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Computation of Surface-Enhanced Infrared Absorption Spectra of Particles at a Surface through the Finite Element Method Ganesh Vasan,† Ying Chen,‡ † and Andreas Erbe*,† ,
†
Max-Planck-Institut f€ur Eisenforschung GmbH, Max-Planck-Str. 1, 40237 D€usseldorf, Germany; ABSTRACT: The finite element method (FEM) was used to solve the time-harmonic Maxwell equations to calculate full infrared (IR) absorption spectra of organic substances near particles adsorbed to a silicon/air interface mimicking a surface-enhanced infrared absorption spectroscopy (SEIRAS) experiment in internal reflection geometry. An enhancement factor attributed to the presence of the metal has been defined, and its dependence on the particle size and arrangement has been studied. The enhancement factor was found to increase with increasing particle size. Contributions of parts of the surface to the overall absorbance has been analyzed through the electric field patterns and by placing a probe material in the region to be analyzed. For isolated particles, the gap between particle and surface has a large contribution to the overall spectrum. Further, large contributions were found from the regions between touching particles. An experiment probing gold particles on a silicon surface using attenuated total internal reflection infrared (ATR-IR) spectroscopy with a silicon crystal was performed and was compared to computations. The experimentally derived enhancement factor in the given geometry was of the same magnitude as the computed enhancement factor.
’ INTRODUCTION Since their discovery in the 1980s, many applications have successfully used surface-enhanced infrared absorption spectroscopy (SEIRAS), that is, infrared absorption spectroscopy in the vicinity of metal island structures.1 The surface enhancement has been achieved in different reflection and transmission geometries.1 Here, the focus is on the attenuated total internal reflection (ATR) geometry as the usually infrared (IR) absorbing solvent does not need to be traversed by the light in the ATR geometry.2-4 For in-situ ATRIR studies using surface enhancement, the metal island layer is deposited on a high refractive index material, for example, silicon or germanium.1,5 Adsorption of metal colloids is a further method of preparation.1,6,7 Self-assembly leads to particle arrays with a large packing density.8,9 Large enhancements are obtained with excitation of plasmons in the mid-IR range on elongated metal structures or hole arrays so far mainly in transmission geometries.10,11 SEIRAS has been popular especially for the analysis of biological samples and in electrochemistry.1,12 Using the local reaction centers presented by the small metal particles, proteins can be attached to the surface and can be studied with high sensitivity.13,14 Fundamental aspects of electrochemical reactions have been studied as well.15-18 The enhancement in SEIRAS is frequently discussed in terms of plasmon excitation around small metal particles and is treated in the framework of effective medium theory.1,19 In a previous work on SEIRAS, the excitation of plasmons in the rough metal r 2011 American Chemical Society
was hypothesized to explain the effect observed.20 Later, calculations treated the metal island film as an effective medium.21-23 Even recent work on characterizing the metal films still use such models as a basis.5,24,25 Such simple effective medium models have problems in quantitatively explaining spectra as well as in predicting effects on the band shape6 and baseline because they do not take into account near-field interactions. Several open problems limiting the more widespread application of SEIRAS have been formulated but have not been fully addressed.19,23 The challenge to date is the fact that results are extremely difficult to quantify. The development of SEIRAS has been paralleled by that of surface-enhanced Raman scattering (SERS).26-30 For SERS, early theoretical work showed the need for an enhanced nearfield scattering efficiency of particles exhibiting SERS.31 Strong near fields are encountered especially in surface mode excitation.28,32 Therefore, a large family of plasmonic structures have been generated and have been used for surface-enhanced spectroscopies with excitation in the visible or near-IR range.28,33 The applications of SERS by far outnumber those of SEIRAS. With the advances in computational resources in recent years, more sophisticated computation of spectra has become possible.34,35 Analytical solutions exist for the fields around spherical particles and their aggregates illuminated by an evanescent wave near an interface.36-38 The discrete source method39 has been Received: September 11, 2010 Revised: January 5, 2011 Published: February 2, 2011 3025
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used to compute scattering patterns from evanescent waves by small metal particles and nanoholes. The boundary element method (BEM), also known as method of moments, has been used to compute electric fields around small metal particles in full three-dimensional geometries.34,35,40 The BEM has also been used to predict fields in IR-plasmonic structures.11,41 The finite-difference time domain (FDTD) method has been successfully applied to solve several electromagnetic problems,42 including applications to predict the field enhancement of spherical and nonspherical particles for SERS.29,43,44 In this work, the frequency-domain-based finite element method (FEM) is used, a further popular and powerful method to compute electromagnetic fields for arbitrary geometries.45 For SEIRAS computations involving metals, FEM is particularly useful as the computational domain size does not need a significant increase for wavelengths much larger than the nanostructures to be simulated as needed in the case of SEIRAS.35 Furthermore, the handling of the dielectric function of a real metal is simple. In this computational study, results on FEM computations of full SEIRA spectra on the basis of continuum electrodynamics models are reported. Trends with particle size and polarization of light are explored, and the field patterns around the particles are analyzed to obtain an insight into how molecules adsorbed on different parts of the surface contribute to the spectrum. The focus here is to show that in SEIRAS, without excitation of surface modes and using inherent properties of small metal particles, enhancement is encountered in regions far away from the plasmon excitation and also to discuss implications for quantification of the results.
’ MATERIALS AND METHODS Computations. The commercially available FEM-based solver JCMsuite (http://www.jcmwave.com/)46 was used. It has been proved to be effective in solving electromagnetic problems involving complex geometries.47-49 It applies a numerical solution of the Maxwell equations to the given geometry while utilizing an adaptive mesh refinement technique which provides both speed and accuracy. Different geometry setups have been studied. In all cases, a plane wave is incident from a medium with refractive index n2 = 3.3, which is equivalent to silicon, at an angle of incidence of 45°. This angle of incidence is above the critical angle of total internal reflection. The exit medium is air with n1 = 1.0. On the silicon/air interface, particles with the optical constants of gold were placed. The frequency-dependent optical constants of gold between wavelengths of 2.5 and 10 μm were approximated by a polynomial.50 The complex refractive index m with real part n and imaginary part k, that is,
m ¼ n þ ik
ð1Þ
as a function of wavenumber ν ~ of organic material near the interface has been modeled using the optical constants of poly(ethylene).51 From the point of view of optical constants, it resembles the absorption spectra of organic molecules with methylene groups and, therefore, the material used in the experiments thus providing the possibility of a direct comparison between the computational and the experimental results. In the computations, organic layers were investigated in two different
Figure 1. (a) 2D computational domain for a 3D problem with rotational symmetry. The three-dimensionality is achieved by having a cylindrical boundary on the rotational symmetry axis. (b) 2D computational domain for multiple particles scenario with perfectly matched layers (PML) on top and bottom and periodic boundary conditions on the sides. The direction of the x-axis is shown in red, the y-axis in green. The mesh drawn around the particle represents the points at which fields are computed.
positions: bound to the silicon surface and bound to the gold particle. As the computation of a full 3-dimensional structure is computationally very expensive, all computations here were done in 2-dimensional geometries but in different coordinate systems as shown in Figure 1. If a rotationally symmetric structure is realized as shown in Figure 1a, a 2D computation in cylindrical coordinates can be used to obtain the solution for a problem which is 3D in Cartesian coordinates.47 The simplest geometry in the frame of this work is a single sphere on a surface. 3026
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Using such a setup in cylindrical coordinates, only single particles can be calculated, and the interaction between particles cannot be studied. Such interaction studies are possible in Cartesian coordinates as depicted in Figure 1b. The Cartesian coordinate system corresponds to a situation with infinite cylinders on the surface. As depicted, periodic boundary conditions were applied in the direction parallel to the surface, while on top and bottom of the system, a perfectly matching layer (PML) was placed. In this geometry, the sample is in effect an infinite array with a certain particle density. In the Results section, different layouts are labeled by a combination of two letters. Capital letters indicate the coordinate system used. “A” indicates a computation in Cartesian coordinates, while “B” indicates a computation in cylindrical polar coordinates. Lower case letters indicate where an organic layer has been placed: “a” is used when the layer is present on the planar silicon surface, and “b” means the absorbing layer is present around the particle. The coordinate system used throughout this work has the xaxis in the silicon/air interface. The direction of the x-axis is indicated as a red arrow in Figure 1b. The surface normal of the silicon/air interface is the y-axis, which is indicated as a green arrow in Figure 1b. The z-direction is therefore pointing outward of the image plane to form a right-handed Cartesian coordinate system. As the linear polarizations of light are the eigenpolarizations of a planar interface, these will be discussed throughout this work. For perpendicular (s-)polarization (transverse electric, TE), the electric field amplitude has components in the z-direction only, while for parallel (p-) polarization (transverse magnetic, TM), the electric field has components in x- and y-directions. Computations for both linear polarizations were carried out simultaneously by defining multiple sources. As depolarization can be neglected in these structures, such a simplification can be used to decrease the computation time to half of the time taken with both independent modes. The source originates from the PML/incidence medium interface. The light impinges on the particle through the incidence medium, and the reflected beam electric field amplitude Erefl is calculated by calculating the propagating Fourier modes in the medium in the appropriate direction in a postprocess on the electric field distribution. The reflectivity R is obtained as R ¼
jErefl j2 jEinc j2
ð2Þ
with the source amplitude Einc. As in experiments, the reflected field is superimposed with the field scattered into the same direction as the reflected beam points. The scattered amplitude in directions other than the reflected light is neglected for the further analysis, that is, it will only contribute to the overall extinction of light. However, the contributions of scattering are rather small: in the case of the spherical particle on the surface, scattered fields are at least 7 orders of magnitude weaker than the reflected field. For computations of full spectra, the FEM solver was controlled through a script performing calculations on a wavelengthby-wavelength basis. To obtain absorbance values, the reflectivity Rref of a reference system without the particles was performed. The absorbance A
Figure 2. Experimental setup with gold nanoparticles immobilized on a thin layer of organic material over a silicon substrate (internal reflection element).
was then calculated as A ¼ -log10
Rsmp Rref
ð3Þ
Because a total internal reflection geometry is used, Rref = 1. The FEM computations for the silicon/air interface in the respective geometry yield values which are O (10-7) different from 1, which is an acceptable disagreement in the numeric solution. The FEM computations have furthermore been compared to absorbance spectra computed by matrix methods based on analytical solutions of the 1D Maxwell equations.52,53 The system used for these validations consisted of zinc selenide medium of incidence on which a germanium layer of 1 μm thickness followed by a gold layer of 20 nm thickness was deposited.54 The sample was the organic solvent acetonitrile. Differences in the computed absorbance of O (10-6) were found between the two computations, which is still acceptable as the differences show up only as a baseline shift. To compute the spectral enhancement factor EF due to the presence of the metallic nanostructures, a computation of the absorbance A0 of the organic layers without metallic particle was done in the same manner as described above, and the absorbance was compared to the absorbance Aw calculated around metal particles with EF ¼
Aw A0
ð4Þ
For computations in cylindrical coordinates, a computation of a hollow shell was performed without metal, while for computations in Cartesian coordinates, a rectangle of the same area as the shell around the circles in the presence of metal was placed on a planar surface. The resulting computed spectra were baseline corrected using a procedure similar to that commonly performed in the analysis of experiments to obtain the absorbance for use in eq 4. In some cases, the uncorrected spectra are displayed for reference. Experimental Procedures. Fourier transform infrared spectroscopy (FTIR) experiments in attenuated total internal reflection (ATR) geometry were carried out on a commercial Nicolet Nexus 870 FT-IR spectrometer with a liquid nitrogen cooled mercury cadmium telluride detector using a homemade setup. The setup uses the optical base of a SpectraTech Model 0001100 horizontal ATR unit (SpectraTech, Stamford, CT). A sketch of the setup is shown in Figure 2. During the experiment, light enters an ATR crystal, which is mounted on a homemade plate, from the bottom. The light probes the silicon/air interface on the probe side at an incidence angle of 60° approximately 100 times because of multiple reflections occurring inside the crystal. It then emerges through the other edge, passes through the polarizer, and is finally collected by the detector. The ATR crystals used here were cut from double side polished, N-type boron-doped silicon wafers of a thickness of 0.5 mm (Si-Mat, Kaufering, Germany). The rectangular pieces 3027
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Figure 3. (a) Computed absorbance spectra in p-polarization without baseline correction from particles with a 1 nm thick organic coating with different separations between the coated particles as indicated in the graph. (b) Plot of enhancement factor with the separation distance between the coated particles for p- (circles, left scale) and s-polarization (squares, right scale). Lines are drawn to guide the eye. Layout type “Ab”.
with dimensions 52 20 0.5 mm3 were polished on the short sides at an angle of 30° to use in the ATR setup. To ensure a binding of the gold particles to the silicon wafer, a silane layer was adsorbed on the surface of the silicon piece.55 The silicon was immersed in a solution of 2.0 mM (3-aminopropyl)-triethoxysilane (APTES) (ABCR, Karlsruhe, Germany) in chloroform for 30 min. Subsequently, the treated wafers were subjected to an ammonia atmosphere overnight and finally were rinsed with deionized water to remove excess unbound silane. The success of the modification was checked by ellipsometry on a spectroscopic ellipsometer Sentech SE-800 (Sentech Instruments, Berlin/Krailling, Germany). The change in the ellipsometric parameter Δ before and after the surface modification was correlated to a layer thickness using the instrument’s fitting software assuming a layer with a refractive index of 1.420. Layer thicknesses of (0.7 ( 0.1) nm were found. Gold nanoparticles were prepared according to Frens and Ji et al. with slight modifications.56,57 In a typical synthesis, a 50 mL aqueous solution of HAuCl4 (0.25 mM) was prepared in a three-neck flask equipped with a condenser, and the solution was heated in an oil bath to boil while being stirred. Then, 0.05 mL 5 wt % aqueous sodium citrate solution was added. The reaction was continued until the solution became wine red color thus indicating that the reaction was completed. Initially, background spectra of the unmodified silicon wafers were measured. After modification with APTES, single-channel spectra with the APTES layer were recorded. Subsequently, one side of the wafers was put in contact with the aqueous gold particle dispersion in a cell for 15 min, was rinsed with water, and was dried,
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Figure 4. Color-coded distribution of electric field strength |Exy|, corresponding to p-polarization, in V m-1 around two coated gold particles with a distance of (a) 0 nm and (b) 10 nm between the coated particles of a diameter of 30 nm with a coating of 1 nm thickness. The medium of incidence is shown on the bottom, while the gold particles are the circles in the middle. The electric field strength at the source is 1 V/m. Layout type Ab.
and spectra with adsorbed particles were recorded. For the single channel spectra, 1000 scans were averaged. The absorbance was calculated as in eq 3. All measurements were performed with both s- and p-polarization at ambient temperature of 23 °C. After finishing the experiment, adsorbed particles were characterized using a field emission scanning electron microscope (SEM) type Zeiss Leo 1550VP Gemini (Carl Zeiss SMT AG, Germany).
’ RESULTS Computation in 2D Cartesian Coordinates. The effect of interactions between neighboring particles was investigated by performing 2D computations in Cartesian coordinates. All computations in this section use poly(ethylene) optical constants in the coating. Figure 3a shows the comparison of absorbance spectra with p-polarization, where the spacing distance between two coated spherical gold particles was varied from 0 nm (touching) to 10 nm (layout type “Ab”). Such a variation corresponds to a variation in particle density with periodic boundary conditions. It was observed that the absorbance was high when the particles were touching each other and that it went down drastically when the spacing distance became bigger. The distance refers to the distance between the particles including the coating. When the distance between the particles is 0 nm, the coatings are touching, while the metal parts are still separated. The corresponding enhancement factors are plotted in Figure 3b for both s- and p-polarization. In p-polarization, EF ∼ 70 for touching particles. At separations of 10 nm and above, 3028
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Figure 5. (a) Absorbance spectra with s- (inset) and p-polarization for particles coated with a 1 nm thick organic layer on the top and bottom half. (b) Electric field strength |Exy| in V m-1 (p-polarization) around a single coated gold particle of diameter 30 nm with a complete shell of 1 nm thickness. (c) As in b but |Ez| (s-polarization). The contour of particle, coating, and interface have been added for clarity as black lines. Total electric field strength at the source in both cases is 1 V/m. Modified layout type Ab.
EF ∼ 2. In s-polarization, EF ∼ 0.1 independent of size. Being smaller than 1, the results correspond not to an enhancement but to a reduction in absorbance. The differences between the geometries is also reflected by the field images that show a plot of field strength for p-polarized incident light around two coated particles with 0 nm (Figure 4a) and 10 nm spacing (Figure 4b) between them. When the coated particles are touching each other, there exists a region with high field strength at the point of contact between the particles. At the point of contact, the field is approximately 10 times stronger than the strongest fields encountered in particle assemblies with a separation of 10 nm. When inspecting the field images, it becomes obvious that the enhancement is not uniform over the particle surface. The contribution of regions with different field strengths to the overall spectrum is illustrated by computations of absorbance spectra in the cases when the 1 nm thick coating is only placed on the top half of the circle and when the coating is placed on the bottom half of the circle (modified layout Ab). The coating on the bottom half of the circle is in direct contact with the medium of incidence, while the coating in the top half is not. Figure 5 illustrates the contribution of top and bottom surfaces of the spherical particle in the signal enhancement. The absorbance (Figure 5a) obtained in the spectrum for the coating on the bottom half of the circle is by a factor of ∼1.5 higher than the absorbance on the top half for both polarizations showing that the bottom half contributes more strongly to the overall
Figure 6. Comparison of absorbance spectra from 0.73 nm organic layer on the interface between a silicon substrate and a spherical gold particle with radius varying from 10 to 300 nm for (a) p- and (b) s-polarization. (c) Size dependence of enhancement factor for p(circles, left scale) and s-polarization (squares, right scale). Layout type “Ba”.
spectrum. The reasons for the different contributions to the overall spectrum are visualized in the electric field in Figure 5b and 5c. There are three regions with high field strength: the bottom in direct vicinity of the incidence medium and the left and the right of the particle, where the field is mainly governed by the particles in the neighboring computational cell of the lattice. On top of the particle, there is a region of exceptionally low electric field strength, and molecules in this position do not contribute to a large extent to the overall spectrum, while those in regions with large field strength do. Computation in 2D Cylindrical Coordinates. The effect of the size of the particles on the overall absorbance of an organic layer with poly(ethylene) optical constants on a silicon substrate was investigated by calculating in a 2D geometry in cylindrical coordinates. The results are shown in Figure 6, where absorbance spectra of particles with the diameter varying from 10 to 300 nm 3029
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Figure 7. Enhancement factor with variations in diameter of a 1 nm thick organic layer under a 30 nm gold nanoparticle (layout type Ba). Figure 9. SEM image of gold particles immobilized on an APTES layer over a silicon substrate.
Figure 8. Effect of the particle shape on the enhancement factor. (a) Layout with six edges in a problem with cylindrical symmetry. (b) Plot of the enhancement factor for p-(left scale) and s-polarization (right scale) with different number of edges. Layout type Ba.
for s- and p-polarization are displayed. The organic layer thickness on the silicon surface is maintained at 0.73 nm with no layer around the particles (layout type “Ba”). The results portray an increase in absorbance with an increase in particle size. With increasing size, the total amount of material on the surface does, however, also increase. With the definition of the enhancement factor in eq 4, these effects are already compensated. The enhancement factor was plotted against the radius of the particle as shown in Figure 6c. For p-polarization, EF > 1 and increases with increasing size, while for s-polarization, a more differentiated behavior is found. Below 150 nm diameter, EF < 1, corresponding to a reduction in absorbance. For particles with diameter >150 nm, actual enhancement is found (EF > 1). To study the limits of the enhancement factor in the region of large fields, which is the gap between the gold particle and the silicon incidence medium, a layer with poly(ethylene) optical constants in the shape of a disk with a thickness of 1 nm and a diameter of 2-50 nm was placed between silicon and gold
(layout Ba). The results of the computation are shown in Figure 7. On the lowest diameters, EF ∼ 60 in p-polarization, increasing with increasing size of the disk. In s-polarization, EF < 1. Real metal particles are rarely exactly spherical but show facets which are inclined with respect to each other. The effect of different shapes has been computationally investigated as well. For these investigations, the semicircle in Figure 1a has been replaced by a polygon with E edges starting from six edges (E = 6). An example for the cylindrically symmetric layout is shown in Figure 8a. The distance from top to bottom of the metal particle, that is, the diameter, was kept constant at 70 nm. An absorbing layer with 1 nm thickness was present on the surface. The effect of E on the enhancement factor is shown in Figure 8 for layout type Ba. The variations in the enhancement factor were not very pronounced. There is some deviation only for the lower number of edges, but in p-polarization, all enhancement factors are between 1.5 and 2, while for s-polarization, the enhancement is 1 to 1.2. Experimental Results. The computational results should be subjected to experimental tests. Because the absorbance from organic coatings around particles in the spectra of randomly aligned particles varies vastly with the distance between particles, such arrangements are not suitable for experimental tests unless a full 3D computation is done over a large area. Therefore, to compare experiment and theory, the second introduced geometry was chosen. An organic coating was placed on a silicon surface, and gold particles were immobilized on top of that surface. Because of this, the effect of the interaction between different particles is not very prominent in the spectra, while the interaction between particle and surface should be uniform in first approximation for all particles. Scanning electron microscopy (SEM) images (Figure 9) of the immobilized particles show an inhomogeneous spreading of particles with no major clusters and with particle radii between 30 and 35 nm. The particles are slightly aspherical. IR spectra were recorded before and after the particle immobilization, and both were background corrected using a reference spectrum of an unmodified silicon substrate. A comparison of the two spectra (Figure 10) was made after baseline corrections to study the signal enhancement of the CH2 stretching modes around 2850 cm-1 and 2920 cm-1. In the presence of the particles, the 3030
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which are randomly generated between neighboring particles do not contain the probe molecules, and the touching of particles has only a minor effect on the field between the particle and the surface. The field images enable an analysis of the local contributions to the overall absorbance. As A µ jEj2
Figure 10. Experimental spectra showing the CH2 stretching modes from the APTES layer in the presence (straight lines) and the absence (dashed lines) of gold nanoparticles with s- (red) and p-polarizations (black).
observed peaks in s- and p-polarization are a factor of ∼2 larger than in the absence of the particles. The parameters obtained from the experiments such as the particle radius (35 nm) and the thickness (0.7 nm) were utilized in the computation to create an approximate model of the experimental setup. Optical constants of poly(ethylene) were used for the layer on the silicon substrate, while in the spectral range studied here, the coatings on the particles have been modeled as IR-transparent coatings with n = 1.4.
’ DISCUSSION Analysis of the SEM images yields a particle density on the surface for the example described here as ∼1.1 particle/μm2. With the total area of 5.6 10-4 m2 covered, we can estimate the increase of peak absorbance at 2920 cm-1 per particle as 2.4 10-11 in p-polarization and as 1.6 10-11 in s-polarization. As the area per particle is ∼0.9 μm2, the absorbance without the particles in such an area is 2.4 10-11 in p-polarization and 1.6 10-11 in s-polarization. For both polarizations, the experimentally determined enhancement factor obtained here is ∼2. Simulations of absorbance spectra for an APTES layer on silicon with a particle on top of the layer in a geometry close to the experiment have been performed. The native oxide layer of silicon with a thickness of 1.5 nm was taken into account. Different thicknesses of the citrate coating, varying from 1 to 10 nm, were used. The results yield an enhancement factor per particle for p-polarization of ∼1.6, and for s-polarization of ∼1.2, with only little dependence on the thickness of the citrate shell. As all computations performed here were effectively done in two dimensions (either in cylindrical or in Cartesian coordinates), only ratios of the absorbance value can be compared to the experiments because of effects of the beam size in real experiments which are not accounted for in the computation here. If a beam of larger size is incident on a single sphere, the absorbance of a finite organic layer will become lower. In that way, the absorbance itself scales with the beam size, while the ratio between absorbance with and without metal structure does not. The differences to the experimental values can be attributed to the anisotropy of the thin organic film. The region from which the enhanced absorption is observed is a region between a semiconductor and a metal, which is different from the hot spot between two metal particles. Between two metal particles, the absorbance decays much faster, as shown above. The assumption of the noninteracting particles should be justified to first approximation, as is obvious from the field images in Figure 4, because in the experimental situation, the hot spots
ð5Þ
the regions with larger field strength contribute more to the overall absorbance spectrum than the regions with lower field do. Analyzing the field patterns, therefore, shows the different contributions of different regions to the spectrum. Even for a single particle on the surface, there is a hot spot in the region between particle and surface. The presence of this hot spot between the metal and the silicon surface shows that in a thorough analysis of surface enhancement, the effects caused by the proximity of the substrate must always be included. Electromagnetic methods which yield only fields around particles without the presence of the surface must be used with extreme care when interpreting results of reflection spectra.11 Furthermore, a computation of the fields around the particle in an evanescent wave without the inclusion of the surface will also lead to misleading results. The field images of p- and s-polarizations (Figure 5b and 5c) show characteristic differences. Figure 5b is very similar to the radiation pattern of an oscillating dipole distorted by the surface. Large fields and consequently large enhancement is obtained in regions between particles and between particle and surface. The same behavior was used in SERS experiments using gap modes to achieve a field enhancement.58-61 In s-polarization, the presence of the particles leads to a decrease in the field compared to undistorted situation giving a reduction in absorbance compared to the situation without metal in 2D computations. However, this reduction is not observed in computations in cylindrical coordinates, which reflect a 3D problem. This behavior can again be understood from the radiation pattern of a distorted dipole in the case of the computations in cylindrical coordinates. In the 2D case, no dipole can be excited with s-polarization. Comparing both polarizations, one of the characteristic features of SEIRAS is that the sample regions largely contributing to the spectrum with p-polarization and those largely contributing to the spectrum with s-polarization are different. Overall, the computational observations here correspond to experimental observations.7 Very strong enhancement is found for citrate-coated gold colloids before they touch each other. A careful inspection of Figure 4a shows that even with coated particles, the field strength in the region where particles are touching is lower than in the regions where particles are about to touch. The size dependence of the enhancement factor for p-polarization can be understood on the basis of the confinement of electromagnetic fields inside a narrow gap.61 The larger the radius of a particle the lower the curvature. Consequently, the sample area exposed to the high field in the gap between particle and substrate surface increases, and therefore, a larger part of the sample senses the higher fields. The enhancement factors do not follow a simple power law scaling with size. Size effects have been studied extensively for SERS. The behavior found here can be explained similarly to off-resonance enhancement for SERS.62 In general, the size dependence for SERS is more involved when considering the high enhancements 3031
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The Journal of Physical Chemistry C obtained near a plasmonic resonance. In this regime, the resonance shifts with particle size, which is reflected in the spectral enhancement.62,63 Size dependence in regions away from a plasmonic resonance follows a trend similar to those obtained here.63 A further important difference to SERS is the fact that in SERS experiments, the enhancement is frequently determined by one excitation field given by the laser used for excitation. In SEIRAS, the enhancement factor is not uniform over the spectrum and it depends on the wavelength used. This wavelength dependence is the reason why larger particles show strong changes in the baseline. Similar effects have been noted in experiments.6 As the overall spectral response of the system is simulated, such baseline effects are seen in the simulations as well, though detailed discussions are beyond the scope of this work. The major difference is that to compute results of SEIRAS experiments, the small differences in reflectivity because of the presence of the organic layer need to be captured, while for most work computing, for example, transmission spectra or SERS enhancement, the overall picture of the optical properties of the particles is most important. In SERS, special cases arise for molecules with an electronic absorption near the excitation wavelength. In these cases, enhancement is not uniform over the Raman spectrum. A discussion of this is, however, beyond the scope of this work. All comparisons made here with geometries where the metal is replaced by the ambient medium show that the absorption indeed increases because of the presence of the metal. It must be stressed that when working in the mid-infrared wavelength range one is far away from surface modes, for example, surface plasmons. For spherical gold particles and arrays of spherical gold particles in the size ranges studied here, these modes are excited in the visible. As the dielectric function of many metals is similar in the mid-IR, SEIRAS should in principle work for almost all metals, though the difficulty is to produce stable structures of the right size. Though the largest enhancements encountered here are just below 100, the use of wavelengths far away from surface mode excitations has advantages. Near surface mode excitations, larger enhancement factors are found,11 but there is also the possibility of coupling between these surface modes and the vibrational or electronic states of the involved sample molecules. Such a coupling is very difficult to quantify. In general, the enhancement factors reported here agree with those reported in literature for SEIRAS without plasmon excitation.1 In SEIRAS, the enhancement factors are in general lower than in SERS.1,19,27 On the other hand, IR absorption cross sections are in general higher than Raman cross sections, which is why SEIRAS with its lower enhancement factor is useful. Quantification is still difficult for these simulations on a continuum electrodynamics basis. Changes in the structure of molecules near the surface, for example, because of adsorption, lead to a change in polarizability and, hence, in refractive index with respect to the bulk species. Such effects can be computationally addressed only on an atomistic level. The electrodynamic models as used here could be the bridge between such atomistic models and large scale models, for example, based on diffraction theory, in multiscale simulations.
’ CONCLUSIONS AND OUTLOOK The results shown here on the basis of computations of full absorbance spectra from a thin organic layer on or near gold
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particles clearly display the ability of predicting the enhancement factors on the basis of surface morphologies in a total internal reflection geometry. The size dependence of the enhancement factor shows that larger particles yield stronger enhancements in SEIRAS. Besides the well-known enhancement in regions where two particles are almost touching, the field localization between particles and substrate offers interesting perspectives for spectroscopic studies of the hardly accessible region under a particle. If a quantitative agreement between computation and experiment should be achieved, methods which can include the presence of the surface need to be used, as otherwise the field strength in the gap between surface and nanostructure is not accounted for properly. For computation of complex threedimensional surfaces, large computational efforts will be needed. However, a full, quantitative computation of spectra from surfaces with complex surface morphologies is within reach. As gold particles can be produced with good control over size and with narrow size distribution, arrays of well-defined particles offer a further playground for comparison between experiment and theory.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected],
[email protected]. Tel.: þ49 211 6792 890. Fax: þ49 211 6792 218. Present Addresses ‡
Center for Electrochemical Sciences - CES, Analytische Chemie, Ruhr-Universit€at Bochum, 44801 Bochum, Germany.
’ ACKNOWLEDGMENT We thank Dr. Sven Burger, JCMWave GmbH, for his kind assistance setting up initial computations. Furthermore, we thank Prof. Stratmann for his support. Part of Y.C.’s contribution to this work was supported by the European Union and by the state of North Rhine-Westphalia in the frame of the HighTech.NRW program. ’ REFERENCES (1) Tolstoy, V.; Chernyshova, I.; Skryshevsky, V. Handbook of Infrared Spectroscopy of Ultrathin Films; Wiley: Hoboken, NJ, 2003. (2) Plieth, W.; Wilson, G.; Gutierrez de la Fe, C. Pure Appl. Chem. 1998, 70, 1395–1414. (3) Plieth, W.; Wilson, G.; Gutierrez de la Fe, C. Pure Appl. Chem. 1998, 70, 2409–2412. (4) Vigano, C.; Ruysschaert, J.-M.; Goormaghtigh, E. Talanta 2005, 65, 1132–1142. (5) Busk€uhl, M.; Korte, E.-H. Anal. Bioanal. Chem. 2002, 374, 672– 675. (6) Heaps, D. A.; Griffiths, P. R. Vib. Spectrosc. 2006, 42, 45–50. (7) Enders, D.; Nagao, T.; Nakayama, T.; Aono, M. Langmuir 2007, 23, 6119–6125. (8) Wang, H.; Kundu, J.; Halas, N. J. Angew. Chem., Int. Ed. 2007, 46, 9040–9044. (9) Fan, M.; Brolo, A. G. ChemPhysChem 2008, 9, 1899–1907. (10) Williams, S. M.; Rodriguez, K. R.; Teeters-Kennedy, S.; Stafford, A. D.; Bishop, S. R.; Lincoln, U. K.; Coe, J. V. J. Phys. Chem. B 2004, 108, 11833–11837. (11) Neubrech, F.; Pucci, A.; Cornelius, T. W.; Karim, S.; GarcíaEtxarri, A.; Aizpurua, J. Phys. Rev. Lett. 2008, 101, 157403. (12) Ataka, K.; Heberle, J. Anal. Bioanal. Chem. 2007, 388, 47–54. 3032
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