sign in sign up
<<
Home , Dispersion relation, Evanescent field, Nanophotonics, Phonon, Plasma oscillation, Plasmon

Surface (SPPs) - Introduction and basic properties

- Overview - - interaction - SPP and properties

Standard textbook: - Heinz Raether, Surface on Smooth and Rough Surfaces and on Gratings Springer Tracts in Modern , Vol. 111, Springer Berlin 1988

Overview articles on : - A. Zayats, I. Smolyaninov, Journal of A: Pure and Applied Optics 5, S16 (2003) - A. Zayats, et. al., Physics Reports 408, 131-414 (2005) - W.L.Barnes et. al., 424, 825 (2003) OverviewOverview

Replace ‘slow’ electronic devices with ‘fast’ photonic ones Photonic SPs is the way to concentrate and channel light using subwavelength structures

SPs structures at the interface between a and a material Transverse magnetic in character Æ normal to the surface Perpendicular direction Æ mismatch in the SP dispersion curve OverviewOverview optics (SPP) Band-gap effects, SPP waveguiding along straight and bent line,

Gold stripes OverviewOverview enhanced optical transmission ElementaryElementary excitations excitations and and polaritonspolaritons

Elementary excitations: • (lattice ) • Plasmons (collective oscillations) • Excitions ( between an excited electron and a hole)

Polaritons: Commonly called coupled state between an elementary excitation and a = light-matter interaction

Plasmon polariton: coupled state between a plasmon and a photon.

Phonon polariton: coupled state between a and a photon. PlasmonsPlasmons

A plasmon is the of the collective excitation of free in . Electron effects are most pronounced in free-electron-like . εω The dielectric constant of such materials can be expressed as 2 ⎛⎞ω p m ()=− 1⎜⎟ ⎝⎠ω

ω < ωp Æ εm < 0 Æ wavevector of light in the medium is imaginary Æ no propagating electromagnetic modes

ω > ωp Æ εm→1 Æ altered by intraband transitions in noble metals

A combined excitation consisting of a surface plasmon and a photon is called a surface plasmon polariton (SSP). Æ different nature, such as phonon–polariton, –polariton, etc. PlasmonsPlasmons

23 −3 • free electrons in metal are treated as an electron of high density n ≈10 cm

• longitudinal density fluctuations (plasma oscillations) at eigenfrequency ω p ω 4πne2 • quanta of volume plasmons have , in the order 10eV h p = h m0 Volume plasmon polaritons

propagate through the volume for ω > ω p

Surface plasmon polaritons

Maxell´s theory shows that EM surface can propagate also along a metallic surface with a broad spectrum of eigen frequencies ω from ω = 0 up to = ω p 2

Particle (localized) plasmon polaritons PlasmonPlasmon resonance positions positions in in vacuumvacuum

+ + + + Bulk ω p metal ε = 0 ----

Metal ++ -- ++ -- ++ -- surface ε = −1 ω p 2

+ Metal sphere + + ε = −2 ω p 3 localized SPPs - - drude - model SurfaceSurface Plasmon Plasmon Photonics

Optical technology using Also called: - propagating surface plasmon polaritons • Plasmonics - localized plasmon polaritons • Plasmon photonics • Plasmon optics

Topics include: Localized / - nanoscopic local field enhancement - near-field tips

Propagation and guiding - photonic devices - near-field probes

Enhanced transmission - aperture probes - filters

Negative index of - perfect and

SERS/TERS - surface/tip enhanced

Molecules and - enhanced fluoresence quantum dots NanophotonicsNanophotonics using using plasmonic plasmonic circuits circuits

Nanoscale plasmon

• Proposal by Takahara et. al. 1997

hω

Metal Diameter << λ

• Proposal by Quinten et. al. 1998

hω

Chain of metal Diameter and spacing << λ

Atwater et.al., MRS Bulletin 30, No. 5 (2005) First experimental observation by Maier et. al. 2003 Subwavelength-scaleSubwavelength-scale plasmon plasmon waveguides waveguides

Krenn, Aussenegg, Physik Journal 1 (2002) Nr. 3 SomeSome applications applications of of plasmonplasmon resonant resonant nanoparticles nanoparticles

• SNOM probes T. Kalkbrenner et.al., J. Microsc. 202, 72 (2001) • Sensors

• Surface enhanced Raman scattering (SERS)

• Nanoscopic waveguides for light

S.A. Maier et.al., Nature Materials 2, 229 (2003) Light-matter interactions in solids

EM waves in matter Lorenz oscillator Isolators, Phonon polaritons Metals, Plasmon polaritons

Literature:

- C.F.Bohren, D.R.Huffman, Absorption and scattering of light by small particles

- K.Kopitzki, Einführung in die Festkörperphysik

- C.Kittel, Einführung in die Festkörperphysik EM-wavesEM-waves in in mattermatter (linear(linear media)media) -- definitions definitions

Polarization Complex κ P = ε 0 χE with suszeptibility χ N = n + i = ε

Complex dielectric function

Electric displacement ε = ε′ + iε′′ = 1+ χ

D = ε 0E + P = ε 0 ()1+ χ E = ε 0εE

Relationship between N and ε ε ′ 2 2 ε = n +κ ′′ = 2nκ EM-wavesEM-waves in in mattermatter (linear(linear media)media) -- dispersion dispersion 2π E = E ei()kr −ωt wavevector k = 0 λ i()kr −ωt ω = 2πf B = B0e

Dispersion in transparent media without absorption:

ω c c k and ε are real = k = k ε n ε >0

Dispersion generally:

2 ω ε(ω) and thus k = k´ + ik´´ are complex numbers! ()k⋅ k = 2 ε c ω

i(kr− t )()i k′r−ωt −k′′r E = E0e = E0e e

propagating exponential decay of HarmonicHarmonic oscillator oscillator (Lorentz) (Lorentz) modelmodel

- iωt m&x& + bx& + Kx = eE = eE0e ω0

+ ω e m iΘ eE x = 2 ω 2 E = Ae 0 − − iγω m

ω0 p = ex = ε 0αE ε P = Np = 0 χE ε χ ω 2 ω p =1+ =1+ 2 ω 2 0 − − iγω 2 ω0 = Km γ = bm 2 2 ωp = Ne / mε0 plasma frequency One-oscillatorOne-oscillator (Lorentz) (Lorentz) modelmodel

κ N = n + i = ε

(n −1)2 +κ 2 R = ()n +1 2 +κ 2

from Bohren/Huffman WeakWeak and and strongstrong molecular molecular vibrations vibrations

weak oscillator: ε‘> 1 strong oscillator: ε‘< 0 2 1 25 25 eps' eps'' eps' eps'' 1,6 0,8 20 20

1,2 0,6 15 15

0,8 0,4 10 10

0,4 0,2 5 5

0 0 0 0

-0,4 -0,2 -5 -5

-0,8 -0,4 -10 -10 860 880 900 920 940 860 880 900 920 940 / cm -1 wavenumber / cm -1

caused by: molecular vibrations lattice vibrations

examples: PMMA, PS, SiC, Xonotlit, Calcite, Si3N4 OpticalOptical properties properties of of polarpolar crystalscrystals

note: lattice has transversal T and longitudinal L γ = 0 oscillations but only transversal phonons can be excited by light

κ κ A n = 0, = ε N = n + i = ε B n = ε , κ = 0

ω ω 2 A k = i ε c (n −1)2 +κ 2 ()k ⋅k = 2 ε R = c ω ()n +1 2 +κ 2 B k = ε c BBA

i()kr−ωt A: total E = E0e B: transmission and reflection SiCSiC - - single single oscillator oscillator model model

γ > 0 GeneralGeneral dispersiondispersion for for nonconductor nonconductor

ω = 0 resonances 0 restoring permanent ω > 0 e.g. water 0

phonons molecular vibrations DispersionDispersion inin polarpolar crystals crystals - - phonon phonon polaritons polaritons ω γ = 0 Dispersionε relationω for γ = 0 ω ε ω 2 2 2 2 −ω between TO and LO k 2 = = ()ωL there is no solution for 2 2 s 2 2 real values ω and k c c T −ω

frequency gap: E ∝ e−iωte−kr

no propagation

reflection

strong of mechanical and electromagnetic waves, polariton like photon like MetalsMetals --Drude Drude modelmodel

(e/m)E m&x&+ bx& + Kx = eE → x = ω2 − ω2 −γωi γω θ 0 1 γω A = ; = tan −1 2 2 2 2 ω2 − ω2 (ω0 − ω ) + ( ) 0

ω 2 ωεp 0 E px=→==eNeε P x 22 ωω0 −−iγω εω Ne2 K ω 2 22==; ω ()=+ 1 p p mm0 ωω22−−iγω ωεεω0 ωε ω ω 0

PE()ω = εχω ()() ω 0 →=+ε ()ωχω 1 () DEEP()==+00 ()() () () MetalsMetals -- drude drude model model

- Intraband transitions ω ω - longitudinal plasma oscillations γ = 0 k = i ε k = ε - ω0 = 0 i.e. no restoring c c i(kr−ωt ) - ωεP = plasma frequency E = E e ω 0 κ 2 n = 0, = ε n = ε , κ = 0 ω p =1+ 2 ω 2 0 − − iγω

ω0 = 0 ε ω 2 =1− ω p 2 − iγω

ε total reflection transmission ω ω 2 ω 2 ′ =1− pγ ≈1− p 2 + 2 ω 2 ε ω ω >> γ = 1/τ (1/ collision time) γ collisions usually by electron-phonon scattering ω 2 2 ω ω γ ′′ = p γ ≈ p ()2 + 2 ω 3 generally: γ > 0 leads to of transmitted wave n > 0, κ > 0 P

ω = L

ω Aluminium Aluminium = 0 T

ω MetalsMetals --Drude Drude modelmodel

εm (ωp)= 0 Æ longitudinal field Æ bulk caused by Coulomb forces.

if γ≠0, εm (ω)= 0 Æ ω = ωp+iγ/2 Æ plasmon is the quantum of plasma oscillation with energy ħ ωp and lifetime τ=2/γ. plasmon is not an electron but a collection of electrons.

The damping constant γ is related to the average collision time 1/ τ Æ interactions with the lattice vibrations: electron-phonon scattering. non conductor = metals at high frequencies: intraband transitions acts mainly at low frequencies Æ Drude model as well.

at frequencies >> ωp metals are transparent: transparency. MetalsMetals --Drude Drude modelmodel

αεω The Drude model needs to consider the effect of bound electrons Æ lower lying shells Equation of motion has to include also the “restoring force” ω 2   p mx++ bx x = eE →Interband () =+ 1 22 ωω0 −−iγω α ω2 = 0 m FreeFree andand boundbound electronselectrons inin metalsmetals

Bound electrons contribute like a Lorenz oscillator

ε metal = ε drude + ε bound ε bound ε where ω 2 =1− ω p,d ε drude 2 γ − i dω ε ω drude ω 2 = ω p, j bound ∑ 2 2 γ j 0 − − i jω MetalsMetals -- plasmon plasmon polaritons polaritons

Plasmon polariton dispersion (γ = 0)

ω ω 2 2 2 2 = ω p + c k

ω = ck ω p E ∝ e−iωte−kr

no propagation

0 k

reflection DielectricDielectric function function of of metalsmetals andand polarpolar crystalscrystals

Metal Polar crystal collective free electron oscillations (plasmons) strong lattice vibrations (phonons) no restoring force

Reststrahlenband

plasma frequency transversal optical longitudinal optical (longitudinal oscillation) phonon frequency, TO phonon frequency, LO Surface Plasmon Polaritons (SPPs) - Introduction and basic properties

- Overview - Light-matter interaction - SPP dispersion and properties

Standard textbook: - Heinz Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings Springer Tracts in Modern Physics, Vol. 111, Springer Berlin 1988

Overview articles on Plasmonics: - A. Zayats, I. Smolyaninov, Journal of Optics A: Pure and Applied Optics 5, S16 (2003) - A. Zayats, et. al., Physics Reports 408, 131-414 (2005) - W.L.Barnes et. al., Nature 424, 825 (2003) SPPSPP atat metal/dielectricmetal/dielectric interfacesinterfaces

μ From Maxwell’s equations, combining the two curl eq. with Jext=ρext=0 ∂ 2 D ε ∂ 2 D ∇×∇× E = − → ∇2 E - = 0 0 ∂t 2 c 2 ∂t 2

Assuming inω general a harmonic time dependence

−itω 22 Er( ,-0te)= Er( ) →∇ E k0 ε E =

k = Helmholtz equation 0 c SPPSPP atat metal/dielectricmetal/dielectric interfacesinterfaces

In the defined geometry ε = ε(z) Æthe propagating waves can be described as EEz(xy,,ze)= ( ) ixβ

β = kx

Æinserted into the Helmholtz equation gives the

∂2 E (z) + ()k 22εβ−=E 0 ∂z2 0

which is the starting point for any general guided EM modes

TM or p modes Æ Ex, Ez and Hy ≠ 0 Two set of solutions

TE or s modes Æ Hx, Hz and Ey ≠ 0 SPPSPP atat metal/dielectricmetal/dielectric interfacesinterfaces

TM or p modes

ixβ −ik2,z z Hzy ()= Aee2

() 1 ix −kz2,z ε2(ω) Ezx = iA22 kee ωε ε02 β

1 ix −kz2,z Ezz ()=− A1 ee ωε02 ε β

ixβ kz1,z Hzy ()= Aee1 1 E ()ziAkee=− ix kz1,z ε (ω) x 11ωε ε 1 01 β

1 ix kz1,z Ezz ()=− A2 ee ωε01 ε β

where Re[ε1]<0 and 1/|kz| defines the evanescent decay length ┴ to the interface Æ wave confinement SPPSPP atat metal/dielectricmetal/dielectric interfacesinterfaces

β AA= Continuity of εiEz Æ 12 ε12ε = k0 ε12+ ε k ε Continuity of E Æ 2,z = − 2 i,x k ε 2 1,z 1 εi kkiz,0= 22 2 ε12+ ε kk1,z + 0εβ 1 −=0 H has to fulfill the wave eq.Æ i,y 22 2 Dispersion relations kk2,z + 0 2εβ−=0 Valid for both real and complex ε TE or s modes

Continuity of Ey Hx Æ Ak11,2,( zz+ k)=→00 A 1 == A 2

SPP only exist for TM (p) SPPSPP atat metal/dielectricmetal/dielectric interfacesinterfaces

The interface mode have to fulfill some conditions in order to exist

Im[()]Re[()]εiiωεω<

Propagating interface waves (The is valid) Æ β real⇒∧+<εε εε12 ε ε ε ()0 1 2 or 12 ∧+> ()0 1 2

β complex Æ damped propagation along the interface Bound solution Æ vertical components are imaginary Æ ε12+ ε < 0

εωεω12( ) ( )∧ εω⎣⎦⎡⎤ 1( εω)+< 2( ) 0

One of the dielectric functions must be negative and > than the other DispersionDispersion relationrelation of of SPPsSPPs

± i()kx x±kz z−ωt ESP = E0 e ω 2 ε ω photon ⎛ ⎞ ε k 2 = m d in air x ⎜ ⎟ ε ω = ck ⎝ c ⎠ m + ε d x ω ω 2 ε 2 ω p 2 ⎛ ⎞ ε′ → −ε k = ⎜ ⎟ ε d ω d zd c ()+ ε 1 ⎝ ⎠ m d SP = p ω 1+ ε d ωsurface plasmon polariton 2 ε 2 ε ⎛ ⎞ +ε k 2 = m = m d ck zm ⎜ ⎟ ε ε x ⎝ c ⎠ ()m + ε d mε d ε kx ε Re()m < 0 and m > ε d → real kx

kzd and kzm are imaginary SPPSPP atat metal/dielectricmetal/dielectric interfacesinterfaces

surface plasmon polaritons are bound waves Æ SPP excitations lie on the right of the light line

Radiation into metal occurs if ω > ωp

Between the bound and the radiative regime β is imaginary Æ no propagation

for small k (

1/2 1/2 for large k, ωsp = ωp / (1+ε2) ~ ωp / (2)

in the limit Im[ε1(ω)] = 0, β → ∞ and vg → 0 Æ the mode acquire an electrostatic character Æ Surface Plasmon

but real metals suffer also from intraband transitions Æ damping and ε1 is complex Æ the quasibound regime is allowed

at ω ~ ωsp Æ better confinement of the SPP but small propagation length ( Æ increased damping) VolumeVolume vs. vs. surfacesurface plasmon plasmon polariton polariton

ω ω = ckx ω photon 2 = ω 2 + c2k 2 p x in air plasmon polariton

ωp volume plasmon with damping ω surface plasmons ω 1 ε′ → −ε d non-propagating = SP p 1+ ε collective oscillations d of electron plasma surface plasmon polariton near the surface ω ε +ε = m d ck ε x mε d

kx 1 x − k → ′ ε surface phonon polariton photon in air phonon phonon 2 vs. vs. SiC, SiO III-V, II-VI-

ω Phonon polaritons: Light-phonon coupling polar in crystals • • TO LO

ω ω 1

+ ε ε x ck = 1 x − ω k → ′ ε photon in air SP dispersion - plasmon SP dispersion - plasmon surface plasmon polariton metals semiconductors

ω p Plasmon polaritons: Light-electron coupling in • •

ω SPSP propagationpropagation length length

ω εε d =1 metal/air ε ω 2 ′ ′′ m interface kx = kx + ikx = p Drude c ε +1 m =1− ω 2 m + iγω model γ = 0.2

′ ′′ ik x x ikx x −kx x 10 E()x = E0e = E0e e 7.5 5 ε m′′ 2.5 propagating term exponential decay 0.4 0.6 0.8 1 ω ω p in x-direction -2.5 -5 -7.5 ε m′ -10 propagation ω 1 p length ωSP = Lx = Lx (λvac ) 2 2k′′ 200 x ! 100 ε = −1 λ 50 20 λ 10 5 Example silver: = 514.5 nm : L = 22 μm 2 x 1 ω ω =1060 nm : Lx = 500 μm 0.2 0.4 0.6 0.8 1 p SPPSPP fieldfield perpendicular perpendicular to to surfacesurface

dielectric z 1 − Imk z ε ω z Lz = d () E(z)= E0e Im kz kx E z z-decay length (skin depth): metal ε ()ω z m x

Examples:

silver: λ = 600 nm : Lz,m = 390 nm and Lz,m = 24 nm

: λ = 600 nm : Lz,m = 280 nm and Lz,m = 31 nm SPPsSPPs have have transversal transversal andand longitudinallongitudinal el.el. fieldsfields

E z ω ω Themag. fieldH is SP p H y parallel to surface Ex ⊗+ + - - and perpendicular to −1 ω z propagation x ε m′

kx El. field Ez = i Ex kz

E −ε E ε zd = i m zm = −i d E −ε Ex ε d x m

At large k , i.e. close to ε = - ε , At large ε m ′ values, x d both components become equal the el. field in air/diel. has a strong Ez = ±iEx (air: +i, metal: -i) transvers Ez component compared to the longitudinal component Ex

In the metal Ez is small against Ex DispersionDispersion andand excitationexcitation ofof SPPSPP

SPP are 2D EM waves propagating at the interface conductor-dielectric (bound waves)

β > kd Æ evanescent decay at both interfaces Æ confinement

Æ SPP dispersion curve lies to the right of the light line

Æ excitation by 3D light beams is not possible

Æ -matching techniques are required

Excitation by charged impact

R. H. Ritchie, Phys. Rev. 106, 874 (1957) Æ theoretical investigations of plasma losses in thin metallic films εω1/2 ε εω ω Æ predicted an additional loss at ħωp/(2) Æ measured by C. J. Powell and J. B. Swan, Phys. Rev. 118, 640 (1960) ωω2 Quasi-static electromagnetic pp md()+=∧01 m =− ()2 ⇒ sp = surface modes Æ ω 1+ ε d ⇒→∞β DispersionDispersion andand excitationexcitation ofof SPPSPP

Bulk plasmon Surface plasmon

Progressive oxidation DispersionDispersion andand excitationexcitation of of SPPsSPPs

in a dielectric k of the photon is increased

photon in air k of photon in dielectric can equal k of SPP photon in ω dielectric SPP can be excited by p-polarized light (SPP has longitudinal component) SPP dispersion

Kretschmann configuration kx

kx k of photon in air is always < k of SPP z thin metal film x dielectric θ0 no excitation of SPP is possible E R ExcitationExcitation by by ATR ATR

Kretschmann configuration Otto configuration

ε m k ε d kx x z z ε ε d m x x ε ε 0 0 θ θ0 0 E E R R

total reflection at prism/metal interface total reflection at prism/dielectric medium -> evanescent field in metal -> evanescent field excites surface plasmon -> excites surface plasmon polariton at at interface dielectric medium/metal interface metal/dielectric medium usful for surfaces that should not be damaged metal thickness < skin depth or for surface phonon polaritons on thick crystals

distance metal – prism of about λ

ATR: Attenuated Total Reflection ExcitationExcitation by by ATR ATR

SPP excitation requires kphoton,x = kSP,x

k k ε d SP,x SP,x

ε m k k ε 0 photon,x θ photon,x θ z R x R

kphoton,x < kSP,x kphoton,x = kSP,x

no SPP excitation SPP excitation ExcitationExcitation by by Kretschmann Kretschmann configurationconfiguration

ω z ε = c / kx 0 sin(θ0 ) x ω ω = ck ω ε photon photon in kx = 0 sin()θ0 ε d c in air dielectric

ε m ω ε 0 k = ε ω0 SPP dispersion c 0 ω θ0 ε m kx = c ε m +1

ω k = c 0 kx ω kx ε ε +1 c Resonance 0 = c m = ε 0 condition kx m 0 sin()θ0 KretschmannKretschmann configurationconfiguration – – angle angle scanscan

illumination freq. ω0= const.

s-polarized θ0 R -> no excitation of SPPs R p-polarized photon ω in air θ0

ω0

θ0

kx DispersionDispersion andand excitationexcitation ofof SPPSPP

64 nm thick Au film 810nm p-polarized, Δ~0.207º 1160nm p-polarized, Δ~0.099º HeNe SPP excitation at Au-air interface (ATR) 632nm p-polarized, Δ~0.98º

0.9

0.6 41.92º Reflectivity [arb.]

0.3 42.57º 43.92º

36 40 44 48 Incident angle [degrees]

64 nm thick Au film 633nm p-polarized, data 1.0 633nm p-polarized, theory

0.9

0.8

0.7

0.6

0.5 Reflectivity [arb.] 0.4

0.3

0.2 43.92º

36 38 40 42 44 46 48 Incident angle [degrees] MethodsMethods of of SPPSPP excitationexcitation

nprism > nL !! DispersionDispersion andand excitationexcitation ofof SPPSPP

Excitation by grating coupling

k// θ π k νν kk= sinθ kg // 2 →=β k sinθν +g kg== g a

E. Devaux, T. W. Ebbesen, J.C. Weeber, A. Dereux, Appl. Phys. Lett. 83, 4936 (2003) DispersionDispersion andand excitationexcitation ofof SPPSPP

Periodicity a = 760 nm & p-pol beam Periodicity a = 760 nm & s-pol beam

Periodicity a = 700 nm & p-pol beam DispersionDispersion andand excitationexcitation ofof SPPSPP

Excitation by highly focused optical beams

⎛⎞β θSPP=>arcsin ⎜⎟θ c ⎝⎠nk0

A. Bouhelier and G. P. Wiederrecht, Opt. Lett. 30, 884 (2005) DispersionDispersion andand excitationexcitation ofof SPPSPP

Near-field excitation

Small aperture Æ wavevector

components k0 < β < k Æ phase matched excitation

B. Hecht, H. Bielefeld, L. Novotny, Y. Inouye, D. W. Pohl Phys. Rev. Lett. 77, 1889 (1996) LocalizedLocalized SurfaceSurface PlasmonPlasmon

Overview

True color image of a sample containing gold and silver nanospheres as well as gold nanorods photographed with a dark field microscope. Each dot corresponds to light scattered by an individual at the plasmon resonance. The resonance varies from blue (silver nanospheres) via green and yellow (gold nanospheres) to orange and red (nanorods). LocalizedLocalized SurfaceSurface PlasmonPlasmon

Overview

Illuminated from inside Illuminated from outside

Lycurgus cup LocalizedLocalized SurfaceSurface PlasmonPlasmon

Overview

Localized Surface Plasmon

SPP are 2D, dispersive EM waves propagating at the interface conductor-dielectric

SP are non-propagating collective oscillations of electron plasma near the surface

LSP are non-propagating excitations of the conduction electrons of a metallic nanostructure coupled to an EM field. LocalizedLocalized SurfaceSurface PlasmonPlasmon

The curved surface of the nanostructure act as a restoring force Æ immobile positively charged core

The curved surface of the nanostructure allows the excitation of the LSP by 3D light

The resonance falls into the visible region for Au and Ag nanoparticles LocalizedLocalized SurfaceSurface PlasmonPlasmon

Particle in an electrostatic field Mie Theory Æ nanoparticle acts as an Æ rigorous electrodynamic electric approach ∂∂2rr v Drude mm+Γ = eeE −iwt Γ = F ee∂tt2 ∂ 0 l

νF Æ Fermi velocity (about 1.4 nm/fs in the case of Au and Ag) l is the electron mean free path (lAu=42 nm and lAg=52 nm at 273K )

εω 222 ωωωΓppp ()=−11222 =− +i ωΓω++i ωΓω ()ωΓ22+

it is not possible to ignore the contribution to the dielectric function of the interband transitions LocalizedLocalized SurfaceSurface PlasmonPlasmon

Sub-wavelength metal particles

P E0 a θ z εd ε(ω)

electrostatic approach Æ 2 Laplace equation for the electric ∇ Φ=0 →E=- ∇Φ potential + boundary conditions εε 3ε d Φ=−in Er0 cosθ + 2 d −εε cosθ Φ=−Ercosθ + d Ea3 out 00+ 2 r 2 εεd

Applied field Oscillating dipole field LocalizedLocalized SurfaceSurface PlasmonPlasmon

3ε EE= d in εε+ 2 0 E=-∇Φ → d 3(nnp⋅− ) p 1 EEout =+0 3 4 0 d r πε ε

pr⋅ Φ=−out Er0 cosθ + 3 4 0 d r πε ε πε ε pE= ε 00εαd 3 −εεd pE= 4 00d a +εε2 d LocalizedLocalized SurfaceSurface PlasmonPlasmon

απ ε ()ωε− = 4 a3 d ε ()ωε+ 2d

Complex of a sub-wavelength diameter in the electrostatic approximation LocalizedLocalized SurfaceSurface PlasmonPlasmon

εω() ε+→ 2d 0 Complex polarizability enhancement

Re[]εω εω ( ε )=− 2d if Im[ ( )] is small

Fröhlic condition LocalizedLocalized SurfaceSurface PlasmonPlasmon

Corresponding absorption & scattering cross sections calculated via the Pointing vector σαfrom the full EM field associated with an oscillating dipole

πεωεπ 2 σαπ4 k 2 8 46 εω() ε− d σ ext= σσ sca+ abs sca ==ka 63()2+ d

3 ⎡ ε ()ωε− d ⎤ I0 (ω) abs ==kkaIm() 4 Im ⎢ ⎥ Iext()ω = σω ext () ⎣ε ()ωε+ 2d ⎦ S LocalizedLocalized SurfaceSurface PlasmonPlasmon

Corresponding absorption & scattering cross sections calculated via Mie Theory in the case of large particles where the electrostatic approx. brakes down and the retardation effects are to be considered

∞ 2π 22 σ =++21nab sca2 ∑()( n n ) k n=1 2 ∞ =++21Renab extσ 2 ∑()() n n k πn=1

σ abs= σσ ext− sca LocalizedLocalized SurfaceSurface PlasmonPlasmon

Damping mechanism In a quasi-particle picture, damping is described as population decay Æ radiative (by emission of a photon), or nonradiative

Drude-Sommerfeld model Æ plasmon is a superposition of many independent electron oscillations

Nonradiative damping Æ due to a dephasing of the 2= oscillation of individual electrons (scattering events with Γ = phonons, lattice , other conduction or core electrons, the T metal surface, impurities, etc.) 2 111 Pauli-exclusion principle, the electrons can only be excited =+* into empty states in the conduction band Æ inter- and TTT2122 intraband excitations by electron from either the d-band or the conduction band Decay time (radiative & non Pure dephasing time energy loss processes) (elastic collisions)

* TT21 →= TT 212

510fs≤≤ T2 fs LocalizedLocalized SurfaceSurface PlasmonPlasmon

Damping mechanism for very small nanoparticles (< 10nm)

v ΓΓ=+A F bulk R

Experimental methods

Total Internal Reflection Dark Field Microscopy Dark Field Microscopy (TIRM) in reflection in transmission LocalizedLocalized SurfaceSurface PlasmonPlasmon

Detection and of Gold Nanoparticles Using Supercontinuum White Light Confocal Microscopy K. Lindfors, T. Kalkbrenner, P. Stoller, V. Sandoghdar, PRL 93, 037401-1 (2004)

−π 2 ErEeri= ληαληε EsEseE==iϕ si iπ 3 D ε ()ωε− d →=s() () =d 2()2ε ωε+ d

222 2 IEEErsrsmrs=+ = i{ +−2sinϕ}

2 IImr()λλ− () 2 σ ()λαλαλϕ==−2 () 2 ()sin Irr () ηη r λ LocalizedLocalized SurfaceSurface PlasmonPlasmon

(a) 60 ± 12 nm (b) 31 ± 6 nm (c) 20 ± 4 nm (d) 10.3 ± 1.0 nm (e) 5.4 ± 0.8 nm LocalizedLocalized SurfaceSurface PlasmonPlasmon

20nm Au particles

6,6x105

6,4x105

6,2x105 240 nm

5 cps 6,0x10 250 nm

5,8x105

5,6x105

5,4x105

012345 Position (μm)