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**Fundamentals & applications of plasmonics**

Svetlana V. Boriskina

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**Plasmonics in EE engineering**

tens-to-hundreds nm

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**Plasmonics in EE engineering**

Image credit: M. Brongersma & V. Shalaev

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**Plasmonics in chemistry & biotechnology**

Image: Jain et al, Nano Today, 2(1) 2007, 18–29 Particle synthesis Image: D. Pacifici, Brown University Sensing Image: Reinhard group, Boston University Spectroscopy Theragnostics Image: Nanopartz Inc

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**Plasmonics in art & architecture**

Rayonnat Gothic rose window of north transept, Notre-Dame de Paris (Jean de Chelles, 13th century A.D.) Lycurgus Cup: Roman goblet, 4th century A.D

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**Overview: lecture 1 Drude model Theoretical models for plasmonics**

Surface plasmon polariton (SPP) waves Localized SP resonances - plasmonic atoms Component miniaturization Sub-resolution imaging Temporal & spatial coherence of SP modes Q-factor enhancement mechanisms Plasmonic antennas & arrays Plasmonic atoms & molecules Plasmonic nanorulers & nanosensors

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**Drude theory Material response to electric field: Collision frequency**

electron velocity mean free path Image credit: Wikipedia Electrons in thermal equilibrium with the surrounding No restoring force (free ideal electron gas) No long-range interaction between electrons & ions No short-range interaction between electrons Instantaneous collisions with ions with a fixed probability per unit time dt: dt/τ. (τ - relaxation time; ) Electrons move with constant velocity Although the major result of the theory as originally developed by Drude was establishing the relation between electrical and thermal conductivity of metals, here we mostly focus on the response of metals to the external electric field, which is given by the Drude dielectric function. We will return to the discussion of the specific definition of the electron velocity a few slides later. Electrons are accelerated by the applied electric field & collide with ions with a fixed probability per unit time. Collective long-range Coulomb interactions between free electrons displaced by the external field generate a force to restore the neutrality of the plasma (i.e., displaced electrons are pulled back to their equilibrium positions and oscillate with a characteristic frequency - plasma frequency). e.g., N.W. Ashcroft and N.D. Mermin “Solid state Physics” (Saunders College, PA 1976)

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**Drude theory Frequency-domain solution (monochromatic fields):**

Macroscopic polarization (dipole moment per unit volume): Definition of the dielectric constant: Although the major result of the theory as originally developed by Drude was establishing the relation between electrical and thermal conductivity of metals, here we mostly focus on the response of metals to the external electric field, which is given by the Drude dielectric function. We will return to the discussion of the specific definition of the electron velocity a few slides later. Electrons are accelerated by the applied electric field & collide with ions with a fixed probability per unit time. Collective long-range Coulomb interactions between free electrons displaced by the external field generate a force to restore the neutrality of the plasma (i.e., displaced electrons are pulled back to their equilibrium positions and oscillate with a characteristic frequency - plasma frequency). Drude permittivity function:

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**Drude-Lorentz theory ω0 Au:**

Damping factor (mostly radiative) Drude frequency of metals is in the ultra-violet range Interband transitions should be taken into account In the classical model, they are treated as the contribution from bound charges

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**Results Bulk plasmon (SP) oscillation is a longitudinal wave**

Light of frequency above the plasma frequency is transmitted, with frequency below that - reflected (electrons cannot respond fast enough to screen light) Plasmon - a quasiparticle resulting from the quantization of plasma oscillations: Permittivity Reflectance

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**Popular Drude-like materials**

Noble metals (Ag, Au, Pt, Cu, Al …) Drude frequency in the ultra-violet range Applications from visible to mid-IR Ordal, M.A. et al, Appl. Opt., (7): p Doped silicon Drude frequency in the infra-red range Ginn, J.C. et al, J. Appl. Phys (4): p Oxides and nitrides Al:ZnO, Ga:ZnO, ITO: near-IR frequency range Transition-metal nitrides (TiN, ZrN): visible range Naik, G.V. et al, Opt. Mater. Express, (6): p Graphene IR frequency range Jablan, M. et al, Phys. Rev. B, (24): p Vakil, A. & Engheta, N. Science, (6035): pp

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**Theoretical models for plasmonics**

‘The oversimplification or extension afforded by the model is not error: the model, if well made, shows at least how the universe might behave, but logical errors bring us no closer to the reality of any universe.’ Truesdell and Toupin (1960) Classical electromagnetic theory Local response approximation Quasi-static approximation Antenna-theory design Circuit-theory design Quantum theory Drude model modifications Ab initio density functional theory Hydrodynamical models Hydrodynamical model for electrons: non-local response Hydrodynamical model for photons e.g. D. C. Marinica, e.g., Nano Lett. 12, (2012). Next lecture

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**Quantum-mechanical effects**

Velocity definition: electron velocity mean free path Classical Drude model of an ideal electron gas: Maxwell-Boltzmann statistics of energy distribution Drude-Sommerfeld model: Fermi-Dirac statistics of energy distribution Fermi energy Quantum size effects (particle size below the mean free path): Discretized energy levels in conduction band Free electron gas constrained by infinite potential barriers at the particle edges transitions from occupied (Ei) to excited (Ef ) energy levels J. Scholl, A. Koh & J. Dionne, Nature 483, 421, (2012)

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**Surface plasmon-polariton wave**

Planar interface between two media: Eigensolutions of the Helmholtz equation: Solution:

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**Surface plasmon-polariton wave**

Planar interface between two media: < λ Dispersion equation for a surface plasmon-polariton (SPP) wave: Should be negative! Propagating along the interface: real kx Exponentially decaying away from it: imaginary kz

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**Surface plasmon-polariton wave**

ω Re(kx) Propagating: real kz Surface: imaginary kz ω Re(kx) Experimental Au P. B. Johnson & R. W. Christy, Phys. Rev. B 6, 4370 (1972) High DOS: ρ(ħω)∝(dω/dk)-1

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**SPP excitation a Via prisms: Via gratings:**

Via localized sources (e.g. tips, molecules):

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**Miniaturization of photonic components**

Gramotnev & Bozhevolnyi, Nature Photon 4, (2010)

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**Localized SPs on metal nanoparticles**

+ boundary conditions Multi-polar Mie theory formulation: Exact series solution: Sphere (cluster of spheres) – fields expansion in the spherical-wave basis Circular cylinders - fields expansion in the cylindrical-wave basis More complex geometries require numerical treatment (FDTD, FEM, BEM …) Object much smaller than the light wavelength: all points respond simultaneously Helmholtz equation reduces to the Laplace equation Quasi-static limit: Plasmon hybridization method (quasi-static): deformations of a charged, incompressible electron liquid expanded in a complete set of primitive plasmon modes (Peter Nordlander, Rice University) C.F. Bohren & Huffman, Absorption and Scattering of Light by Small Particles (Wiley) Novotny, L. & B. Hecht. Principles of Nano-Optics, Cambridge: Cambridge University Press

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**Localized SPs on metal nanoparticles**

Modes with different angular momentum: analogs of electron orbitals of atoms Higher-order modes have lower radiation losses; do not couple efficiently to propagating waves (dark plasmons) Image: Wikimedia commons (author: PoorLeno) K.L. Kelly et al, J. Phys. Chem. B 2003, 107, Extinction=scattering+absorption 30nm Ag 60nm

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**Tuning LSP resonance Cscatt Particle shape: Nanosphere size:**

W. A. Murray, W. L. Barnes, Adv. Mater. 19, 3771 (2007) . Particle shape: Nanosphere size: B. Yan, S.V. Boriskina &B.M. Reinhard J Phys Chem C 115 (50), (2011) Cscatt

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**Applications: sub-resolution imaging**

S. Kawata, Y. Inouye & P. Verma, Nat Photon 3, (2009). Image:

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**SP modes characteristic lengthscales**

W.L. Barnes 2006 J. Opt. A: Pure Appl. Opt. 8 S87

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**Coherence of SP modes Solutions of the SP dispersion equation:**

T.B. Wild, et al, ACS Nano 6, (2012) Solutions of the SP dispersion equation: complex-k solution: a complex wave number (k+iα) as a function of real frequency ω SP propagation length: complex-ω solution: a complex frequency (ω+iγ) as a function of real wave number. SP lifetime: 6-10fs T. Klar, et al, Phys. Rev. Lett. 80, (1998).

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**Q-factor as a measure of temporal coherence**

Q - the number of oscillations that occur coherently, during which the mode sustains its phase and accumulates energy For eigenmode: From experimental spectra: Why large Q-values are important? Local fields enhancement: ~ Q Spontaneous emission rate enhancement: Purcell factor ~ Q Stimulated emission & absorption rates enhancement ~ Q Spectral resolution of sensors: ~ Q Enhancement of Coulomb interaction between distant charges ~ Q

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**Coherence enhancement**

Coupling to photonic modes: Blanchard, R. et al, Opt. Express, (22): See also: Y. Chu, et al, Appl. Phys. Lett., (18): ; S. Zou, J. Chem. Phys., (23): Ahn, W., et al. ACS Nano, (1): p See also: Boriskina, S.V. & B.M. Reinhard, Proc. Natl. Acad. Sci., (8): p ; Santiago-Cordoba, M.A., et al. Appl. Phys. Lett., : p Fano resonance engineering: Fan, J.A., et al. Science, (5982): 1135 also: Luk'yanchuk, B., et al. Nat Mater, (9): 707; Verellen, N., et al. Nano Lett., (4): 1663 SP gain amplification: Grandidier, J., et al. Nano Lett (8): p also: Noginov, M. A. et al. Opt. Express 16, 1385 (2008); De Leon, I. & P. Berini, Nat Photon, (6):

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**Antenna-theory design of SP components**

Alu & Engheta, Phys. Rev. B, (19): ; Nature Photon., (5): Plasmonic nanodimer as a Hertzian dipole Au particle analog of a dipole antenna Review: P. Bharadwaj, B. Deutsch & L. Novotny, Optical antennas. Adv. Opt. Photon., (3): p

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**Antenna-theory design of SP components**

Phased nanoantenna arrays: Constructive/destructive interference between dipole fields of individual nanoparticles Y. Chu, et al, Appl. Phys. Lett., (18): p Curto, A.G., et al. Science, (5994): p QD

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**Circuit-theory design of SP components**

Au particle Engheta, N. Science, (5845): p

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**Chemical analogs: plasmonic molecules**

P. Nordlander, et al, Nano Lett. 4, (2004). Credit: Capasso Lab, Harvard School of Engineering & Applied Sciences Bonding LSP mode Anti-bonding mode

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Spectra shaping B. Yan, S. V. Boriskina, & B. M. Reinhard, J. Phys. Chem. C 115, (2011); J. Phys. Chem. C 115,

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**Local field enhancement**

Diatomic plasmonic molecule: Cscatt |E|2 Spectroscopy applications (next lecture) B. Yan, S. V. Boriskina, & B. M. Reinhard, J. Phys. Chem. C 115, (2011)

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**Applications: plasmon nanorulers**

Measuring distances below diffraction limit Stable probes (no photobleaching) N. Liu, et al, Science 332, (2011) Alivisatos group, UC Berkeley; C. Sonnichsen, et al, Nat Biotech 23, (2005)

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**Applications: cell surface imaging**

Quantification of cell surface receptors, which are important biomarkers for many diseases Wang, Yu, Boriskina & Reinhard, Nano Lett., Article ASAP, DOI: /nl , 2012

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**Overview: lecture 2 Refractive index, fluorescence & Raman sensing**

SP-induced nanoscale optical forces Optical trapping & manipulation of nano-objects Near-field heat transfer via SPP waves Plasmonics for photovoltaics Hydrodynamical models Hydrodynamical model for electrons: non-local response Hydrodynamical model for photons Magnetic effects Plasmonic cloaking Quantum effects Further reading & software packages

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