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Fundamentals & applications of plasmonics
Svetlana V. Boriskina
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S.V. Boriskina, 2012
Plasmonics in EE engineering
E light
current
tens-to-hundreds nm
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S.V. Boriskina, 2012
Plasmonics in EE engineering
Image credit: M. Brongersma & V. Shalaev
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S.V. Boriskina, 2012
Plasmonics in chemistry & biotechnology
Image: Jain et al, Nano Today, 2(1) 2007, 18–29
Particle synthesis
Image: D. Pacifici, Brown University
Sensing
Theragnostics
Image: Nanopartz Inc
Image: Reinhard group, Boston University
Spectroscopy
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S.V. Boriskina, 2012
Plasmonics in art & architecture
Lycurgus Cup: Roman goblet, 4th century A.D
Rayonnat Gothic rose window of north transept, Notre-Dame de Paris (Jean de Chelles, 13th century A.D.)
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S.V. Boriskina, 2012
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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S.V. Boriskina, 2012
Drude theory
Material response to electric field:
• 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
e.g., N.W. Ashcroft and N.D. Mermin “Solid state Physics” (Saunders College, PA 1976)
Image credit: Wikipedia
Collision
frequency
1v
electron velocity
mean free path
lv 1
)()()(
2
2
tet
tm
t
tm ee E
rr
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S.V. Boriskina, 2012
Drude theory
)()()(
2
2
tet
tm
t
tm ee E
rr
Frequency-domain solution (monochromatic fields):
tie
)()(
)(2
Er
im
e
e
Macroscopic polarization (dipole moment per unit volume):
)( 2
2
im
nene
e
ErP
Definition of the dielectric constant:
EP 10
)(1)(
2
2
i
p
ep mne 022
Drude permittivity function:
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S.V. Boriskina, 2012
Drude-Lorentz theory
• 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
Au:
ti
e eett
m
0
2
02
2
Errr
i
p
IB
)(
1)(22
0
2Damping factor (mostly radiative)
ω0
Hz10075.1 ,Hz108.13 1415 p
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S.V. Boriskina, 2012
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: ppE
Permittivity Reflectance
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S.V. Boriskina, 2012
• 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., 1983. 22(7): p. 1099-1119.
• Doped silicon • Drude frequency in the infra-red range • Ginn, J.C. et al, J. Appl. Phys. 2011. 110(4): p. 043110-6.
• 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, 2011. 1(6): p. 1090-1099.
• Graphene • IR frequency range • Jablan, M. et al, Phys. Rev. B, 2009. 80(24): p. 245435. • Vakil, A. & Engheta, N. Science, 2011. 332(6035): pp. 1291-1294.
Popular Drude-like materials
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S.V. Boriskina, 2012
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
),(),(),( rErrD
Next lecture
e.g. D. C. Marinica, e.g., Nano Lett. 12, 1333-1339 (2012).
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S.V. Boriskina, 2012
Quantum-mechanical effects electron velocity
mean free path
lv 1
Velocity definition:
Quantum size effects (particle size below the mean free path):
eB mTkv 3TkE
MBBeEf
)(
Classical Drude model of an ideal electron gas:
Maxwell-Boltzmann statistics of energy distribution
1
1)(
)(
TkEEFD Bfe
Ef
Drude-Sommerfeld model:
ef mEv 2
Fermi-Dirac statistics of energy distribution
Fermi energy
• Discretized energy levels in conduction band • Free electron gas constrained by infinite potential barriers at the particle edges
)( )(
22
2
)()(
i f if
if
pIBi
S
transitions from occupied (Ei) to
excited (Ef ) energy levels
J. Scholl, A. Koh & J. Dionne, Nature 483, 421, (2012)
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S.V. Boriskina, 2012
Surface plasmon-polariton wave
• Planar interface between two media:
• Eigensolutions of the Helmholtz equation:
0),(),(),(2
2
rErrEc
Solution: ziktixikj
xx
jzx eeEE
)()(
dielmetalj or
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S.V. Boriskina, 2012
Surface plasmon-polariton wave
• Planar interface between two media:
• Dispersion equation for a surface plasmon-polariton (SPP) wave:
21
dm
dmx
ck
212
)()(
dm
dmdm
zc
k
Should be negative! Propagating along the interface: real kx
Exponentially decaying away from it: imaginary kz
< λ
dmxk if
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S.V. Boriskina, 2012
Surface plasmon-polariton wave
ω
Re(kx)
d
p
1
d
xck
Propagating: real kz
Surface: imaginary kz
0 ,Hz108.13 15 p
High DOS: ρ(ħω)∝(dω/dk)-1
ω
Re(kx)
Experimental Au
P. B. Johnson & R. W. Christy, Phys. Rev. B 6, 4370 (1972)
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S.V. Boriskina, 2012
SPP excitation SPPx
photon
x kk
Via gratings:
ankk photonxSPP
x 2
a
Via prisms:
p
xck
p
Via localized sources (e.g. tips, molecules):
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S.V. Boriskina, 2012
Miniaturization of photonic components
Gramotnev & Bozhevolnyi, Nature Photon 4, 83 - 91 (2010)
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S.V. Boriskina, 2012
Localized SPs on metal nanoparticles ),(or 0),(),(),(
2
2
rErErrE inc
+ 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
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
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:
0 , 2 E
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)
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S.V. Boriskina, 2012
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)
K.L. Kelly et al, J. Phys. Chem. B 2003, 107, 668-677.
Extinction=scattering+absorption
30nm Ag
60nm Ag
Image: Wikimedia commons (author: PoorLeno)
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S.V. Boriskina, 2012
Tuning LSP resonance
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), 24437-24453 (2011)
Cscatt
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S.V. Boriskina, 2012
Applications: sub-resolution imaging
Image: http://www.xenophilia.com
S. Kawata, Y. Inouye & P. Verma, Nat Photon 3, 388-394 (2009).
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S.V. Boriskina, 2012
SP modes characteristic lengthscales
W.L. Barnes 2006 J. Opt. A: Pure Appl. Opt. 8 S87
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S.V. Boriskina, 2012
Coherence of SP modes 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: 21SPL
6-10fs T. Klar, et al, Phys.
Rev. Lett. 80, 4249-4252 (1998).
2-20μm T.B. Wild, et al, ACS Nano 6, 472-482 (2012)
1
• complex-ω solution: a complex frequency (ω+iγ) as a function of real wave number.
SP lifetime:
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S.V. Boriskina, 2012
Q-factor as a measure of temporal coherence
• 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
Q - the number of oscillations that occur coherently, during which the mode sustains its phase and accumulates energy
resQ
From experimental spectra:
nnn i nnQ 2
For eigenmode:
Why large Q-values are important?
http://www.nanowerk.com/spotlight/spotid=24124.php
http://www.nanowerk.com/spotlight/spotid=24124.phphttp://www.nanowerk.com/spotlight/spotid=24124.php
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S.V. Boriskina, 2012
Coherence enhancement Coupling to photonic modes:
Blanchard, R. et al, Opt. Express, 2011. 19(22): 22113. See also: Y. Chu, et al, Appl. Phys. Lett., 2008. 93(18): 181108-3; S. Zou, J. Chem. Phys., 2004. 120(23): 10871.
Ahn, W., et al. ACS Nano, 2012. 6(1): p. 951-960. See also: Boriskina, S.V. & B.M. Reinhard, Proc. Natl. Acad. Sci., 2011. 108(8): p. 3147-3151; Santiago-Cordoba, M.A., et al. Appl. Phys. Lett., 2011. 99: p. 073701.
Fano resonance engineering:
Fan, J.A., et al. Science, 2010. 328(5982): 1135 also: Luk'yanchuk, B., et al. Nat Mater, 2010. 9(9): 707; Verellen, N., et al. Nano Lett., 2009. 9(4): 1663
SP gain amplification:
Grandidier, J., et al. Nano Lett. 2009. 9(8): p. 2935-2939. also: Noginov, M. A. et al. Opt. Express 16, 1385 (2008); De Leon, I. & P. Berini, Nat Photon, 2010. 4(6): 382-387.
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S.V. Boriskina, 2012
Antenna-theory design of SP components
Au particle
analog of a dipole antenna
Alu & Engheta, Phys. Rev. B, 2008. 78(19): 195111; Nature Photon., 2008. 2(5): 307-310
Plasmonic nanodimer as a Hertzian dipole
Review: P. Bharadwaj, B. Deutsch & L. Novotny, Optical antennas. Adv. Opt. Photon., 2009. 1(3): p. 438-483.
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S.V. Boriskina, 2012
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., 2008. 93(18): p. 181108-3
http://www.haarp.alaska.edu/haarp/
Curto, A.G., et al. Science, 2010. 329(5994): p. 930-933.
QD
http://www.ehow.com/info_12198356_yagi-antenna.html
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S.V. Boriskina, 2012
Circuit-theory design of SP components
Au particle
Engheta, N. Science, 2007. 317(5845): p. 1698-1702.
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S.V. Boriskina, 2012
Chemical analogs: plasmonic molecules
Credit: Capasso Lab, Harvard School of Engineering & Applied Sciences
P. Nordlander, et al, Nano Lett. 4, 899-903 (2004).
Bonding LSP mode Anti-bonding mode
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S.V. Boriskina, 2012
Spectra shaping
B. Yan, S. V. Boriskina, & B. M. Reinhard, J. Phys. Chem. C 115, 4578-4583 (2011); J. Phys. Chem. C 115, 24437-24453
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S.V. Boriskina, 2012
Local field enhancement Diatomic plasmonic molecule:
Cscatt |E|2
B. Yan, S. V. Boriskina, & B. M. Reinhard, J. Phys. Chem. C 115, 24437-24453 (2011)
Spectroscopy applications (next lecture)
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S.V. Boriskina, 2012
Applications: plasmon nanorulers
N. Liu, et al, Science 332, 1407-1410 (2011)
• Measuring distances below diffraction limit • Stable probes (no photobleaching)
Alivisatos group, UC Berkeley; C. Sonnichsen, et al, Nat Biotech 23, 741-745 (2005)
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S.V. Boriskina, 2012
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: 10.1021/nl3012227, 2012
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S.V. Boriskina, 2012
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