ARTICLE pubs.acs.org/JPCC
Optical Absorption of Plasmonic Nanoparticles in Presence of a Local Interband Transition Tavakol Pakizeh* Faculty of Electrical and Computer Engineering, K. N. Toosi University of Technology, Tehran 16315, Iran, and Department of Applied Physics, Chalmers University of Technology, G€oteborg S-412 96, Sweden ABSTRACT: The influences of a local interband transition on optical absorptions of plasmonic nanoparticles are investigated. It is found that the optical absorption associated with a local interband transition is remarkably enhanced because of the presence of a localized surface plasmon resonance. On the basis of the quasistatic theory, Mie calculations, and the classical models, the electromagnetic interaction of plasmon- and interband-dipole is studied. It is shown that two modes with different characters are generated because of the strong inherent interaction. As a consequence of this interaction, the two modes are repulsed, and an avoided crossing zone is constructed where the particle plasmon energy coincides with the energy of a local interband transition. The energies of the two modes are interchanged as the crossing zone traversed.
1. INTRODUCTION It is known that optical properties of metal nanoparticles are mainly characterized by the collective oscillation of their free electrons, so-called the localized surface plasmon resonance (LSPR).13 LSPRs have been the subject of enormous research in nanooptics, aiming at manipulating and tailoring the optical properties of nanostructures and nanoparticles in different applications such as nanobiosensors,46 photonics nanodevices and nanoantennas,4,7,8 photovoltaic,9 magneto-optics,10 and optical metamaterials.11,12 In general, the plasmon resonance of a single metallic nanoparticle is affected by the size, geometry, and surrounding environment as well as the constituent mater.2,1315 The latter certainly affects the energy and broadening or the quality factor of the plasmon resonance. Besides, it has been recently shown that the electronic structure of the constituent metal considerably affects the line shape of LSPR.1618 For instance, the broad optical responses with asymmetric line shapes are observed for the Pd nanoparticles. In this case, the free electrons region is covered by a structureless background, describing the overlapped interband (IB) transitions along the interested energy range. Interestingly, this exhibits a Fano-type resonance19 due to interference between the broad LSPR and the background.16,17 This type of resonance is also observed in the strong interaction of LSPR with excitons.20,21 In contrast, the optical constants of some other metals such as nickel (Ni), iron (Fe), copper (Cu), and aluminum (Al) contain features that can be assigned to the local IB transitions. For instance, these features are seen at ∼4.7 eV for Ni, ∼1.5 eV for Al, ∼2.1 eV for Cu, and ∼2.5 eV for Fe.22,23 These IB transitions may considerably affect on the plasmonic properties of nanoparticles. Accordingly, it has been shown that the optical extinction of Al nanodisks contains features that are attributed to the local IB transition at ∼1.5 eV. 24 Furthermore, it has been reported that when a r 2011 American Chemical Society
plasmon overlaps the IB transition threshold, a doublepeaked resonance appears.25 The geometric control of the LSPR energy in the Cu nanoshells allowed investigation of the electronic IB-transition influences on the optical resonances. In this contribution, the electromagnetic interference of an LSPR with a local IB transition is investigated in a single nanoparticle. It is shown that the plasmonic properties are remarkably influenced because of the presence of a local IB transition. In particular, the optical absorption associated with the local IB transition is remarkably enhanced because of the interaction of the IB transition and a dipolar LSPR. On the basis of the quasistatic theory, Mie calculations, and the classical models, the interaction of plasmon- and IB-dipole is studied. It is demonstrated that two modes are generated and repulsed because of the inherent interaction. Consequently, an avoided crossing zone is constructed where the particle plasmon energy coincides with the energy of a local IB transition. Moreover, the energies of the two modes are interchanged over traversing the zone.
2. THEORETICAL BASES AND MODELS The dielectric constant or permittivity of a metal can be described using a classical model consisting of a Drude term and a few Lorentzian terms.14 The former term is deduced from the free-electrons behavior, and the later models the electromagnetic response of bound electrons. The presence of a local IB transition can be well-described by the classical Lorentzian oscillator model. Throughout the Article, this combined model, expressed in eq 1, is called the Drude-Lorentz (DL) model, and the nanoparticle described by the DL model is called DL Received: August 3, 2011 Revised: September 2, 2011 Published: September 08, 2011 21826
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nanoparticle. εðωÞ ¼ 1
ω2p ωðω þ iγc Þ
þ
G0 ω20 ω20 ω2 iΓω
ð1Þ
where ωp and γc are the plasma frequency and the collision rate in the Drude term and ω0, Γ, and G0 are the central frequency, damping factor, and gain of the Lorentzian term, respectively.14 The Drude function, the first two terms, is responsible for an LSPR, and the Lorentizan function, the last term, describes the response due to an IB transition at energy ω0. The optical response of a metal nanospheroid with a dispersive and complex dielectric constant ε = Re(ε) + i Im(ε) immersed in a medium εm can be obtained by using the quasistatic theory. The optical absorption cross section σabs = k Im(α) is proportional to the polarizability of the nanoparticle αðωÞ ¼ V
ðε εm Þ ½εm þ Lðε εm Þ
ð2Þ
where V is a constant that represents the volume of the nanoparticle with a shape factor L.14 The optical absorption efficiency is Qabs = σabs/A, where A is the area defined by geometrical optics. According to the polarizability α of a particle, in the quasistatic dipole approximation, an LSPR is approximately defined by the criteria Re(εm + L(ε(ω) εm)) = 0, and for a spherical nanoparticle in air (L = 1/3, εm = 1), an LSPR is defined by the Fr€ohlich condition Re{ε(ω)} = 2.3,14 Two new resonances may be formed due to the interaction or interference of a LSPR and a local IB transition. The new resonances can be obtained by inserting eq 1 into eq 2. For a spherical nanoparticle, immersed in air (εm = 1), the resonances should satisfy the following equation RðxÞ ¼ A3 x3 þ A2 x2 þ A1 x þ A0 ≈ 0
ð3Þ
where x = ω2, and 8 > A3 ¼ 3 > > > < A2 ¼ 3Γ2 ω2 ω2 ðG0 þ 6Þ p 0 A1 ¼ ω40 ðG0 þ 3Þ þ ω2p ð2ω20 Γ2 Þ > > > > : A4 ¼ ω2 ω4 p 0 This polynomial equation (eq 3) is obtained by simplifying the Drude model (eq 1). This is done by ignoring the collision rate of free electrons, γc = 0. It is worth mentioning that the roots of eq 4, which are energies of the new modes, could be complex numbers because of the damping factor Γ. In this case, the local minimums of the left-hand side of eq 3 (min{|R(x)|}) are considered to be the energies of the coupled LSPR-IB resonances. The expression in eq 3 can be further simplified by approximating the Lorentzian term in eq 1. Therefore, at the vicinity of the IB transition energy, that is, for ω ≈ ω0, G0ω02/[ω02 ω2 iΓω] ≈ G0ω0/[ω0 ω iΓ], eq 3 becomes RðωÞ ¼ 3ω3 ω0 ð3 þ G0 Þω2 ω2p ω þ ω2p ω0 ¼ 0
ð4Þ
Considering only the Drude term in eq 1, by assuming G0 = 0, the valid resonance energy obtained from eq 4 or eq 3 is the expected LSPR energy of a Drude nanosphere, that is, ωLSPR = ωp/31/2, referring to the Classius-Mossotti relation.2
√ Figure 1. Dielectric constant: (a) Composed DL-model with ωp = 3 3 eV, γc = 0.5 eV, G0 = 0.25, ω0 = 3.3 eV, and Γ = 0.5 eV; (b) Drude model; and (c) Lorentz model.
3. RESULTS AND DISCUSSIONS To investigate theoretically the influences of a local IB transition on the optical response of a plasmonic nanoparticle, we used eqs 14. The energy-dependent real and imaginary parts of the permittivity or dielectric constant, expressed in the DL model (eq 1), are shown in Figure 1a. The dielectric constant can be decomposed to the described Drude (Figure 1b) and Lorentz (Figure 1c) parts. The response that is proportional to Im{α(ω)} takes its maximum values at the energies where the main part of the denominator, R(x), is zero or has a local minimum value along the interested energy range. When an LSPR and a local IB transition interfere, two distinguishable local minimums can be mathematically obtained. Figure 2 shows the roots and minimums of |R(x)| for two cases based on eqs 3 (thin lines) and 4 (thick lines). In the first case, it is assumed that G0 ≈ 0 (Drude metal), shown with the dashed lines. For the second case, G0 = 0.25, the dipolar LSPR and the local IB-transition interfere, and thereby two distinguishable minimums are clearly observed, as denoted by the solid lines in Figure 2. However, both curves obtained from eqs 3 and 4 show similar trends; their minimums are at different values. This might be expected because a rough approximation has been made in obtaining eq 3. The new minimums noticeably separated from the primary roots at ωLSPR and ω0, where ωLSPR denotes the LSPR energy due to the contribution of free electrons and ω0 is the central energy of a local IB transition. Therefore, the interaction of an LSPR and a local IB transition 21827
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Figure 2. Roots or minimums of R(ω) obtained based on eqs 3 (thick lines) and 4 (thin lines) with G0 = 0.25 (solid lines) and 0 (dashed lines) and Γ = 0.5 eV.
Figure 3. Energies of the two coupled modes, that is, the LSPR-like mode (green thick line) and the IB-like (red thin line) mode, when the IB transition energy ω0 is varied (a), assuming ωLSPR = 3 eV and G0 = 0.25, and when the LSPR energy ωLSPR is varied (b), assuming ω0 = 3 eV and G0 = 0.25.
in a single nanoparticle leads to two new coupled energies, different from their primary values (ωLSPR and ω0) in the noninteracting case. More precisely, stronger interaction of the coupled resonances exhibits the wider separation or repulsion of energies of the new hybridized resonances. This property is inherent to the hybridized and strongly coupled systems in different configurations,12,2628 although the described interaction can barely be understood as an interaction of two
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Figure 4. Optical absorption of a DL nanosphere immersed in air (εm = 1). Varying the LSPR energy ωLSPR = 1 (dashed-dotted line), 2.8 (blue dashed line), 3.0 (thick solid line), and 3.2 eV (green solid-line); γc = 0 eV, G0 = 0.25, Γ = 0.5 eV, and ω0 = 3 eV. The optical IB absorption is remarkably enhanced because of the presence of an LSPR of ∼3 eV compared with the noninteracting IB transition case (thick dashed line) and the very weak interaction case (ωLSPR = 1 eV).
separate phenomena. Nevertheless, in the quasistatic limit, it is convenient to consider the interaction of a plasmon dipole and an IB dipole, leading to a meaningful hybridized system. In Figure 3a, the energies of two peaks in optical response, associated with the new coupled modes, are plotted versus the IB√transition energy (ω0) at a given plasma frequency, ωp = 3 3 eV. Therefore, it is assumed that ωLSPR of the considered metal nanosphere is 3 eV, for which the IB transition is ignored. In the noninteracting case, the energies of the LSPR and IB transition follow the dashed lines depicted in Figure 3. However, their interaction leads to the different behaviors, as shown in Figure 3. The thick lines are associated with the LSPR-like energy, and the thin lines are associated with the energy of the IB-like mode. The influence of the IB transition is remarkable when its energy approaches to the considered LSPR energy, that is, 3 eV. At the vicinity of this energy, there is ambiguity to define the LSPR because it is overdamped by the high probability of the IB transitions (electron energy loss). Hence the collective-oscillation of free electrons may be overdamped, and thus an avoided crossing zone for the resonance is constructed at those energies of coincidences of the two phenomena. This situation is reminiscent of diatomic avoided crossing (as a function of bond distance) with nonadiabtic interactions. Roughly, this zone is denoted by the central gray area in Figure 3. There, by increasing ω0 in Figure 3a, an interesting switching process occurs for the hybridized LSPR-like energy (green thick line) from the higher energy side (>3 eV) to the lower energy side ( ω0. In the next step,
for the same DL nanoparticle, the energy of LSPR is tuned based on its dependence to the surrounding medium. This particular energy tuning allows us to control and manipulate the LSPRIB interactions and its influences on the optical absorption of a DL nanoparticle. For this purpose, the refractive index of the surrounding medium is varied from nm =1.0 to 2.2 in Figure 7 to tune effectively the LSPR energy. However, the LSPR energy considerably red-shifted by increasing nm (Figure 7c), and the energy shift of the IB absorption is negligible. (See Figure 7d.) Therefore, the IB absorption is less-sensitive to the environment. It is seen that the LSPR energy approaches 3 eV for nm = 1.5 (solid lines, Figure 7b). In this case, the optical IB absorption is remarkably enhanced because of the LSPR-IB interaction, shown by the solid line in Figure 7b. As also seen in Figure 5, the two excited modes, that is, LSPR-IB1 and LSPRIB2, have different characters. The first mode that is excited at lower energies (ω0), is the IB-like mode. Obviously, the former is more sensitive to nm; however, the latter with a dominant IB character is less-sensitive (Figure 7). Additionally, the avoided crossing zone is constructed about the IB energy ω0 = 3.0 eV, as seen in Figure 7a, denoted by a dashed ellipse; clearly, the absorption is featureless in this zone. Because the dielectric constants of some ferromagnets such as Ni, Co, Fe, and metals like Al contain the local IB transitions in the optical energy range of interest, the physics and results discussed here are generally applicable. Specifically, the IB transition features can be seen at about 4.7, 1.5, and 2.5 eV in the dielectric constants of Ni, Al, and Fe, respectively. However, to try to keep the generality of the study here, the findings are certainly useful in understanding and manipulating the optical properties of transition-metal nanoparticles. Moreover, this should be of relevance to the works in this field, which are aimed at fundamental understanding and simple modeling of optical 21830
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4. CONCLUSIONS The presented results show that the optical absorption of a nanoparticle is severely influenced because of the interaction of an LSPR with a local IB transition. In particular, the optical IB absorption of a single nanoparticle is remarkably enhanced. This leads to the excitation of two coupled modes with new hybridized energies, different from their primary noninteracting values. The interaction strength defines the repulsion of two new energies associated with the coupled LSPR-IB1 and LSPRIB2 modes. Qualitatively, an avoided crossing zone is constructed around the IB transition energy. The energies of the two modes are interchanged as the crossing zone is traversed. Finally, tuning the dipolar LSPR allowed controlling and manipulating the LSPRIB interaction and its resulting effects on the optical absorption of a DL nanoparticle. The findings pave the way for better understanding of the physics of the optical properties of those metallic nanoparticle for which the distinct local IB transitions occur somewhere along the interested optical energy range. Most of the 3d metals such as Ni, Co, Fe, Cu, and Al nanoparticles are good examples of such cases, and they favor catalytic, photothermal, nanomagnetics, and nanophotonics applications.
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(16) Pakizeh, T.; Langhammer, C.; Zoric, I.; Apell, P.; K€all, M. Nano Lett. 2009, 9, 882–886. (17) Hao, Q.; Juluri, B. K.; Zheng, Y. B.; Wang, B.; Chiang, I.-K.; Jensen, L.; Crespi, V.; Eklund, P. C.; Huang, T. J. J. Phys. Chem. C 2010, 114, 18059–18066. (18) Pinchuk, A.; von Plessen, G.; Kreibig, U. J. Phys. D: Appl. Phys. 2004, 37, 3133–3139. (19) Fano, U. Phys. Rev. 1961, 124, 1866–1878. (20) Sturm, K.; Sch€ulke, W.; Schmitz, J. R. Phys. Rev. Lett. 1992, 68, 228–231. (21) Manjavacas, A.; Garcia de Abajo, F. J.; Nordlander, P. Nano Lett. 2011, 11, 2318–2323. (22) Johnson, P. B.; Christy, R. W. Phys. Rev. B 1974, 9, 5056–5070. (23) Palik, E. D. Handbook of Optical Constants of Solids; Academic Press: New York, 1991. (24) Langhammer, C.; Schwind, M.; Kasemo, B.; Zoric, I. Nano Lett. 2008, 8, 1461–1471. (25) Wang, H.; Tam, F.; Grady, N. K.; Halas, N. J. J. Phys. Chem. B 2005, 109, 18218–18222. (26) Halas, N. J.; Lal, S.; Chang, W.-S.; Link, S.; Nordlander, P. Chem. Rev. 2011, 111, 3913–3961. (27) Pakizeh, T.; Abrishamian, M. S.; Granpayeh, N.; Dmitriev, A.; K€all, M. Opt. Express 2006, 14, 8240–8246. (28) Prodan, E.; Radloff, C.; Halas, N. J.; Nordlander, P. Science 2003, 302, 419–422.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT I thank A. Dmitriev and C. Langhammer for fruitful discussions. The financial support by the Iran National Science Foundation (INSF) is acknowledged. ’ REFERENCES (1) Garcia, M. A. J. Phys. D: Appl. Phys. 2011, 44, 283001. (2) Novotny, L.; Hecht, B. Principles of Nanooptics; Cambridge University Press: New York, 2006. (3) Kreibig, U.; Vollmer, M. Optical Properties of Metal Clusters; Springer: Berlin, 1995. (4) Lal, S.; Link, S.; Halas, N. J. Nat. Photonics 2007, 1, 641–648. (5) Jain, P. K.; Huang, X.; El-Sayed, I. H.; El-Sayed, M. A. Acc. Chem. Res. 2008, 41, 1578–1586. (6) Dmitriev, A.; H€agglund, C.; Chen, S.; Fredriksson, H.; Pakizeh, T; K€all, M.; Sutherland, D. S. Nano Lett. 2008, 11, 3893–3898. (7) Curto, A. G.; Volpe, G.; Taminiau, T. H.; Kreuzer, M. P.; Quidant, R.; van Hulst, N. F. Science 2010, 329, 930–933. (8) Knight, M. W.; Sobhani, H.; Nordlander, P.; Halas, N. J. Science 2011, 332, 702–704. (9) H€agglund, C.; Kasemo, B. Opt. Express 2009, 17, 11944–11957. (10) Sepulveda, B.; Gonzalez-Diaz, J. B.; Garcia-Martin, A.; Lechuga, L. M.; Armelles, G. Phys. Rev. Lett. 2010, 104, 147401. (11) Valentine, J.; Zhang, S.; Zentgraf, T.; Avila, E. U.; Genov, D. A.; Bartal, G.; Zhang, X. Nature 2008, 455, 376–380. (12) Pakizeh, T.; Dmitriev, A.; Abrishamian, M. S.; Granpayeh, N.; K€all, M. J. Opt. Soc. Am. B 2008, 25, 659–667. (13) Kelly, K. L.; Coronado, E.; Zhao, L. L.; Schatz, G. C. J. Phys. Chem. B 2008, 107, 668–677. (14) Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light by Small Particles; John Wiley & Sons: New York, 1998. (15) Zoric, I.; Zach, M.; Kasemo, B.; Langhammer, C. ACS Nano 2011, 5, 2535–2546. 21831
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