Nanoscale optical circuits: controlling light using localized surface plasmon resonances

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Valentine Wade
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1 Nanoscale optical circuits: controlling light using localized surface plasmon resonances T. J. Davis, D. E. Gómez and K. C. Vernon CSIRO Materials Science and Engineering Localized surface plasmon (LSP) resonances are surface-charge oscillations that can be excited on metallic nanoparticles. The oscillation frequencies occur in the infrared and visible regions of the optical spectrum and can be excited by light. This results in the light energy being converted into electrical energy. Because the LSP oscillations are the result of the collective motion of the conduction electrons in the metal they produce strong electric fields that penetrate the region about the nanoparticle and can interact with nearby particles. Although surface plasmons have been studied for a long time, there has been renewed interest in them due to the development of nanofabrication techniques that allow researchers to fabricate metallic structures at the nanoscale. In essence the surface plasmon provides a means by which light can be captured and converted into an electrical signal. In this article we discuss the idea of using metallic nanoparticles as elements in optical circuits for controlling and manipulating light at the nanoscale. The nanoparticles play several roles: they act as optical antennas for capturing optical energy and for efficiently radiating it; they also act in a manner similar to passive electrical components, controlling the flow of the energy and its resonances. When combined with non-linear materials, we have the possibility of creating devices for compact optical signal processing at optical frequencies. We will discuss the use of metallic nanoparticles supporting LSP resonances as circuit elements, drawing on the work we are doing at CSIRO Materials Science and Engineering. Figure 1. Examples of the first five resonant modes of a gold nanorod in the shape of a rectangular prism 20 nm thick, 30 nm wide and 100 nm long on a glass substrate (as depicted at the top). The colours on the rod represent the relative strength of the surface plasmon eigenfunction (as discussed later in the article), with blue positive and red negative. γ is the eigenvalue and the wavelengths of the light that will excite the resonances are also shown. 1

2 The LSP exist as surface-charge waves that reflect from the boundaries of the metallic nanoparticles, leading to a discrete set of resonances, or resonant modes. The modes are strongly dependent on the shape of the metal particle and its electric permittivity. LSP resonances arise in many different metals, but gold and silver generally have the lowest losses in the visible spectrum. Moreover, the LSP are an electrical phenomenon and are sensitive to the electric permittivity in the vicinity of the nanoparticle. An example of the first five resonant modes of a nanoscale rectangular prism is shown in Fig. 1. The colours represent the relative strength of the surfacedipole distributions with red negative and blue positive. These distributions are the eigenfunction solutions to the resonant modes that we discuss below. Associated with each eigenfunction is an eigenvalue that depends on the shape of the nanoparticle. The eigenvalue is related to the resonance frequency via the electric permittivity of the metal. In Fig. 1, the wavelengths of the incident light that will excite the resonances are also given, assuming the nanoparticle is made from gold and is on a glass substrate. As can be seen, the resonant modes can be quite complex. However, the high-order modes resonate at high frequencies, which are often close to the bulk plasma frequency of the metal where the absorption losses are high. In most cases it is the fundamental mode of the nanoparticle which is of most interest, and this is usually a dipole mode. Figure 2. The scattering cross section of the metal prism of Fig. 1. Only the fundamental dipole mode is dominant. The dipole mode of the metallic nanoparticle is similar to the dipole mode of a resonant dipole antenna. In this regard we can think of the plasmonic nanoparticle as an optical antenna. The scattering cross section of the nanorod in Fig.1 is shown as a function of wavelength in Fig. 2. The rod was modelled as a rectangular prism of width 30 nm, length 100 nm and thickness 20 nm. For light incident from above, the nanorod presents a geometric cross sectional area of μm 2. However, at resonance the nanorod has an optical cross sectional area of μm 2 which is some 25 times larger. This means the nanoparticle can efficiently capture optical energy, 2

3 which is an important feature of an optical antenna. The calculation was done with a background permittivity of 1.3 which mimics the nanorod on a glass substrate[1]. The large scattering cross section is a result of the resonance in the metallic nanoparticle. Associated with the resonance is a strong electric field that exists in the vicinity of the nanoparticle. This field is responsible for surface-enhanced Raman scattering (SERS). In this effect, many orders of magnitude increase in the Raman scattering spectrum of a molecule is observed when it is in the presence of a plasmonic nanoparticle. In essence, the nanoparticle acts as an optical antenna, efficiently coupling the incident light energy into the nearby molecule. Because the molecule has a polarizability that is modulated slightly by the molecular vibrations, a fraction of the incident light is scattered with wavelength shifts equal to the molecule vibration frequencies. This scattered light is then efficiently out-coupled by the plasmonic particle, leading to the large enhancement which can be observed experimentally. Figure 3. This shows the formation of a subradiant mode (dark mode) as the separation between two nanoparticles supporting LSP decreases. The fundamental dipole modes couple creating a symmetric and an antisymmetric pair of modes. The antisymmetric mode has a low scattering cross section because the opposing dipoles interfere. However, the electric fields can still be strong, as indicated by the magnitudes of the amplitudes of the excitations. The plasmonic nanoparticle has an important role as a component in optical circuits. Because the electric charges associated with the LSP in the nanoparticle interact strongly with the electric fields, it is possible to modify the behaviour of the resonances by placing two or more nanoparticles in close proximity. Their mutual electric fields overlap and influence the resonances in a variety of ways, sometimes leading to unexpected results. One example is the formation of dark modes [2, 3], which are closely related to an effect known as the plasmonic equivalent of electromagnetically-induced transparency [4, 5]. When two of the rectangular prisms 3

4 in Fig. 1 are placed close to one another, their resonant modes split into two, giving rise to a symmetric and an antisymmetric mode. The antisymmetric mode has two dipole moments that are anti-aligned and this mode is difficult to excite when the nanoparticles are very small. However, if we introduce a slight asymmetry, such as making one rod longer than the other, then the antisymmetric mode has a small net dipole moment and can be excited by the incident field (Fig. 3). When the two rods are brought into close proximity to one another, the overlap of the electric fields from the LSP resonances alters the flow of charge and changes the resonant frequencies. The resonance of the symmetric mode shifts towards the blue end of the spectrum while the antisymmetric mode shifts to the red, the shift increasing with decreasing distance between the nanoparticles. The scattering from the symmetric mode is strong, because the dipole moments of the resonances are aligned. The scattering from the antisymmetric mode is small because the dipole moments oppose one another. This leads to a large drop in the scattering cross section for this mode. Surprisingly, if we look at the strength of the resonances we find that the antisymmetric mode is strongly excited, even though its scattering is small. In effect, by placing two strongly scattering particles together we can create an object which is strongly excited by the incident light field but which barely scatters any light. This is quite remarkable, and it is known as a dark mode, or a subradiant mode. An experimental observation of this effect was reported recently on a three-nanorod structure[5]. The resonances of this type form part of the class of Fano resonances [6]. As we have shown, the interaction between plasmonic nanoparticles can lead to unexpected effects. The modelling of these systems can be quite complex, because of the difficulty in solving Maxwell s equations for the electromagnetic fields. However, for nanoparticles much smaller than the wavelength of light, the electric and magnetic fields decouple and Maxwell s equations take the same form as in electrostatics, even though the fields are oscillating at optical frequencies. In this electrostatic approximation the problem of determining the LSP resonances takes the form of an integral eigenvalue problem, where the eigenfunctions are the surface-charge and surface-dipole distributions of the resonant modes and the eigenvalues determine the resonant frequencies [7, 8]. The eigenfunctions for the rod are shown in Fig. 1. By describing the interactions between nanoparticles in terms of their eigenfunctions, the mathematics of the coupling/interaction problem is the same for all plasmonic nanoparticles, independent of their shape. For an arbitrary nanoparticle the eigenvalues and eigenfunctions are kept in the theory as parameters the interactions between them as mediated through their evanescent electric fields determine the properties of the nanoparticle ensembles. This is very similar to quantum mechanics or quantum optics where, very often, the actual wavefunctions or state vectors are unknown but many useful properties can be determined from the relationships between them. We have developed a theory of the coupling of nanoparticles through their evanescent fields that provides very useful, although approximate, analytical expressions that capture much of the underlying physics [2, 3, 6]. The importance of this theory is that by using relatively simple algebra we can gain insight into the coupling mechanisms, how the relative phases of the resonances can be controlled, and the theory allows us to design nanoparticle circuits to achieve desired optical properties, similar to how we use electric circuit theory to design electronic devices. 4

We have used this method to design the plasmonic equivalent of the Wheatstone bridge circuit in electronics to produce a nanoscale optical circuit that is sensitive to phase shifts[9].

5 We have used this method to design the plasmonic equivalent of the Wheatstone bridge circuit in electronics to produce a nanoscale optical circuit that is sensitive to phase shifts[9]. An example of the circuit is shown in Fig. 4 along with the equivalent electrical circuit. The plasmonic circuit is excited by light polarized in the y direction. When both arms of the bridge are balanced (that is, they have the same resonant frequency), the LSP in the left and right nanoparticles resonate in phase so that the electric fields at each end of the optical antenna are the same. This means that no resonances can be excited in the antenna. If the resonance of one of the arms changes, such as with the binding of a molecule, the two arms no longer resonate in phase and an imbalance occurs in the electric field across the antenna. This imbalance excites a surface plasmon in the antenna which radiates x polarized light. Note that this output signal is polarized perpendicular to the incident light and can be easily distinguished from it. An analysis of the circuit suggests that it could be capable of detecting the presence of a single large molecule such as a protein. We are in the process of fabricating this structure to test its properties. Figure 4. The plasmonic equivalent of the Wheatstone bridge circuit. The plasmonic circuit consists of three metal structures that support localized surface plasmon resonances (as shown in the centre and on the right). In the equivalent circuit on the left, Z l and Z r are the impedances of the left and right nanoparticles and the voltmeter represents the optical antenna. The capacitances C c represent the electric coupling from the arms to the antenna and the C E represent the electric coupling to the incident light. As we have discussed, metallic nanoparticles can be used as optical antennas and as components in plasmonic circuits. These particles provide a means by which light energy can be captured, manipulated and re-emitted. The systems we have described are analogous to passive electrical components that react in a linear fashion to the applied light fields. Modern electronics relies heavily on non-linear devices, such as diodes for controlling the frequency content of signals or transistors for providing gain and electrical modulation of signals. Can we create devices that do the same for light? One of the goals in plasmonics research is to find suitable materials for creating plasmonic transistors that can be used in high frequency optical signal processing. Optical control of light is relatively difficult because light does not interact with itself and optical non-linearities are usually only achieved with very intense light fields. Surface plasmons, on the other hand, are an electrical phenomenon mediated by the conduction electrons in metals that can interact strongly with one another and with electrons in nearby materials. By converting the light energy into electric energy we 5

6 open up the possibility of creating devices that can manipulate this energy and reradiate it as a modified light signal. At CSIRO Materials Science and Engineering we are investigating materials that can be used to manipulate the surface plasmons. One of the systems we have been studying consists of semiconductor nanocrystals that support excitons (electron-hole excitations) that can be excited by light or any other electric field oscillating at optical frequencies. In some recent experiments [10, 11], we generated surface plasmons in a thin silver film and showed that it was possible to create a coherent superposition of surface plasmon and excitonic states when the film was densely coated with CdSe nanocrystals (also known as quantum dots QDs) (Fig. 5). The chemically synthesised nanocrystals are typically 3 nm in diameter. The energy levels of the electrons in the conduction band can be likened to those of the particle-in-a-box problem, where the size of the box determines the spacing of the levels. The QD are an ideal system for studying the interactions because they appear as simple two-level systems, essentially isolated from the surrounding environment. The coherent superposition leads to an avoided crossing in the dispersion diagram for the surface plasmons. The energy gap in our experiments is about 112 mev and leads to a Rabi oscillations associated with the interchange of energy between the surface plasmons and the excitons. Figure 5. a) Semiconducting CdSe nanocrystals on a silver film are used to demonstrate the coherent coupling between surface plasmons in the film and excitons in the nanocrystals. There is a resonant exchange of energy between the LSP and the QDs which occurs at the Rabi frequency. b) The coupling leads to an avoided crossing in the dispersion diagram (adapted from ref.[9]) and an energy gap of 112 mev. Here the angle determines the in-plane wavenumber of the surface plasmon. Since we now know that it is possible to obtain coherent mixed states with a thin metal film, we are working towards demonstrating this effect with metal nanoparticles. As an example we show in Fig. 6 the coupling to a silver nanowire of excitons in CdSe QDs. The silver nanowire rests on a glass substrate and has been coated with a film of QDs. The figure shows a light beam focussed on the QD at the centre of the wire. When the excitons decay to the ground state, their energy is captured by the wire in the form of surface plasmons that propagate to both ends of the wire. At the ends the surface plasmons radiate the energy as light, which can be seen in Fig. 6 highlighted by the arrows. Combinations of these systems hold promise for creating gain media for surface plasmons leading to the possibility of optical 6

7 transistors and surface plasmon lasers. Eventually we could have optical-plasmonicexcitonic circuits that could be used for signal processing, optical computing, new materials, sensors and many other applications. Figure 6. Progress towards creating all-optical nanocircuits. a) A microscope image of a silver nanowire coated with CdSe quantum dots. b) The excitons generated by light focussed at the centre of the wire decay into surface plasmons that propagate to the ends of the wire and re-emit the light highlighted by the arrows. c) an artists impression of a hybrid plasmonic-excitonic circuit where plasmonic nanostructures act as optical antennas and passive circuit components, and denselypacked QD structures provide nonlinear interactions though their excitonic states. Acknowledgements The authors would like to thank Alison Funston for assistance with fabricating the silver nanowires and Paul Mulvaney for the use of his laboratory. References 1. K. C. Vernon, A. M. Funston, C. Novo, D. E. Gomez, P. Mulvaney, and T. J. Davis, "Influence of Particle-Substrate Interaction on Localized Plasmon Resonances," Nano Letters 10, (2010). 2. T. J. Davis, K. C. Vernon, and D. E. Gomez, "Designing plasmonic systems using optical coupling between nanoparticles," Physical Review B (Condensed Matter and Materials Physics) 79, (2009). 3. T. J. Davis, K. C. Vernon, and D. E. Gómez, "Designing plasmonic systems: applications to dark modes in nanoparticle pairs and triplets," Proceedings of the SPIE 7394, (2009). 4. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, "Plasmon-Induced Transparency in Metamaterials," Physical Review Letters 101, (2008). 5. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, "Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit," Nature Materials 8, (2009). 6. T. J. Davis, D. E. Gomez, and K. C. Vernon, "Simple Model for the Hybridization of Surface Plasmon Resonances in Metallic Nanoparticles," Nano Letters 10, (2010). 7. F. Ouyang, and M. Isaacson, "Surface plasmon excitation of objects with arbitrary shape and dielectric constant," Philosophical Magazine B 60, (1989). 8. I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, "Electrostatic (plasmon) resonances in nanoparticles," Physical Review B 72, (2005). 7

8 9. T. J. Davis, K. C. Vernon, and D. E. Gomez, "A plasmonic ``ac Wheatstone bridge'' circuit for high-sensitivity phase measurement and single-molecule detection," Journal of Applied Physics 106, (2009). 10. D. E. Gómez, K. C. Vernon, P. Mulvaney, and T. J. Davis, "Surface Plasmon Mediated Strong Exciton-Photon Coupling in Semiconductor Nanocrystals," Nano Letters 10, (2010). 11. D. E. Gomez, K. C. Vernon, P. Mulvaney, and T. J. Davis, "Coherent superposition of exciton states in quantum dots induced by surface plasmons," Applied Physics Letters 96, (2010). 8

Nanophysics: Main trends

Nano-opto-electronics Nanophysics: Main trends Nanomechanics Main issues Light interaction with small structures Molecules Nanoparticles (semiconductor and metallic) Microparticles Photonic crystals Nanoplasmonics