provided the best overall trade-off between computational time and accuracy A. Single Nanoshell in the Far Field Region. In Figure 4, we show a plot of the extinction cross section for a silver nanoshell with an inner radius of 39 nm and an outer radius of 48 nm with a silica core. In the following, we will use the notation (39, 48) nm to refer to the geometry of such a shell. The solid line is the FDTD result, and the dashed line is an exact Mie calculation using the fitted Drude dielectric function. The features at 443 and 550 nm are the quadrupole and dipole plasmons, respectively. The agreement between the FDTD and Mie calculations is quite good with a 3% difference in the plasmon peak height and a 0.2% difference in the plasmon peak location (the wavelength of maximum absorption). To estimate the numerical errors introduced in our implementation of the FDTD algorithms, we investigate how the extinction cross section depends on mesh size. In Figure 5, we plot the calculated plasmon wavelength and peak height as a function of mesh size. The shape of the curve in Figure 5 closely resembles that found elsewhere.38 Finer grid resolutions come at a computational expense. The calculation at a resolution of 3.0 nm used 0.2 GB of ram and took 3 CPU hours to complete on a 723 grid. In contrast, the 0.5 nm calculation used 28.3 GB of ram and took 7920 CPU hours to complete on a 4323 grid. To perform the 0.375 nm

Figure 6. Two-dimensional slice of the near field enhancements, at a wavelength of 542 nm, for a (39, 48) nm silver nanoshell with a silica core. Upper left: the FDTD results. Upper right: the results from Mie theory. The plot on the lower left is the difference between the FDTD and Mie results. The excitation filed is incident along the y-axis and polarized along the x-axis.

grid calculation, we relaxed our constraints on other parameters by increasing the Courant number and reducing the object-toUPML distance. This greatly reduced the run time but increased our error in the peak height, as seen in Figure 5. The reduced 0.375 nm calculation used 11.4 GB of ram and took 1896 CPU hours to complete on a 2963 grid. The linearity of the normalized error shown in Figure 5 is due to staircasing effects,39 that is, the rough surface resulting when representing a sphere on a Cartesian grid. The inset of Figure 10 illustrates the staircasing effect of a 48 nm nanoshell on a 0.75 nm grid. B. Single Nanoshell in the Near Field Region. In this section, we will investigate the electric field enhancement on the surface of a nanoshell. By enhancement, we mean the field at a particular point divided by what the field would have been if the nanoshell was not there, that is, Etotal/Eincident. In Figure 6, we see near field plots for a (39, 48) nm silver nanoshell with a silica core calculated using both FDTD and Mie theory. The lower left plot shows the difference between the two calculations. Overall, there is quite good agreement with typical


Figure 7. Calculated near field enhancements, at a wavelength of 542 nm, as a function of distance from the center of the nanoshell. The nanoshell shown here is a (39, 48) nm silver nanoshell with a silica core. The polarization of the incident light is parallel to the x-axis of this plot. The solid line is the result from the FDTD calculation, and the dashed line is from Mie theory.

Figure 8. Plot of the average electric field enhancement at a wavelength of 542 nm in a 3 nm3 region 3 nm outside the nanoshell surface along the axis of polarization as a function of mesh size. The solid line is the result from the FDTD calculation, and the dashed line is from Mie theory.

differences hovering around 1-5% except at the shell surfaces where larger errors arise from the discrete nature of the FDTD method. In general, the FDTD calculation underestimates the electric field enhancement; however, as seen in Figure 6, at material boundaries, we find small regions of very large

Oubre and Nordlander enhancements. These highly localized results are nonphysical and are an artifact of staircasing. Figure 7 provides a closer look at the peak electric enhancements as a function of distance from the nanoshell. We see that, at a wavelength of 542 nm, the electric field at the nanoshell surface is enhanced by approximately a factor of 12 and is reduced by a factor of 2 over a distance of 10 nm. Again, there is excellent agreement between the FDTD approach and classical Mie scattering. As shown in Figure 8, the calculation approaches the analytical result as the mesh size approaches zero. Additionally, we see that the FDTD calculation underestimates the near field enhancements. As a result, quick calculations using large mesh sizes can be used to set lower bounds on the near field enhancements. C. Single Nanoshell Charge Distributions. The local charge densities can be calculated by taking the divergence of the near field. Figure 9 shows the amplitude of the local charge density oscillation for a (39, 48) nm silver nanoshell at the frequencies of the two peaks shown in Figure 4. From the symmetries of the charge distributions in Figure 9, these peaks can clearly be labeled as the dipole and quadrupole resonances of this nanoshell. Furthermore, the dipole moments of the inner and outer surfaces are aligned. This is in agreement with the prediction of plasmon hybridization.40 The ability to examine the induced charge distributions as a function of wavelength provides a powerful tool for the microscopic interpretation of features in the far field spectra. A similar technique has been successfully used to elucidate the nature of plasmons and electromagnetic field enhancements in other nanosystems.41 The fine structure in Figure 9 is a result of staircasing and the finite nature of the discrete Fourier transform used to calculate the electric fields. D. Nanoshells with Defects. We now investigate nanoshells with surface defects. The presence of defects breaks the spherical symmetry of the system, preventing the use of classical Mie scattering. In Figure 10, we show the extinction cross section of a (39, 48) nm nanoshell with a dimple and a bump. Both features have the same radius, 6 nm, to conserve the number of electrons. The inset in Figure 10 is a visualization of this geometry. The features can be seen in the lower right side of the nanoshell. The figure shows that the spectra from the defect free and defective shells are very similar. The dashed line in Figure 10 is the absolute value of the difference between the two extinction cross sections. We see that the presence of defects has a minimal impact on the far field results. This agrees well with previous results on the effect of small pinholes on the

Figure 9. Two-dimensional slice of the amplitude of the charge distribution of a (39, 48) nm silver nanoshell at 550 nm (left) and 443 nm (right). The excitation field is incident along the y-axis and polarized along the x-axis.


Figure 10. Extinction cross section as a function of wavelength for a perfect (39, 48) nm silver nanoshell (solid line), for a (39, 48) nm silver nanoshell with a bump and a dimple defect (dashed line), and the absolute value of the difference between the two curves (dash-dotted line). The dashed and solid lines coalesce on the scale of this figure. The bump extends 6 nm out of the nanoshell, and the dimple extends 6 nm into the shell. The inset in the upper right of the plot is a visualization of the system simulated. The additional fine structure in this inset is due to staircasing effects. Note that the y-axis is logarithmic.

optical properties of nanoshells.42 From size arguments, one expects the change in extinction cross section to be proportional to the number of shifted electrons over the total number of electrons (Nshifted/Ntotal). This number is roughly 0.02%. The difference calculated by the FDTD method is around 2-4%, indicating that the surface defects are dipole active. In Figure 11, we show the near field of this system for two different polarizations. In the plot on the left, the features are mostly aligned with the incident pulse. At the edge of the bump, we see near field enhancements roughly twice as large as those seen in Figure 6. In the plot on the right, the features are mostly perpendicular to the excitation field. In this case, it is in the dimple that we find large electric field enhancements. This enhancement is again approximately twice as large as the same region for the defect free nanoshell shown in Figure 6. These enhancements are in good agreement with the results from a recent investigation of the effects of pinholes in nanoshells.42 We now examine the effects of a larger number of surface defects. Figure 12 shows the extinction cross section for various numbers of three different types of surface defects. In part A,

J. Phys. Chem. B, Vol. 108, No. 46, 2004 17745 we show the effect of a nanoshell covered with a random mixture of 70% bumps and 30% dimples with radii ranging from 1.5 to 3.75 nm. It can be seen that the calculated spectra are very similar. A small red shift and broadening is visible for the nanoshell with the largest number of defects. Part B shows the effect of larger surface defects. Specifically, these defects range from 3.75 to 9 nm in radius and are comparable in size to the thickness of the shell. Compared to the curves in part A, those in part B show a slightly larger red shift and more pronounced broadening. For the nanoshell with the largest number of bumps, the extinction cross section is broadened by roughly 20%. The figure also shows that, as we add bumps, the magnitude of the extinction cross section increases. This is expected because we are adding material to the system. It has been shown that increasing the thickness of a shell leads to a blue shift40 of the dipole plasmon resonance. For this reason, one would expect that, as bumps are added to the system, the plasmon resonance should blue shift. However, for the two lowest defect concentrations, the plasmon peak red shifts with increasing defect concentration. Only for the largest defect concentration does the plasmon energy begin to blue shift. These shifts are analogous to those seen in incomplete shells during nanoshell synthesis1 and can be understood using dimer plasmon hybridization.43 The large bumps can be thought of as small colloids on the surface of the nanoshell. These colloids will have a plasmon energy at higher energy, that is, at a smaller wavelength than that of the nanoshell. The interaction between these two plasmons will lead to an overall red shift in the nanoshell plasmon energy. The curve representing 360 defects in part B is blue shifted compared to the plasmon peak of the nanoshell with 180 defects. This is because the surface is almost completely covered by defects, resulting in a thicker nanoshell with a blue shifted plasmon resonance. In part C, we plot the extinction cross section for nanoshells with different numbers of large dimple defects. The figure shows that the extinction spectra are very sensitive to the number of defects. A strong red shift of the plasmon resonance is observed as the number of defects is increased. For the largest defect concentration, a red shift of 152 nm is observed with negligible broadening. The reason for the monotonic red shift of the plasmon resonance with defect concentration is the thinning of the nanoshell, which leads to a red shift. As in the case with the bump defects, a red shift caused by hybridization between the bare defect plasmon and nanoshell plasmons is also present.

Figure 11. Two-dimensional plot of electric field enhancements at 544 nm for the (39, 48) nm silver nanoshell in Figure 10. The surface plotted bisects the surface features. The excitation field is incident in the positive y direction and polarized along the x-axis.


Figure 12. The graphic in each part is the most roughened example of each type of defect distribution. (A) A plot of 0 (solid line), 160 (dotted line), and 320 (dashed line) randomly distributed small surface features. These features are 70% bumps and 30% dimples with radii ranging from 1.5 to 3.75 nm. (B) A plot of 0 (solid line), 10 (dotted line), 180 (dashed line), and 360 (dash-dotted line) large randomly distributed bumps. The bumps range from 3.75 to 9 nm in radius. (C) A plot of 0 (solid line), 10 (dotted line), 45 (dashed line), 115 (dashdotted line), 180 (dash-dot-dot-dotted line), and 360 (long dashed line) large randomly distributed dimples. The dimples range from 3.75 to 9 nm in radius.

E. Ellipsoidal Nanoshells. Another type of imperfection that may influence the plasmon resonance of a nanoshell is the distortion of its shape. In this section, we explore the optical properties of slightly ellipsoidal nanoshells. Figure 13 shows the extinction cross sections for which (39, 48) nm silver nanoshells have been compressed in one direction and elongated a similar amount in the other direction. In part A, the incident electric field is polarized along the short axis. The extinction cross section is increasingly blue shifted with increasing distortion. In part B, the incident electric field is polarized along the long axis. In this case, the extinction cross section is increasingly red shifted with increasing distortion. Both of these results can be explained using plasmon hybridization. The dipolar plasmon resonance is highly localized, as seen in Figure 6; thus, each ellipsoid can be replaced by an “effective” shell with a radius to the length of the axis parallel to the incident electric field. With this approximation, the shells in part A have a decreasing aspect ratio, resulting in a blue shift. The shells in part B have an increasing aspect ratio, which results in a red shift. Because the direction of the shift of the plasmon peak depends on the polarization, far field measurements of randomly oriented ensembles of nanoshells with elliptical distortions

Oubre and Nordlander

Figure 13. Plot of the extinction cross section of ellipsoidal (39, 48) nm silver nanoshells. The long and short axes of each shell have been increased and decreased, respectively, by a given percentage from the spherical nanoshell. The level of distortion is given in the legend. In part A, the incident electric field is polarized along the short axis. In part B, the incident electric field is polarized along the long axis.

would have a larger full width at half-maximum (fwhm) than undistorted nanoshells. For example, the 12.6% distortion shown in Figure 13 would broaden the peak by ∼20%. No significant local electric field enhancements, as compared to the undistorted nanoshell, were found. F. Nanoshells with an Offset Core. Another possible nanoshell imperfection may be a nonuniform shell thickness. In this section, we model such a system by off-setting the core of the nanoshells. A schematic of such nanoshells can be found in Figure 14. The nanoshell plasmons can be viewed as resulting from hybridization of the surface plasmons on the outer surface of the shell with the cavity plasmons on the inner surface. For concentric surfaces, the interaction is diagonal.44 For a subwavelength-sized nanoparticle, the dominant excitation caused by light is the dipolar plasmon. The probabilities for excitation of higher angular momentum plasmons are very small because of their lack of a dipole moment. In the offset nanoshell, unlike a perfectly spherical nanoshell system, plasmons of different angular momenta on the inner and outer surfaces of the shell will couple. The larger angular momentum plasmons will therefore contain a finite admixture of the dipole plasmon and couple efficiently to light. The extinction spectra will therefore exhibit several different plasmon resonances. As seen in the excitation spectra shown in Figure 14, the amount of coupling depends primarily on the size of the offset, with the incident polarization mostly affecting the height of the peaks. The coupling between plasmons of different angular momenta results in an overall red shift of the spectrum. The red shift is particularly strong for the plasmon originating from the l ) 1 nanoshell plasmon, since it has the lowest energy. The results in this section show that the extinction spectra depend relatively


Figure 14. Extinction cross sections for (39, 48) nm silver nanoshells with offset silica cores for two different incident polarizations. The amount which the silica core is offset from the center of the nanoshell is given in the legend.

strongly on the nonuniformities of the shell thickness. Even small inhomogeneities may cause shifts of the plasmon peaks which could be mistaken for a broadening in a bulk measurement of the extinction cross section. V. Conclusions In this paper, we have presented an application of the FDTD method to calculate the optical properties of metallic nanoshell systems. The method was used to analyze the near and far field response for single nanoshells, nanoshells with defects, elliptical nanoshells, and nanoshells with nonuniform shell thicknesses. We found excellent agreement with Mie theory in the case of a perfect spherical shell. We then used the method to investigate the optical properties of distorted and defective nanoshells. The far field extinction spectra of nanoshells were found to be remarkably insensitive to defects. Near field properties such as electric field enhancements were found to be strongly influenced by localized defects. Future work includes an investigation of nanoshell dimers and nanoshells near surfaces. Acknowledgment. This work was supported by the Robert A. Welch foundation under grant C-1222, the Texas Advanced Technology Program, the Multi-University Research Initiative of the Army Research Office, and the National Science Foundation under grants EIA-0216467 and EEC-0304097. It is a pleasure to acknowledge valuable discussions with J. B. Jackson, E. Prodan, S. Yamamoto, and A. Lee. References and Notes (1) Oldenburg, S.; Averitt, R. D.; Westcott, S.; Halas, N. J. Chem. Phys. Lett. 1998, 288, 243-247. (2) Jackson, J. B.; Halas, N. J. J. Phys. Chem. B 2001, 105, 27432746. (3) Oldenburg, S. J.; Jackson, J. B.; Westcott, S. L.; Halas, N. J. Appl. Phys. Lett. 1999, 75, 2897-2899.

J. Phys. Chem. B, Vol. 108, No. 46, 2004 17747 (4) Prodan, E.; Nordlander, P.; Halas, N. J. Nano Lett. 2003, 3, 14111415. (5) Baer, R.; Neuhauser, D.; Weiss, S. Nano Lett. 2004, 4, 85-88. (6) Teperik, T. V.; Popov, V. V.; de Abajo, F. J. G. Phys. ReV. 2004, B69, 155402. (7) Jackson, J. B.; Westcott, S. L.; Hirsch, L. R.; West, J. L.; Halas, N. J. Appl. Phys. Lett. 2003, 82, 257-259. (8) Charnay, C.; Lee, A.; Man, S.-Q.; Moran, C. E.; Radloff, C.; Bradley, R. K.; Halas, N. J. J. Phys. Chem. B 2003, 107, 7327-7333. (9) Futamata, M.; Maruyama, Y.; Ishikawa, M. J. Phys. Chem. B 2003, 107, 7607-7617. (10) Li, K.; Stockman, M. I.; Bergman, D. J. Phys. ReV. Lett. 2003, 91, 227402-1-4. (11) Alivisatos, P. Nat. Biotechnol. 2003, 100, 13549-13554. (12) Sun, Y.; Xia, Y. Anal. Chem. 2002, 74, 5297-5305. (13) Hirsh, L. R.; Jackson, J. B.; Lee, A.; Halas, N. J.; West, J. L. Anal. Chem. 2003, 75, 2377-2381. (14) Sershen, S.; Westcott, S. L.; Halas, N. J.; West, J. L. J. Biomed. Mater. Res. 2000, 51, 293-298. (15) Loo, C. H.; Lin, A.; Hirsch, L. R.; Lee, M. H.; Barton, J.; Halas, N. J.; West, J. L.; Drezek, R. A. Technol. Cancer Res. Treat. 2004, 3, 3340. (16) West, J. L.; Halas, N. J. Annu. ReV. Biomed. Eng. 2003, 5, 285294. (17) Hirsch, L. R.; Stafford, R. J.; Bankson, J. A.; Sershen, S. R.; Rivera, B.; Price, R. E.; Hazle, J. D.; Halas, N. J.; West, J. L. Proc. Natl. Acad. Sci. USA 2003, 22, 47-52. (18) O’Neal, D. P.; Hirsch, L. R.; Halas, N. J.; Payne, J. D.; West, J. L. Cancer Lett. 2004, 209, 171-176. (19) Sershen, S.; Westcott, S. L.; Halas, N. J.; West, J. L. Appl. Phys. Lett. 2002, 80, 4609-4611. (20) Palkar, S. A.; Ryde, N. P.; Schure, M. R. Langmuir 1998, 14, 3483492. (21) Tamaru, H.; Kuwata, H.; Miyazaki, H. T.; Miyano, K. Appl. Phys. Lett. 2002, 14, 348-3492. (22) Bian, R. X.; Dunn, R. C.; Xie, X. S. Phys. ReV. Lett. 1995, 75, 4772-4775. (23) Scherer, A.; Painter, O.; Vuckovic, J.; Loncar, M.; Yoshie, T. IEEE Trans. Nanotechnol. 2002, 1, 4-11. (24) Maier, S. A.; Kik, P. G.; Atwater, H. A. Phys. ReV. B 2003, 67, 205402. (25) Taflove, A.; Hagness, S. C. Computational Electrodynamics: The finite-difference time domain method; Artech House, Inc.: Norwood, MA, 2000. (26) Westcott, S. L.; Jackson, J. B.; Radloff, C.; Halas, N. J. Phys. ReV. 2002, B66, 155431. (27) Sullivan, D. M. Electromagnetic Simulation Using the FDTD Method; IEEE Press: Piscataway, NJ, 2000. (28) Yee, K. S. IEEE Trans. Antennas Propag. 1966, 17, 585-589. (29) Kunz, K. S.; Luebbers, R. J. The Finite Difference Time Domain Method for Electromagnetics; CRC Press: Boca Raton, FL, 1993. (30) Johnson, P. B.; Christy, R. W. Phys. ReV. 1972, B6, 4370. (31) Coronado, E. A.; Schatz, G. C. J. Chem. Phys. 2003, 119, 39263934. (32) Sullivan, D. M. IEEE Trans. Antennas Propag. 1992, AP-40, 12231230. (33) Oubre, C.; Nordlander, P. Proc. SPIE-Int. Soc. Opt. Eng. 2003, 5221, 133-143. (34) Sacks, Z. S.; Kingskand, D. M.; Lee, R.; Lee, J. F. IEEE Trans. Antennas Propag. 1995, 43, 1460-1463. (35) Schatz, G. C.; Duyne, R. P. V. Electromagnetic mechanism of surface-enhanced spectroscopy; In Handbook of Vibrational spectroscopy; Chalmers, J. M., Griffiths, P. R., Eds.; John Wiley: Chichester, U.K., 2002. (36) Snir, M.; Otto, S.; Huss-Lederman, S.; Walker, D.; Dongarra, J. MPIsThe Complete Reference Volume 1, The MPI Core; MIT Press: Cambridge, MA, 2001. (37) Amdahl, G. M. Validity of the single-processor approach to achieving large scale computing capabilities; AFIPS Conference Proceedings; AFIPS Press: Atlantic City, NJ, 1967; Vol. 30, pp 483-485. (38) Fujii, M.; Lukashevich, D.; Sakagami, I.; Russer, P. IEEE Trans. MicrowaVe Wireless Comput. Lett. 2003, 13, 469-471. (39) Ditkowski, A.; Dridi, K.; Hesthaven, J. S. J. Comput. Phys. 2000, 170, 39-80. (40) Prodan, E.; Radloff, C.; Halas, N. J.; Nordlander, P. Science 2003, 302, 419-422. (41) Kottman, J. P.; Martin, O. J. F.; Smith, D. R.; Schultz, S. New J. Phys. 2000, 2, 27.1-27.9. (42) Hao, E.; Li, S.; Bailey, R. C.; Zou, S.; Schatz, G. C.; Hupp, J. T. J. Phys. Chem. B 2004, 108, 1224-1229. (43) Nordlander, P.; Oubre, C.; Prodan, E.; Li, K.; Stockman, M. I. Nano Lett. 2004, 4, 899-903. (44) Prodan, E.; Nordlander, P. J. Chem. Phys. 2004, 120, 5444-5454.

Optical Properties of Metallodielectric Nanostructures Calculated

Recommend Documents