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Fundamentals &
applications of
plasmonics
Svetlana V. Boriskina
Plasmonics in EE engineering
tens-to-hundreds nm
current
E
S.V. Boriskina, 2012
light
Plasmonics in EE engineering
Image credit: M. Brongersma & V. Shalaev
S.V. Boriskina, 2012
Plasmonics in chemistry & biotechnology
Particle synthesis
Sensing
Image: D. Pacifici, Brown University
Spectroscopy
Image: Jain et al, Nano Today, 2(1) 2007, 18–29
Theragnostics
Image: Reinhard group, Boston University
S.V. Boriskina, 2012
Image: Nanopartz Inc
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
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
S.V. Boriskina, 2012
Drude theory
Material response to electric field:
 2r(t )
r(t )
me
 me 
 eE(t )
2
t
t
electron velocity
 1   v l
Collision
frequency
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;   1  )
• Electrons move with constant velocity v
S.V. Boriskina, 2012
e.g., N.W. Ashcroft and N.D. Mermin “Solid state
Physics” (Saunders College, PA 1976)
Drude theory
 2r ( t )
r(t )
me
 me 
 eE(t )
2
t
t
Macroscopic polarization (dipole
moment per unit volume):
Frequency-domain solution e  it
(monochromatic fields):
e
r ( ) 
E( )
2
me (  i )
Definition of the dielectric
constant:
 ne2E
P  ner 
me ( 2  i )
P   0   1E
 p2  ne2  0me
Drude
permittivity
function:
S.V. Boriskina, 2012
 ( )  1 
 p2
( 2  i )
Drude-Lorentz theory
Au:
 p  13.8 1015 Hz,   1.075 1014 Hz
ω0
r
   2r
2 

me  2  
 0 r   eE0eit
t
 t

Damping factor (mostly radiative)

 p2
 IB ( )  1  2
(0   2 )  i
• 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
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:
Permittivity
S.V. Boriskina, 2012
Reflectance
E p   p
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., 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.
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
D(r,  )   (r,  )  E(r,  )
• Quantum theory
• Drude model modifications
• Ab initio density functional theory e.g. D. C. Marinica, e.g., Nano
Lett. 12, 1333-1339 (2012).
• Hydrodynamical models
• Hydrodynamical model for electrons: non-local response
• Hydrodynamical model for photons
S.V. Boriskina, 2012
Next
lecture
Quantum-mechanical effects
Velocity definition:
electron velocity
Classical Drude model of an ideal electron gas:
 1   v l
f MB ( E )  e  E k BT
mean free
path
v  3k BT me
Maxwell-Boltzmann statistics of energy distribution
Drude-Sommerfeld model:
f FD ( E ) 
1
e
( E  E f ) k BT
1
Fermi energy
v  2 E f me
Fermi-Dirac statistics of energy distribution
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
 ( )   IB  
2
p
 (
(i ) ( f )
S.V. Boriskina, 2012
Sif
2
if
  2 )  i
transitions from occupied (Ei) to
excited (Ef ) energy levels
J. Scholl, A. Koh & J. Dionne, Nature 483, 421, (2012)
Surface plasmon-polariton wave
• Planar interface
between two
media:
• Eigensolutions of the Helmholtz equation:
    E(r,  ) 
Solution:
Ex  E
( j)
x
e
2
c
2
 (r,  )E(r,  )  0
ik x x it
e
ikz( j ) z
j  metal or diel
S.V. Boriskina, 2012
Surface plasmon-polariton wave
• Planar interface
between two
media:
<λ
• Dispersion equation for a surface plasmon-polariton (SPP) wave:
   m d
k x    
c  m  d
12



k
m(d )
z
  
   
c  m  d
Propagating along the interface: real kx
Exponentially decaying away from it: imaginary kz
S.V. Boriskina, 2012
2
m(d )
12




Should be
negative!
k x   if  m   d
Surface plasmon-polariton wave
ck x
Experimental Au
d
ω
ω
Propagating:
real kz
p
1  d
High DOS:
ρ(ħω)∝(dω/dk)-1
Surface:
imaginary kz
Re(kx)
p  13.8 1015 Hz,   0
S.V. Boriskina, 2012
Re(kx)
P. B. Johnson & R. W. Christy,
Phys. Rev. B 6, 4370 (1972)
SPP excitation
Via prisms:
p
k xphoton  k xSPP
Via gratings:
ck x
p
a
k xSPP  k xphoton  2n a
Via localized sources (e.g. tips, molecules):
S.V. Boriskina, 2012
Miniaturization of photonic components
Gramotnev & Bozhevolnyi,
Nature Photon 4, 83 - 91 (2010)
S.V. Boriskina, 2012
Localized SPs on metal nanoparticles
    E(r,  ) 
2
 (r,  )E(r,  )  0 or Ein (r,  ) 
c
+ boundary conditions
2
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 …)
Quasi-static limit:
• Object much smaller than the light wavelength: all points respond simultaneously
• Helmholtz equation reduces to the Laplace equation
E  ,  2  0
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)
S.V. Boriskina, 2012
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
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)
30nm
Ag
60nm
Ag
Extinction=scattering+absorption
K.L.Boriskina,
Kelly et al,2012
J. Phys. Chem. B 2003, 107, 668-677.
S.V.
Image: Wikimedia commons (author: PoorLeno)
Tuning LSP resonance
Particle
shape:
Nanosphere size:
Cscatt
B. Yan, S.V. Boriskina &B.M. Reinhard
J Phys Chem C 115 (50), 24437-24453 (2011)
S.V. Boriskina, 2012
W. A. Murray, W. L. Barnes, Adv.
Mater. 19, 3771 (2007) .
Applications: sub-resolution imaging
Image: http://www.xenophilia.com
S.V. Boriskina, 2012
S. Kawata, Y. Inouye & P. Verma, Nat
Photon 3, 388-394 (2009).
SP modes characteristic lengthscales
W.L. Barnes 2006 J. Opt. A: Pure Appl. Opt. 8 S87
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: LSP  1 2
2-20μm
T.B. Wild, et al, ACS
Nano 6, 472-482 (2012)
• complex-ω solution: a complex frequency
(ω+iγ) as a function of real wave number.
SP lifetime:   1 
6-10fs
S.V. Boriskina, 2012
T. Klar, et al, Phys.
Rev. Lett. 80, 42494252 (1998).
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:
n  n  i n
From experimental spectra:
Q  n 2 n
Q  res 
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
S.V. Boriskina, 2012
http://www.nanowerk.com/spotlight/spotid=24124.php
Coupling to photonic modes:
Coherence enhancement
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.
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
S.V. Boriskina,
2012
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.
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.
Antenna-theory design of SP components
Au particle
Plasmonic
nanodimer as a
Hertzian dipole
Alu & Engheta, Phys. Rev. B,
2008. 78(19): 195111; Nature
Photon., 2008. 2(5): 307-310
analog of a
dipole antenna
S.V. Boriskina, 2012
Review: P. Bharadwaj, B. Deutsch & L. Novotny, Optical
antennas. Adv. Opt. Photon., 2009. 1(3): p. 438-483.
Antenna-theory design of SP components
Phased nanoantenna arrays:
Constructive/destructive
interference between dipole fields
of individual nanoparticles
QD
Y. Chu, et al, Appl. Phys. Lett., 2008.
93(18): p. 181108-3
Curto, A.G., et al. Science, 2010. 329(5994):
p. 930-933.
http://www.haarp.alaska.edu/haarp/
S.V. Boriskina, 2012
http://www.ehow.com/info_12198356_yagi-antenna.html
Circuit-theory design of SP components
Au particle
Engheta, N. Science, 2007. 317(5845): p. 1698-1702.
S.V. Boriskina, 2012
Chemical analogs: plasmonic molecules
P. Nordlander, et al, Nano Lett. 4, 899-903 (2004).
Bonding LSP mode
Anti-bonding mode
Credit: Capasso Lab, Harvard School
of Engineering & Applied Sciences
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
S.V. Boriskina, 2012
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, 24437-24453 (2011)
S.V. Boriskina, 2012
Applications: plasmon nanorulers
• Measuring distances below diffraction limit
• Stable probes (no photobleaching)
Alivisatos group, UC Berkeley;
C. Sonnichsen, et al, Nat Biotech 23, 741-745 (2005)
S.V. Boriskina, 2012
N. Liu, et al, Science 332, 14071410 (2011)
Applications: cell surface imaging
Quantification of cell surface receptors, which are
important biomarkers for many diseases
S.V. Boriskina, 2012
Wang, Yu, Boriskina & Reinhard, Nano Lett., Article
ASAP, DOI: 10.1021/nl3012227, 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
S.V. Boriskina, 2012