Surface Plasmons on Metal Nanoparticles - UNAM

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Feature Article J. Phys. Chem. C, Vol. 111, No. 10, 2007 38111:101:20TABLE 1: Eigenvalues of the SPRs of Prolates and Oblates Spheroids of Different Aspect Ratiosa:c1:21:41:61:8s n L x,L z L x,L z L x,L z L x,L z L x,L z L x,L zprolate 0.1735, 0.4132 0.0754, 0.4623 0.0432, 0.4784 0.0284, 0.4858 0.0203, 0.4898 0.0067, 0.4966oblate 0.2363, 0.5272 0.1482, 0.7036 0.1077, 0.7846 0.0845, 0.8308 0.0695, 0.8608 0.0369, 0.9262In this case, we can easily identify the geometrical factors withthe eigenvalues of the surface proper modes of the spheroid, s n) L γ , and their strength are C γ ) 1/3. Thus, the eigenvaluesare independent of the material properties of the ellipsoid, andthe SPRs are given by the poles of eq 9. Taking into accountthe definition of the spectral variable, we find that these SPRsfulfill the condition ɛ(ω)L γ + ɛ h (1 - L γ ) ) 0. Now, it isnecessary to calculate the depolarization factors L γ .For simplicity, let us consider ellipsoids generated by therotation of an ellipse around its major or minor axes, whichproduce prolate or oblate spheroids, correspondingly. Sinceprolate and oblate NPs have a symmetry axis, they have threeproper modes, where two of them are degenerated. Thegeometrical factors L γ for prolate spheroids (a > b ) c) are 74L x ) 1 - e22e 3 (log 1 + e1 - e - 2e )L y ) L z ) 1/2(1 - L x ) and e ) 1 - b 2 /a 2 (10)Whereas for oblate spheroids (a ) b > c), they areL z ) 1 + e2e 3 (e - tan -1 e)L x ) L y ) 1/2(1 - L z ) and e ) a 2 /c 2 - 1 (11)For a sphere (a ) b ) c), it is evident that the depolarizationfactors are all degenerated, and to satisfy the sum rule condition,they must be equal to L x ) L y ) L z ) 1/3, and the SPR fulfillsthe well-known expression ɛ(ω)L γ + ɛ h (1 - L γ ) ) ɛ(ω) + 2ɛ h) 0.In the previous section, we found a general behavior of theSPRs, where a red shift of the modes is found when the particleis immersed in a host media, and when interband electronstransitions are present. To make a closer analysis of the SPRsof elongated NPs and without loss of generality, let us assumethat the NP is immersed in vacuum and its dielectric functiondoes not contain contributions from interband transitions, andΓ , ω p , such that ω n ) ω p s n . In this case, the frequency ofthe resonance for the sphere is given by the well-knownexpression ω n)1 ) ω p 3. Now, let us analyze the case forprolate particles when the semi-axis a goes to ∞, and a cylinderor needle with its axis along x is obtained. Then, the depolarizationfactors are L x f 0, L y ) L z f 1/2, such that, as aincreases, the mode along the symmetry axis is red-shifted untilit ceases to be visible (zero frequency), while the two perpendicularmodes converge to ω n ) ω p 2. The other limit case isfor oblates when a and b f ∞ that corresponds to a flat plateor disk, and the depolarization factors L x ) L y f 0 and L z f1. Here, the two identical modes along the symmetry axis cease,since their resonances go to zero as a and b f ∞ while theperpendicular one ω n f ω p . In Table 1, we show the depolarizationfactors for prolate and oblates spheroids as a functionof the aspect ratio a:c. We observe that for an aspect ratio a:c) 6:1 (1:6) for prolates (oblates), the limit cases mentionedpreviously have been already attained within a 10%.B. Application to Gold and Silver NPs. To exhibit thepotentiality of the spectral representation, here, we perform asystematic study of the SPRs of ellipsoidal gold and silver NPsthat are immersed in different media. In the following, we definethe aspect ratio of the NP as the length of the major axis dividedby the width of the minor axis. We will also label the modes aslongitudinal (LM) when they are along the symmetry axis ofthe particle, whereas transversal modes (TM) are those excitedin the perpendicular direction. Let us consider gold (Au), andsilver (Ag) NPs with theoretical pω p ) 8.55 and 9.2 eV,respectively. The NPs are embedded in a medium with refractionindex n ) ɛ h . For visible wavelengths, the contributions ofinterband transitions can be approximated by a constant, ɛ inter= 9.9 for Au and ɛ inter = 3.9 for Ag. With these parameters,we calculate the dielectric function according to eq 4, and usingthe eigenvalues n s from Table 1, and substituting in eq 7, thefrequency of the LM and TM modes of prolate and oblates NPscan be obtained with minimum numerical effort.In Figure 2, we plot the position of the resonance as a functionof the aspect ratio for gold prolate (left side) and silver oblate(right side) NPs, embedded in vacuum with n ) 1, water/glycerol with n ) 1.3, dimethyl sulfoxide or silica glass with n) 1.47, sapphire with n ) 1.77 or TiO 2 with n ) 2.79. Ingeneral, we confirm that, as the refraction index of the hostmedia increases, all of the modes are always red-shifted. TheLM of prolate particles behaves as the TM of oblate NPs, wherethe position of the modes is displaced to larger wavelengths asthe aspect ratio increases, although the shift is smaller for oblatesthan for prolates. In the same way, the TM of prolates behavessimilarly to that of the LM of oblates, but now, the modes areshifted to smaller wavelengths when the aspect ratio increases,and this shift is larger for oblates than for prolates. We observethat the LMs and TMs of prolates and oblates, respectively,show a linear behavior independently of the refraction index n,where the slope is modified by n as well as by the particularshape. For instance, the slope of the LMs of prolates is largerthan that of the TMs of oblates. On the other hand, the TM(LM) of prolates (oblates) shows a behavior inversely proportionalto the aspect ratio. The TMs of prolates are always blueshifted,and for aspect ratios between 1 and 3, it occurs rapidly,whereas from 4.5 and larger the limit value is almost reached.Furthermore, the limit value of LMs of oblates is slightly shiftedto large wavelengths as n increases, whereas for prolates thelimit value of TMs is more susceptible to the value of n.The linear behavior of the modes of elongated NPs has beenobserved experimentally and theoretically in nanorods andnanodisks. 59-67 In some of these works, phenomenologicalequations to determine the position of the resonances have beenproposed. However, these empirical models can be applied onlyto a specific system, where the slope of the linear equationdepends on the solution where the particles are dispersed. Onthe other hand, using the expressions for the eigenvalues givenin eqs 10 and 11, we find an exact dependence of the SPRposition as a function of the aspect ratio. For instance, let usconsider the case when the spheroid is nearly spherical (e ,1), such that, the longitudinal eigenvalues of prolates and oblatesNPs are approximately 74
3812 J. Phys. Chem. C, Vol. 111, No. 10, 2007 Noguezs prolate LM ) 1 3 - 215 e2 , and s oblate TM ) 1 3 + 215 e2respectively. Since e 2 is inversely proportional to the aspect ratiofor prolates, and proportional for oblates, we find from eq 7that the frequencies of LMs for prolates are inversely proportionalto the aspect ratio, whereas they are proportional to it foroblates. Also from eq 7, we find that all modes are inverselyproportional to ɛ h . Now, considering that the wavelength isinversely proportional to the frequency, λ ) c/ω, we obtain thatthe position of the LMs of prolates is proportional to the aspectratio, while for oblates is inversely proportional. Additionally,the modes are proportional to the refraction index n in all cases.These explain the behavior of the resonances in Figure 2, aswell as the observations of experimental and theoreticalworks. 59-67 Notice that deviations from a prolate ellipsoidalshape have effects on the optical properties of nanorods. 64,67C. Aligned Elongated Nanoparticles. To show the sensitivityof anisotropic NPs to polarized light, we simulate the opticalabsorbance of prolate spheroids with a small aspect ratio of 1.6and a major axis of 8 nm, which are embedded in silica (n )1.47). The optical absorbance of the nanocomposites has beencalculated by controlling the angle θ of the wavevector kB ofthe incident electromagnetic field with respect to the major axisof the NP. The polarization of the incident electric field wasvaried at different angles φ with respect to the minor axis andperpendicular to kB. In Figure 3, panels a and b, the simulatedabsorbance spectra are shown for the incident electromagneticfield at θ )-45° and 90°, respectively, and different angles ofpolarization. When θ )-45° in Figure 3a, it is observed thatfor φ ) 0° the electric field is along the minor axis excitingonly the surface plasmon at 375 nm. Conversely, when the angleof polarization is φ ) 72° , both resonances are excited, butthe one at 375 nm is weaker than the resonance at 470 nm.Similarly, when θ ) 90° in Figure 3b, the wavevector kB isalmost aligned to one of the minor axes, and as a consequence,the electric field is polarized along the other minor axis at φ )0° and almost along the major axis at φ ) 72°. Finally, if weconsider that θ ) 0° (not shown in the figure), the wavevectorkB is along the major axis, in such a way that the electric fieldmostly excites the resonance at 375 nm for any polarization.Similar conclusions have been found recently for metal nanorods.71Recently, the shape and alignment of metallic NPs embeddedin insulator matrices have been controlled using MeV ion beamirradiation. 69,70 Symmetric NPs were transformed into anisotropicparticles whose larger axis is along the ion beam. Uponirradiation, the surface plasmon resonance of symmetric particlessplit into two resonances whose separation depends on thefluence of the ion irradiation. 70 Simulations of the opticalabsorbance showed that the anisotropy is caused by thedeformation and alignment of the nanoparticles and that bothproperties can be controlled with the irradiation fluence. 70V. Shape Influence on the <strong>Surface</strong> <strong>Plasmons</strong>In the case of metal NPs, many results indicate the presenceof polyhedral shapes with well-defined facets and vertices, likeicosahedral (IH) and decahedral (DH) NPs, as well as fcc relatedmorphologies like cubes and truncated cubes (TC). 28,29,57,75-79To understand the influence of morphology, the SPRs forpolyhedral NPs have been recently studied. 80 A general relationshipbetween the SPRs and the morphology of each NP wasestablished in terms of their vertices and faces. The opticalresponse was investigated for cubes and DHs, as well as fordifferent truncations of them. 80 Here, we show results for cubicand DH silver NPs whose volume is equal to that of a spherewith a radius of 2.2 nm, which are immersed in a media witha refraction index n ) 1.47. The extinction efficiencies, Q ext ,were calculated using DDA with the order of 10 5 polarizableentities, which ensure the convergence of the optical responsefor each NP. We employ the measured bulk dielectric functionfor silver by Johnson and Christy, 49 which is modified accordingto eq 4 to incorporate the surface dispersion effects.A. Cubic Morphology. We first study Q ext for a nanocube,and then, we compare it to those obtained for different TCs,the IH, and the sphere. In Figure 4, we show Q ext for a silvernanocube immersed in a medium with n ) 1.47 (solid line)and in vacuum (dashed line). In both spectra, we observe thatthe optical response below 325 nm follows the same behaviorindependently of the dielectric properties of the surroundingmedia, since at those wavelengths and smaller the mainabsorption mechanism is due to the interband transitions.Therefore, this structure should also be independent of themorphology of the NP, as we will show later when we compareQ ext for different TCs. At larger wavelengths, both spectra showFigure 2. Positions of LM and TM surface plasmon resonances of gold prolate (left side) and silver oblate (right side) NPs with different aspectratios and embedded in various media. The lines are just for guidance.
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