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J. Phys. Chem. C 2008, 112, 7356-7362
Optical Properties of Elongated Noble Metal Nanoparticles A. L. Gonza´ lez, J. A. Reyes-Esqueda, and Cecilia Noguez* Instituto de Fı´sica, UniVersidad Nacional Auto´ noma de Me´ xico, Apartado Postal 20-364, D.F. 01000, Me´ xico ReceiVed: January 16, 2008; In Final Form: March 3, 2008
The importance of anisotropy in the optical response of metal nanoparticles of different end shapes is studied theoretically. The number, behavior, and localization of the plasmon modes in terms of the shape and elongation is revised in detail for silver, gold, and copper nanoparticles. It is found that the position of longitudinal and transversal modes depends not only on the aspect ratio but also on the nanoparticle end shape. Besides, the relation between nanoparticles’ anisotropy and optical macroscopic birefringence is also analyzed for different shapes and materials.
Introduction In recent years, there has been an increasing interest in the physical properties of nanosized systems for their technological applications. In particular, metallic nanoparticles (NPs) have proved to be widely attractive for their promising application on communications, magnetooptics, and catalysis, as well as on optoelectronic, storage, chemical, biosensing, and other devices. This relevance is based on the fact that their optical properties are mainly determined by their surface plasmon (SP) resonances, which, in their turn, are strongly dependent on the NP’s geometry and surroundings. For example, for sensing applications, the presence of sharp edges or tips has been corroborated to be important to enhance the electromagnetic field,1 which has also led to spectroscopies such as surfaceenhanced Raman spectroscopy2 and plasmon-controlled fluorescence.3 Although SPs have been well-known for a long time, such a dependence has opened the possibility of tuning the optical response of metallic NPs by varying their size, shape, environment, etc.4-6 In particular, elongated NPs have attracted attention because their SPs can be tuned systematically by changing the aspect ratio of their axes. Therefore, a large variety of new synthesis methods have been developed to fabricate elongated NPs, such as disks,7 ellipsoids,8 rectangular and decahedral nanorods,9-21 regular octahedra,22 rings, and crescents.23 End-capped elongated NPs, such as ellipsoidal and spherical ends, have been the subject of many theoretical works.1,4,14,17,20,24,25 However, the effect of the geometry on the NP’s ends with realistic motifs, such as decahedral, hexagonal, and rectangular shapes have not been studied previously. When elongated NPs are dispersed in a matrix (solid or liquid), their random orientation leads to an average absorption spectrum containing all the plasmon resonances associated with the corresponding geometry. On the other hand, when all the NPs in a system are oriented in the same direction, it is possible to distinguish between the different resonances by using polarized light.8,26,27 The orientation of the NPs means a macroscopic anisotropy of all the system, which can be related to optical birefringence28,29 and, under some circumstances, to optical noncentrosymmetry. The latter then gives place to the possibility of second-order nonlinear optical responses like second harmonic generation.30 * To whom correspondence should be addressed. E-mail: cecilia@ fisica.unam.mx.
In this paper, we show mainly how the optical response of metallic NPs, namely, Ag, Au, and Cu, shows strong variations when the NP’s shape becomes anisotropic. These variations are a consequence of the modification of the SP resonances of the NP after changing its shape. We also show how to relate the nanocomposite’s measured anisotropy to its birefringence. Fundamentals of Optical Response of Nanoparticles To understand the optical properties of NPs, it is necessary to know the main phenomena involved and their range of action as a function of the NP’s size. It was found31 that NPs of less than 20 nm in diameter only absorb energy through (i) the collective excitations of free electrons due to intraband transitions, giving rise to SP resonances. (ii) due to electron transitions of bound electrons from occupied to empty bulk bands of different index, called interband transitions, and (iii) because surface dispersion or scattering of the free or unbound electrons, when their mean free path is comparable to the dimension of the NP. In particular, SP resonances are determined by the particle’s shape and variations of the dielectric function. Surface dispersion effects do not change the location of the SPs, but make the absorption spectrum wider and less intense since they affect the coupling of the proper modes to the applied field. On the other hand, scattering effects are important only for NPs of more than 30 nm in diameter, where electrons are accelerated due to the electromagnetic field, then radiating energy in all directions. Due to this secondary radiation, the electrons lose energy experimenting a damping effect on their motion. It was found that the spectrum is less intense, wider, and red-shifted when the particle size increases.31 A depolarization field term provokes the shift to larger wavelengths, while radiation damping causes decreasing intensity and widening of the spectrum, as explained in ref 32. Finally, the scattering effects dominate the response of NPs of 100 nm in diameter and larger. Here we have focused our study in particles whose light extinction have both absorption and scattering processes. We have also taken into account that conduction electron motion can be damped by dispersion due to the surface of the particle, as well as interband contributions due to electron transitions from occupied to empty bulk bands separated by an energy gap. We should recall that bound electrons do not participate in the collective motion of the electron cloud; thus, SP resonances are quite independent of the interband contributions to the dielectric
10.1021/jp800432q CCC: $40.75 © 2008 American Chemical Society Published on Web 04/15/2008
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J. Phys. Chem. C, Vol. 112, No. 19, 2008 7357
function. For all the considerations mentioned above, the dielectric function that better resembles the optical properties of the particle of radius a is given by4
[
(ω,a) ) bulk - intra(ω) + 1 -
ωp2 ω(ω + i/τ + i/τ(a))
]
(1)
where bulk ) inter + intra is the measured bulk dielectric function, intra is the contribution to the dielectric function due to the “free electrons”, which can be well represented by the Drude model. Here, ωp is the plasma frequency, 1/τ is a damping constant due to the dispersion of the electrons by the ions, and 1/τ(a) is the damping term due to the dispersion of the electrons due to the surface of the particle.33 We employ the bulk dielectric function of silver, gold, and copper measured by Johnson and Christy.34 Extinction Spectra of Elongated Nanoparticles SP resonances can be tailored by managing the shape of the NP, and nowadays, there are many synthesis techniques. Controlling the time of growth, concentration, surfactants, and other parameters, the synthesis of NPs with a great variety of shapes is now possible,35 such as, icosahedra, cubes, decahedra, octahedra, cuboctahedra, prolates, tetrahedra, etc.5,6,36-41 There are also novel techniques, which make possible modifying the initial shape of a particle into another one. For instance, by using a photoinduced method it is possible to convert nanospheres into triangular prisms, which would scatter at larger wavelengths, making them candidates for diagnostic labeling devices.10,42 There is also the possibility of deforming spherical silver nanoparticles into aligned prolates by using ion irradiation, making it possible to control the aspect ratio of the prolates with the irradiation fluence.8 On the other hand, the evolution of different silver seeds into specific shapes is possible, for example, twinned decahedra that were made to evolve into nanorods with a pentagonal cross section, and controlling the aspect ratio of them by changing the temperature of the chemical reaction, among other considerations.38 This kind of NP can be employed as biological sensors required to detect biomarkers of diseases or to monitor biological and chemical agents.43 To exploit adequately the SP resonances inherent to each geometry it is important to understand the influence of shape on the number, position, and also intensity of these resonances. To better do this, in this work, we study the optical response of a nanoparticle with an initial shape, assuming that it will evolve to a final different shape, for which we will obtain its respective optical response, comparing it then to the initial one. For simplicity, we identify the initial shape of the NP as the seed. We consider mainly five cases: first, we suppose that a spherical seed is elongated along one of its axes, evolving into a prolate. The second case is a seed formed by two hexagonal pyramids joined by the bases, with 12 triangular faces and, for simplicity, called the dodecahedral seed, which is elongated along the C6 axis, obtaining an hexagonal prism. The third case is the evolution of a decahedral seed (or pentagonal bipyramid) along the C5 axis, giving a pentagonal prism. The fourth and fifth cases are an octahedral and a cube seed, which are elongated parallel to one of their C4 axis, evolving into a tetrahedral and a rectangular prism, respectively. Seed and elongated NPs are all shown in Figure 1. The prolate shape can be seen as a prism with an infinite number of rectangular faces (RF), while the hexagonal prism has six RF, the pentagonal one has five RF, and the tetrahedral and rectangular prisms both have four RF.
Figure 1. Seed and elongated shapes of metal NPs.
This variation of the number of RF will show the influence of this parameter into the optical response of the elongated NP. To compare appropriately the optical response obtained for particles of an initial shape to the corresponding elongated ones, we consider that the volume of the NP is kept constant. This volume corresponds to the one of a sphere of 40-nm radius. We also consider an aspect ratio of 2.8 for all the elongated NPs. Furthermore, we suppose that seeds and elongated NPs are embedded in a dielectric medium with refractive index of 1.47, which may be associated with dimethyl sulfoxide or silica matrix, for example. Because of the complexity of the systems being studied here, efficient computational methods capable of determining the optical properties of nonspherical particles are essential. Several numerical methods have been developed for this purpose, such as the discrete dipole approximation (DDA), which has been used to study a wide variety of particles.5,36,44-46 In this work, we employ the numerical implementation of DDA47 to study NPs of regular and elongated shapes. A complete explanation of the DDA method can be found elsewhere,48 here we only discuss the results obtained for metallic elongated NPs. The computer codes to generate seed and elongated NPs are available elsewhere.49 In Figure 2, we show the optical efficiencies of a decahedral seed NP and a pentagonal prism for all the metals considered in this work: Ag, Au, and Cu. Panels a and a′ show the absorption and scattering contribution to the light extinction of Ag decahedral and pentagonal prism NPs, respectively. Panels b, b′ and c, c′ show the same as panels a and a′ but for Au and Cu NPs, respectively. As can be observed, the relative scattering contribution to extinction of light depends on the material, being larger in Ag NPs than in Au or Cu, which makes Ag NPs better scatters.43 The resonances at small wavelengths for all NPs are more due to absorption than to the scattering process. On the other hand, the resonances at larger wavelengths have a completely different behavior that depends on the material’s dielectric properties. For instance, in Ag NPs, the main contributions for this peak are the scattering effects, while in Au absorption and scattering contribute equally, and in Cu the
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Figure 2. Optical efficiencies of Ag (a) decahedron seed, and (a′) pentagonal prism. Panels b, b′ and c, c′ show the same as in panels a′ and a′ but for Au and Cu NPs. Panels labeled with ′ and ′′ show the real and imaginary parts of the dielectric function of Ag (blue), Au (green), and Cu (red) NPs, as well as their inter- and intraband contributions.
Figure 3. Extinction efficiencies by NPs of Ag, Au, and Cu. Panels labeled with a letter refer to seeds and panels labeled with a letter and (′) refer to elongated NPs.
absorption effects are more important. This feature is a consequence of the role that plays the metal’s dielectric function in the absorption and scattering processes, as we explain as follows. In the right-hand side of Figure 2, the real and imaginary parts of the dielectric function for each material are shown in panels labeled with ′ and ′′, respectively. In these panels, the experimental dielectric function, measured by Johnson and Christy,34 was decomposed to show the intra- and interband contributions according to eq 1. One observes, for both ′ and ′′, that interband transitions become important at smaller wavelengths than 320, 550, and 600 nm for Ag, Au, and Cu, respectively. The contribution to ′ from interband transitions only acts as a positive background, changing in some way the
environment of the free electrons and then the location of the SP resonances.4 The imaginary part of inter contributes to the absorption process, sometimes overlapping the wavelength range of the SP resonances of Au and Cu NPs, particularly in Cu. However, they do not make a difference for Ag since SP resonances are usually at larger wavelengths than 320 nm. The behavior of inter explains the different contributions of absorption and scattering to the same SP resonances for NPs of different metals as shown in Figure 2. Figure 3 shows the extinction efficiency for several seeds and elongated NPs made of Ag (blue), Au (green), and Cu (red), where the (′) character refers to elongated NPs. All panels show the average extinction over different orientations, as it would be found in dilute colloidal systems, where interactions between
Optical Properties of Noble Metal Nanoparticles particles are negligible. This allows us to model the colloidal absorbance as the optical response of one suspended particle times the concentration of particles.5 Let us start with the simplest case to then compare with more complex morphologies. Panels a and a′ of Figure 3 show the extinction by spheres and prolates, respectively. The spectra of the Ag sphere shows two peaks, each of them associated with the dipolar and quadrupolar SP resonances, located at 478 and 395 nm, respectively. However, in the case of Au and Cu, we can only identify the dipolar resonance located at 574 and 598 nm, respectively, since the quadrupolar resonances, being less intense than dipolar ones, are hidden by the interband transitions. In general, at larger wavelengths the spectra for Ag NPs show SP resonances that, in this case, can be easily identified. On the contrary, Au and Cu have interband contributions at larger wavelengths, as we can see in Figure 2, resulting into an overlapping of the SP resonance with them. Furthermore, the damping due to the dispersion of the electrons is 1 order of magnitude more important for Au and Cu than for Ag, such that, it is very difficult to recognize the SP resonances from the extinction spectra, being even worse for Cu than for Au. For Ag, it is well-known that the extinction below 325 nm follows the same behavior, independently of the dielectric properties of the surrounding media.50 As we can deduce from the dielectric function, this results from the fact that, at this wavelength and below, the main absorption mechanism is due to the interband transitions only.4 Therefore, this structure is also independent of the morphology of the Ag NPs.48 For elongated NPs, due to its symmetry, their optical response depends on the direction of the incident field. If the electric field is parallel to the major axis, only a few SP resonances at large wavelengths are excited, which are commonly called longitudinal modes (LMs). While if the field is perpendicular to the major axis, then only SP resonances located at small wavelengths are excited, known as transversal modes (TMs). A detailed analytical analysis of the behavior of prolates and oblate NPs can be found in ref 4. The anisotropy in the optical response observed is a consequence only of the shape of the particle. For Ag elongated NPs, we can appreciate both, LMs and TMs; but for Au and Cu elongated NPs, due to the interband effects, we can appreciate LMs, but the TMs are more difficult to observe. Panels b, c, d, and e in Figure 3 show the extinction of dodecahedral, decahedral, octahedral, and cube seeds, respectively, of Ag, Au, and Cu. It is worth remarking that, contrary to the case of the sphere, the extinction spectra of these seeds show a richer structure of peaks, and since the SP resonances are rather separated among them, the spectra become wider. Each one of these SP resonances can be associated with different charge distributions on the seed that depend on the number of the NP faces and recently symmetry as shown in refs 48 and 49. In the case of Ag, it is rather easier locating the SP resonances for each geometry, observing that the number of SP resonances increases when the number of faces decreases. Nevertheless, for Au and Cu, it is rather difficult determining the position of the SP resonances, again as a consequence of interband effects. Finally, panels b′, c′, d′, and e′ in Figure 3 show the extinction spectra of hexagonal, pentagonal, tetrahedral, and rectangular prisms, respectively, of Ag, Au, and Cu, where we observe LMs and TMs in all the cases. In general, in the LMs, we can appreciate a single SP resonance, which is blue-shifted as the number of faces of the NP increases. In fact, the position of this LM is very sensitive to the NP’s morphology. For instance,
J. Phys. Chem. C, Vol. 112, No. 19, 2008 7359
Figure 4. Experimental setup for birefringence measurements.
Figure 5. Index ellipse associated to the anisotropic nanocomposite.
the position of LMs of prolate NPs is ∼80 nm to the left as compared to the same LMs of rectangular or tetrahedral prisms with the same aspect ratio, independently of the dielectric properties of the elongated NP. The TMs also contain information about the geometry of the prism; this is clear when comparing the TM of a prolate to those of the rectangular prism, which shows a structure similar to the cube. Furthermore, TM of prisms is wider as the number of faces decreases, showing also a larger number of SP resonances;49 this is evident for Ag, but interband effects do not allow us to see that clearly for Au and Cu. Therefore, the position of LMs and TMs not only depends on the aspect ratio, but also on the NP’s shape. Next, we will analyze the implications of this anisotropy on the form birefringence of a macroscopic sample. Macroscopic System with Oriented Anisotropic NPs: Birefringence When synthesizing anisotropic NPs, their orientation is also a major issue that has to be controlled in order to get a related macroscopic anisotropic optical response and a potential technological application. Within this objective, the synthesis of oriented Ag nanoellipsoids was achieved,8 where the optical anisotropy or birefringence of the sample has been measured recently29 using an ellipsometric technique.51 Here, we show the theoretical analysis to interpret these ellipsometer measurements. The experimental setup employed in ref 29 is shown in Figure 4, which measures the birefringence experienced by the probe beam, ∆nR ) np(ψ) - ns, when traversing the sample. Here, the subscripts s and p refer to the linear optical eigenpolarization components of the probe beam, when propagating in direction k parallel to the sample’s normal, N ˆ sample, but makes an internal angle ψ of propagation to the normal of the symmetry axis of the NP, N ˆ NP, as shown in Figure 5. The corresponding refraction indices are ns and np, and the superscript R represents the angle between the incident electric field and the x axis, as shown in Figure 5. The linearly polarized incident light contains both of the linear eigenpolarizations, while the analyzer may be oriented parallel or crossed to transmit the linear polarization state parallel or orthogonal to the incident light, respectively. With respect to the axis system of Figure 5, the NP normal can be represented by the vector N ˆ NP ) (sin ψ, 0, -cos ψ).
7360 J. Phys. Chem. C, Vol. 112, No. 19, 2008
Gonza´lez et al. where β ) R + θ represents the angle of the analyzer from x toward y. Then, if θ ) 0, the analyzer is oriented parallel to the polarizer, and all the light linearly polarized is transmitted; on the contrary, if θ ) π/2, the analyzer is crossed with respect to the polarizer, and the light linearly polarized is filtered by the analyzer, making negligible the measurement of the transmitted light. Therefore, the light detected beyond the analyzer has s- and p-polarizations components given by the dot products
ETs (R, θ) ) E′s‚A ˆ (β),
ETp (R, θ) ) E′p‚A ˆ (β) (5)
and
Then,
ETs (R, θ) ) As exp
(
Figure 6. Transmitted intensity through the sample when the analyzer is parallel or perpendicular to the polarizer.
The direction of the s-eigenpolarization is perpendicular to the incidence plane, which is defined by N ˆ NP and the z axis: s ) N ˆ NP × kˆ ) (0, -sin ψ, 0). On the other hand, by rotating s around the z axis by π/2, such that s‚p ) 0, one gets the direction vector of p-polarization: p ) (sin ψ, 0, 0). After normalization, these vectors become
()
0 sˆ ) 1 , 0
()
1 pˆ ) 0 0
and
(2)
E′s(R) ) As exp
(
)
-iπL∆n λ
(
)
Essˆ
(3)
iπL∆nR Epsˆ λ
(4)
and
E′p(R) ) Ap exp
)
-iπL∆nR Essˆ‚A ˆ (β) λ
)
(6)
-iπL∆nR sin R sin β λ
and
ETp (R, θ) ) Ap exp ) Ap exp
(
(
)
iπL∆nR Eppˆ ‚A ˆ (β) λ
)
(7)
iπL∆nR cos R cos β λ
Consequently, the total field at the detector can be written as
E(R, θ) ) ETs (R, θ) + ETp (R, θ)
Since the probe beam propagates parallel to the normal of the macroscopic sample, k|N ˆ sample, which makes a ψ1 angle with N ˆ NP, one can assume that the light is refracted just until it arrives to the NP’s surface according to Snell’s law neff NP sin ψ ) nhost sin ψ1, where neff is the complex effective index of refraction NP of the nanoparticle at a given wavelength, while nhost is the refraction index of the host matrix, which is isotropic. In a similar way, the orientation of the incident electric field, E ˆ in, will be determined by the polarizer orientation, which allows us to resolve it into the two eigenpolarization components of the nanocomposite, Es ) E ˆ in‚sˆ ) sin R and Ep ) E ˆ in‚pˆ ) cos R, where E ˆ in ) (cos R, sin R, 0) is the unitary incident electric field and R is the angle of the polarizer from x to y. Therefore, the birefringence for such a light path will be ∆nR ) np(ψ) ns, where for p-polarization experiences a refraction index np, while for s-polarization experiences ns. To relate this birefringence to the measured transmitted intensity, we can define complex transmission factors for these two eigenpolarizations, which correspond to the measured transmitted electric field, as R
) As exp
(
where As and Ap are the measured amplitude transmissions factors for each eigenpolarization, L is the interaction length, i.e., the thickness of the NPs layer, and λ is the free-space incident wavelength. For the transmitted light, the analyzer is generally oriented in the direction A ˆ (β) ) (cos β, sin β, 0),
) As exp
(
)
-iπL∆nR sin R sin β + λ
(8)
(
Ap exp
)
iπL∆nR cos R cos β λ
For θ ) 0, that is R ) β, the detected intensity, I(R, 0) ) |E(R, 0)|2 is given by
I(R, 0) ) As2 sin4 R + Ap2 cos4 R + 2πL∆nR 1 AsAp sin2 2R cos (9) 2 λ while, for θ ) π/2, the detected intensity, I(R, π/2) ) |E(R, π/2)|2 ,is
( )
I R,
{
}
π 1 2πL∆nR ) sin2 2R As2 + Ap2 - 2AsAp cos 2 4 λ
(10)
Figure 6 shows the intensities according to eqs 9 and 10, obtained for a system consisting of oriented Ag nanoellipsoids, as obtained in ref 8, and considering a birefringence of 0.135, as measured in ref 29. From these curves, one is able to obtain the amplitudes Ap2 and As2, for R ) 0 and R ) π/2, respectively. On the other hand, from eq 10, with R ) π/4, one gets the maximum measured birefringence as
∆nmax )
[
]
2 2 2Imax λ A s + Ap meas 2πL 2AsAp AsAp
(11)
Given the NP’s geometry and orientation, it is evident the nanocomposite’s uniaxial symmetry, which allows us to describe the refractive index anisotropy by the ellipse shown in Figure
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J. Phys. Chem. C, Vol. 112, No. 19, 2008 7361
TABLE 1: Refractive Index Anisotropy and Birefringence (ne - no) for Metallic Anisotropic Nanocomposites of Elongated and Oriented NPs of Ag, Au, and Cu with Different Geometrya prolate neff NP Ag Au Cu a
pentagonal prism n e - no
(SP/nm)
4.84 (711) 1.86 (385) 4.54 (777) 2.17 (519) 4.67 (765) 2.77-2.79 (560-567)
neff NP
0.105 0.145 0.105 0.130 0.105 0.116
(SP/nm)
5.24 (775) 2.657 (447) 5.08 (831) 2.492 (537) 5.19 (819) 2.79 (575-590)
rectangular prism n e - no 0.104 0.118 0.104 0.121 0.104 0.116
neff NP
(SP/nm)
5.24 (777) 3.32 (509) 5.08 (831) 3.28 (609) 5.19 (819) 2.79 (591)
n e - no 0.104 0.111 0.104 0.111 0.104 0.116
SP resonances correspond to those shown in Figure 3,
5 with principal axes ne and no, i.e., the extraordinary and ordinary indices, respectively. For a refracted light path at an angle ψ relative to N ˆ NP, we have
[ ]
1 2 sin2 ψ cos2 ψ ) + np(ψ) no2 ne2
(12)
for the p-polarized light,52 whereas s-polarized light sees an ordinary index no. In consequence, the refractive index anisotropy of the nanocomposite will be related to the measured birefringence at first-order approximation by
n e - no )
∆nR ) cos2 ψ
∆nR nhost cos2 sin-1 eff sin ψ1 nNP
[ (
)]
(13)
refractive index anisotropy, ne - no, calculated with eq 13 for metallic anisotropic nanocomposites when supposing an angle of 45° between the normal to the sample and N ˆ NP (see ψ1 in Figure 5) and an aspect ratio of 2.8. A brief analysis of this table allows concluding that, in general, given like conditions on the interaction layer, the size and the density of the NPs, and in consequence on the measured birefringence at the proper wavelength, the refractive index anisotropy is rather similar for all the three metals and the geometries considered. It can be also observed, and is clearer for the minor axis, that this index anisotropy decreases with the number of edges of the elongated NP. However, we should notice that our theoretical analysis does not take into account the geometrical differences among nanoparticles, which could influence the results somehow. Summary and Conclusions
This previous analysis has been made under the assumption that the wavelength used to perform the measurements is close to the surface plasmon resonance associated with the major axis of the oriented prolate NPs. In such a case, it is rather clear that the index ellipse’s axis superposes with those of the NPs. However, when exciting its minor axis, the index ellipsoid is perpendicular to the NP’s major axis. Nevertheless, the analysis is exactly the same, although the respective (x′, y′, z′) system is rotated π/2 around the z with respect to the previous one, shown in Figure 5. This means that the new angle R′ for this last case makes an angle R ) π/2 - R′ with respect to the former (x, y, z) system, and since we used the same setup for both wavelengths, in order to model properly our measurements, we have to substitute R′ for R. Note that the following relations hold: sin R′ ) cos R, cos R′ ) sin R, and sin 2R′ ) sin 2R, and in consequence, for a external field with a given wavelength exciting the minor axis, the typical measurement is a reflection with respect to a vertical axis located at π/4 of the curves shown in Figure 6. The other difference is that, now, for R ) 0, eq 14 gives As2, while for R ) π/2, it gives Ap2, and ∆nmax is obtained again from eq 11. The above considerations only affect eq 9, transforming it into
We have shown the importance of elongation of NPs into their optical properties by making a revision of these effects for a variety of common shapes currently obtained in several laboratories. We do a systematic study of the richness of the optical response as well as the position of the resonances of a NP, depending on its geometry. At the same time, within the wide palette offered by all the cases shown, depending on the application wanted, one might also use the interband transitions as a modulating tool. As a consequence of the elongation, the resultant nanocomposites are susceptible of presenting optical birefringence. In this sense, we have presented a general analysis to determine this birefringence for ellipsoidal NPs. This analysis is easily extended to other uniaxial oriented structures, observing that the index anisotropy depends on the number of edges of the NP, decreasing as it does.
I(R′, 0) ) I(R, 0) ) As2 cos4 R + Ap2 sin4 R +
References and Notes
2πL∆n 1 A A sin2 2R cos 2 s p λ
R′
(14)
To compare the refractive index anisotropy for metallic NPs, namely, Ag, Au, and Cu, we can suppose a measured birefringence of 0.1 at the wavelength corresponding to the SP resonance associated with the major or the minor axes of the elongated NP, as reported in ref 29. In such a case, we consider the corresponding complex effective index of refraction of the nanoparticle, neff NP, as the one taken from ref 34 at the proper wavelength as explained above, and the index of refraction of the host as that standing for silica, 1.47. Table 1 shows the
Acknowledgment. C.N. and A.L.G. acknowledge partial financial support by CONACyT-Me´xico grant 48521, and DGAPA-UNAM grant IN106408. J.A.R.-E. acknowledges partial financial support from CONACyT grant 42823-F, and DGAPA-UNAM grant IN108807-3. Computer resources from DGSCA-UNAM are gratefully acknowledged.
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