1. Fundamentals & applications of plasmonics
Svetlana V. Boriskina

2. Plasmonics in EE engineering
tens-to-hundreds nm

3. Plasmonics in EE engineering
Image credit: M. Brongersma & V. Shalaev

4. 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

5. 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

6. 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

7. 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)

8. 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:

9. 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

10. 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

11. 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

12. 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

13. 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)

14. Surface plasmon-polariton wave
Planar interface between two media:
Eigensolutions of the Helmholtz equation:
Solution:

15. 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

16. Surface plasmon-polariton waveω
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

17. SPP excitation a Via prisms: Via gratings:
Via localized sources (e.g. tips, molecules):

18. Miniaturization of photonic components
Gramotnev & Bozhevolnyi,
Nature Photon 4, (2010)

19. 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

20. 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

21. 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

22. Applications: sub-resolution imaging
S. Kawata, Y. Inouye & P. Verma, Nat Photon 3, (2009).
Image:

23. SP modes characteristic lengthscales
W.L. Barnes 2006 J. Opt. A: Pure Appl. Opt. 8 S87

24. 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).

25. 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

26. 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):

27. 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

28. 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

29. Circuit-theory design of SP components
Au particle
Engheta, N. Science, (5845): p

30. 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

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

32. 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)

33. 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)

34. 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

35. 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