Fundamentals & applications of plasmonics

Fundamentals & applications of plasmonics

Svetlana V. Boriskina

S.V. Boriskina, 2012

Plasmonics in EE engineering

E light

current

tens-to-hundreds nm


Plasmonics in EE engineering

Image credit: M. Brongersma & V. Shalaev


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


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.)


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


Drude theoryMaterial 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


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

ip

ep mne 022

Drude permittivity function:


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:

tie ee

ttm

0202

2

Errr

i

pIB

)(

1)(22

0

2Damping factor (mostly radiative)

ω0

Hz10075.1 ,Hz108.13 1415 p


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


• 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


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)

‘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).


Quantum-mechanical effectselectron velocity

mean free path

lv 1

Velocity definition:

Quantum size effects (particle size below the mean free path):

eB mTkv 3TkEMB

BeEf )(

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

ifpIB i

S

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:

0),(),(),(2

2

rErrEc

Solution:ziktixikj

xx

jzx eeEE

)()(

dielmetalj or


Surface plasmon-polariton wave• Planar interface

between two media:

• Dispersion equation for a surface plasmon-polariton (SPP) wave:

21

dm

dmx ck

212)()(

dm

dmdmz ck

Should be negative!Propagating along the interface: real kx

Exponentially decaying away from it: imaginary kz

< λ

dmxk if


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)


SPP excitation SPPx

photonx kk

Via gratings:

ankk photonx

SPPx 2

a

Via prisms:

p

xck

p

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


Miniaturization of photonic components

Gramotnev & Bozhevolnyi,Nature Photon 4, 83 - 91 (2010)


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)


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

30nmAg

60nmAg

Image: Wikimedia commons (author: PoorLeno)


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. ReinhardJ Phys Chem C 115 (50), 24437-24453 (2011)

Cscatt


Applications: sub-resolution imaging

Image: http://www.xenophilia.com

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


Coherence of SP modesSolutions 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

6-10fsT. Klar, et al, Phys.

Rev. Lett. 80, 4249-4252 (1998).

2-20μmT.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:


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

resQFrom experimental spectra:

nnn i nnQ 2For eigenmode:

Why large Q-values are important?

http://www.nanowerk.com/spotlight/spotid=24124.php


Coherence enhancementCoupling 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): 1135also: 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.


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.


Antenna-theory design of SP componentsPhased 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


Circuit-theory design of SP components

Au particle

Engheta, N. Science, 2007. 317(5845): p. 1698-1702.


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


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


Local field enhancementDiatomic 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)


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)


Applications: cell surface imagingQuantification of cell surface receptors, which are important biomarkers for many diseases

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

Fundamentals & applications of plasmonics

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