plasmonics - bio-pagebio-page.org/boriskina/Plasmonics_Boriskina_lecture1.pdf · S.V. Boriskina, 2012 Overview: lecture 1 •Drude model •Theoretical models for plasmonics •Surface

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 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


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:


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


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)

• 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 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)


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



• 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



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

photon

x kk

Via gratings:

ankk photonxSPP

x 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

30nm Ag

60nm Ag

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. Reinhard J 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 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:


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


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.


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 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


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 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)


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 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


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

Documents

plasmonics - bio-pagebio-page.org/boriskina/Plasmonics_Boriskina_lecture1.pdf · S.V. Boriskina, 2012 Overview: lecture 1 •Drude model •Theoretical models for plasmonics •Surface