Surface Plasmon Polaritons (SPPs) -

Introduction and basic 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)

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

Elementary excitations and polaritonsElementary excitations and polaritons

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

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

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

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

• free electrons in metal are treated as an electron liquid of high density

• longitudinal density fluctuations (plasma oscillations) at eigenfrequency

• quanta of volume plasmons have energy , in the order 10eV

propagate through the volume for frequencies

0

24mne

pπω hh =

Volume plasmon polaritons

323cm10 −≈n

Surface plasmon polaritons

PlasmonsPlasmons

Maxell´s theory shows that EM surface waves can propagate also along a metallic surface with a broad spectrum of eigen frequencies

from ω = 0 up to 2pωω =

Particle (localized) plasmon polaritons

pωω >

pω

2pω

pω

3pω

++ -- ++ -- ++ --

+ + + +

+++

---

Bulkmetal

Metalsurface

Metal spherelocalized SPPs

Plasmon resonance positions in vacuumPlasmon resonance positions in vacuum

0=ε

1−=ε

2−=εdrudemodel

- - - -

drudemodel

Surface Plasmon PhotonicsSurface Plasmon PhotonicsOptical technology using- propagating surface plasmon polaritons- localized plasmon polaritons

Topics include:

Localized resonances/ - nanoscopic particleslocal field enhancement - near-field tips

Propagation and guiding - photonic devices- near-field probes

Enhanced transmission - aperture probes- filters

Negative index of refraction - perfect lensand metamaterials

SERS/TERS - surface/tip enhanced Raman scattering

Molecules and - enhanced fluoresencequantum dots

Also called:• Plasmonics• Plasmon photonics• Plasmon optics

Nanophotonics using plasmonic circuitsNanophotonics using plasmonic circuits

Atwater et.al., MRS Bulletin 30, No. 5 (2005)

• Proposal by Takahara et. al. 1997

Metal nanowireDiameter << λ

• Proposal by Quinten et. al. 1998

Chain of metal nanoparticlesDiameter and spacing << λ

First experimental observation byMaier et. al. 2003

ωh

ωh

Nanoscale plasmon waveguides

Subwavelength-scale plasmon waveguidesSubwavelength-scale plasmon waveguides

Krenn, Aussenegg, Physik Journal 1 (2002) Nr. 3

Some applications of plasmon resonant nanoparticlesSome applications of plasmon resonant nanoparticles

• SNOM probes

• Sensors

• Nanoscopic waveguides for light

• Surface enhancedRaman scattering (SERS)

T. Kalkbrenner et.al., J. Microsc. 202, 72 (2001)

S.A. Maier et.al., Nature Materials 2, 229 (2003)

EM waves in matterLorenz oscillator

Isolators, Phonon polaritonsMetals, Plasmon polaritons

Light-matter interactions in solids

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-waves in matter (linear media) - definitionsEM-waves in matter (linear media) - definitions

χχε lity suszeptibi with 0 EP =

( ) EEPED εεχεε 000 1 =+=+=

+1= χεεε ′′+′= i

εκ =inN +=

κεκε

nn2

22

=′′+=′

Polarization

Electric displacement

Complex dielectric function

Complex refractive index

Relationship between N and ε

k ⋅ k( ) =ω 2

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

( ) ( ) rkrkkr EEE ′′−−′− == eee titi ωω00

propagating wave exponential decay of amplitude

E = E0ei kr −ωt( )

B = B0ei kr −ωt( )

wavevector k = 2πλ

frequency ω = 2πf

knckc

==ε

ω

Dispersion in transparent media without absorption:

ε > 0

k and ε are real

Dispersion generally:

EM-waves in matter (linear media) - dispersionEM-waves in matter (linear media) - dispersion

Harmonic oscillator (Lorentz) modelHarmonic oscillator (Lorentz) model

ω0

tieeeKbm ω0EExxx ==++ &&&

0

0

EpPExpχε

αε==

==Ne

γωωωω

χεi

p

−−+=+= 22

0

2

11

meAe

ime i EEx Θ=

−−=

γωωω 220

ω0

+

-

ω02 = K m

γ = b mωp

2 = Ne2 / mε0 plasma frequency

One-oscillator (Lorentz) modelOne-oscillator (Lorentz) model

from Bohren/Huffman

( )( ) 22

22

11

κκ

+++−

=nnR

εκ =inN +=

Weak and strong molecular vibrationsWeak and strong molecular vibrations

weak oscillator: ε‘ > 1

examples: PMMA, PS, proteins

wavenumber / cm -1

-0,8

-0,4

0

0,4

0,8

1,2

1,6

2

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

860 880 900 920 940

eps' eps''

caused by: molecular vibrations

wavenumber / cm -1

strong oscillator: ε‘ < 0

SiC, Xonotlit, Calcite, Si3N4

-10

-5

0

5

10

15

20

25

-10

-5

0

5

10

15

20

25

860 880 900 920 940

eps' eps''

crystal lattice vibrations

Optical properties of polar crystalsOptical properties of polar crystalsnote:lattice has transversal T and longitudinal L oscillations but only transversal phononscan be excited by light

( ) εω2

2

c=⋅kk

εκ =+= inNεκ == ,0n

( )( ) 22

22

11

κκ

+++−

=nnR

( )tie ω−= krEE 0

A

0 , == κεnB

AB Bεω

ck =

εωc

ik =A

B

A: total reflectionB: transmission and reflection

γ = 0

SiC - single oscillator modelSiC - single oscillator model

γ > 0

permanent dipolese.g. water

phononsmolecular vibrations

ω0 = 0

ω0 > 0

resonancesrestoring forces

General dispersion for nonconductorGeneral dispersion for nonconductor

Dispersion in polar crystals - phonon polaritonsDispersion in polar crystals - phonon polaritons

( ) 22

22

2

2

2

22

ωωωω

ωεωεω−

−==

T

Lscc

k

Dispersion relation forbetween TO and LO there is no solution forreal values ω and k

photon like

γ = 0

strong coupling of mechanical and electromagnetic waves, polariton like

krti eeE −−∝ ω

no propagation

reflection

γ = 0

frequency gap:

Metals - drude modelMetals - drude model- Intraband transitions- longitudinal plasma oscillations- ω0 = 0 i.e. no restoring force- ωP = plasma frequency

γωωωω

εi

p

−−+= 22

0

2

1

ω0 = 0

γωωω

εip

−−= 2

2

1

2

2

22

2

11ωω

γωω

ε pp −≈+

−=′

( ) 3

2

22

2

ωγω

γωωγω

ε pp ≈+

=′′

ω >> γ = 1/τ (1/ collision time) collisions usually by electron-phonon scattering

εκ == ,0n

( )tie ω−= krEE 00 , == κεn

εωc

k =εωc

ik =

total reflection transmission

generally:γ > 0 leads to damping of transmitted wave n > 0, κ > 0

γ = 0

ωT = 0 ωL = ωP

AluminiumAluminium

Free and bound electrons in metalsFree and bound electrons in metals

Bound electrons contribute like a Lorenz oscillator

where

bounddrudemetal εεε +=

drudeε

boundε

ωγωω

εd

dpdrude i−

−= 2

2,1

∑ −−=

j j

jpbound i ωγωω

ωε 22

0

2,

Metals - plasmon polaritonsMetals - plasmon polaritons

Plasmon polariton dispersion (γ = 0)

ωp

2222 kcp += ωω

0

ck=ωkrti eeE −−∝ ω

no propagation

reflection

k

ω

Dielectric function of metals and polar crystalsDielectric function of metals and polar crystalsPolar crystal

strong lattice vibrations (phonons)Metal

collective free electron oscillations (plasmons)no restoring force

plasma frequency(longitudinal oscillation)

transversal opticalphonon frequency, TO

longitudinal opticalphonon frequency, LO

Reststrahlenband

Surface Plasmon Polaritons (SPPs) -

Introduction and basic 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)

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

Surface polaritons (SPs)Surface polaritons (SPs)

Coupled state between photons and elementary excitations at the interface between a material with ε < 0 and a dielectric.

- radiative surface polaritons are coupled with propagating EM waves.- nonradiative surface polaritons do not couple with propagating EM waves.- for perfectly flat surfaces SPs are always nonradiative!- mixed transversal and longitudinal EM field.

In contrast to TIR the surface polariton field on both sides of the interface are evanescent.

++ -- ++ -- ++ --

surface wave

dielectric ( ) 0>ωε d

( ) 0<ωεzx

z

( ) zkzezE Im−∝

( )tzkxki zxe ω−±±= 0SP EE

SPxk

λπ2

=′

xxx kikk ′′+′=

( )zE

SP fieldsSP fields

( )tzkxki

z

xzxe

E

EE ω−+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= 0

( )tzkxki zdxde ω−+

( )tzkxki zmxme ω−−

dielectric:

metal/polar crystal:

++ -- ++ -- ++ --

++ -- ++ -- ++ --++ -- ++ -- ++ --

++ -- ++ -- ++ --

xE

zE

zx

⊗yH+ + - - xE

zE yH

xk

( )tzkxkiy

zxeHH ω−+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

0

0

Derivation of SPP dispersion – boundary conditionsDerivation of SPP dispersion – boundary conditions

( ) ( )tzkxkizmxmm

zmxmeEE ω−−= ,0,E

( ) ( )tzkxkiymm

zmxmeH ω−−= 0,,0H

( ) ( )tzkxkiydd

zdxdeH ω−+= 0,,0H

( ) ( )tzkxkizdxdd

zdxdeEE ω−+= ,0,E

++ -- ++ -- ++ --

longitudinal surface wave

dielectric

metal

εd ω( )

εm ω( )zx

xdxm EE =

ydym HH =

Boundarybonditions (z=0)

zddzmm EE εε =

xxdxm kkk ==

+ + - -

0=yE

xE

zE

0== zx HH

zx

⊗yH

0>z

0<z

Derivation of SPP dispersionDerivation of SPP dispersion

EHtc ∂

∂=

1 curl ε

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛∂=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛×

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂∂∂

z

x

ty

z

y

x

E

E

cH 01

0

0ε

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂

∂−

z

x

yx

yz

E

E

cH

H00 ωε

Maxwell eq.:

z-component:

Diel.:

Metal:zmmyx E

cHk ωε−=+

zddyx Ec

Hk ωε−=+

x-component:

Diel.:

Metal: ymzmymz HkH +=∂−

ydzdydz HkH −=∂−

xmyzm Ec

Hk ωε−=+

xdyzd Ec

Hk ωε+=+

( )tzkxki zdxe ω−+

( )tzkxki zmxe ω−−

Diel.:

Metal:

xmyzm Ec

Hk ωε−=

xdyzd Ec

Hk ωε=

0=+m

zm

d

zd kkεε

Derivation of SPP dispersionDerivation of SPP dispersion

I:

II:

I / II:d

m

zd

zm

kk

εε

−=

x-component:

Diel.

Metal

dm

dmx c

kεε

εεω+

⎟⎠⎞

⎜⎝⎛=

22

( )dm

dzd c

kεε

εω+

⎟⎠⎞

⎜⎝⎛=

222

( ) xdmm k real and 0Re →>< εεε

kzd and kzm are imaginary

xmxd kk =

Dispersion relation of SPPsDispersion relation of SPPs

at interface metal/dielectric:

2

222

ckk dzdx

ωε=+

2

222

ckk mzmx

ωε=+

dielectric:

metal:

0=+m

zm

d

zd kkεε

2222 kkkk zyx =++

generally:

( )dm

mzm c

kεε

εω+

⎟⎠⎞

⎜⎝⎛=

222

0 2

2

cωε

dm

dmx c

kεε

εεω+

⎟⎠⎞

⎜⎝⎛=

22

( )dm

dzd c

kεε

εω+

⎟⎠⎞

⎜⎝⎛=

222

( ) xdmm k real and 0Re →>< εεε

kzd and kzm are imaginary

Dispersion relation of SPPsDispersion relation of SPPs

( )dm

mzm c

kεε

εω+

⎟⎠⎞

⎜⎝⎛=

222

( )tzkxki zxe ω−±±= 0SP EE

dpSP ε

ωω+

=1

1

ωp

photonin air

kx

ω

xck=ω

dεε −→′

surface plasmon polariton

xdm

dm ckεεεεω +

=

photonin air

kx

ω

dpSP ε

ωω+

=1

1

xck=ω

dεε −→′

surface plasmon polariton

2222xp kc+= ωω

ωp

plasmonpolariton

xdm

dm ckεεεεω +

=

Volume vs. surface plasmon polaritonVolume vs. surface plasmon polariton

surface plasmonsnon-propagatingcollective oscillationsof electron plasmanear the surface

with dampingvolume plasmon

ωp

photonin air

xk

ω

1−→′ε

surface plasmonpolariton

ωLO

photonin air

1−→′εsurface phononpolaritonωTO

SP dispersion - plasmon vs. phononSP dispersion - plasmon vs. phonon

ω

xk

Plasmon polaritons:

Light-electron coupling in • metals • semiconductors

Phonon polaritons:

Light-phonon coupling in polar crystals• SiC, SiO2• III-V, II-VI-semiconductors

εεω 1+

= xck

Drude model

1=dε

SP propagation lengthSP propagation length

pωω

mε ′

mε ′′

)( vacxL λ

γωωω

εip

m +−= 2

2

1

2.0=γ

0.4 0.6 0.8 1

-10-7.5

-5-2.5

2.55

7.510

0.2 0.4 0.6 0.8 1

12

51020

50100200 1

2−=

=

ε

ωω p

SP

( ) xkxkixik xxx eeex ′′−′== 00 EEE

propagating term exponential decayin x-direction

1+=′′+′=

m

mxxx c

kikkε

εωmetal/airinterface

xx k

L′′

=21 propagation

length

intensity !

Example silver: m 22 :nm 5.514 μλ == xLm 500 :nm 1060 μλ == xL pωω

SPP field perpendicular to surfaceSPP field perpendicular to surface

( ) zkzez Im0

−= EEz

z kL

Im1

=

z-decay length(skin depth):

Examples:

silver:

gold:

nm 24 and nm 390 :nm 600 ,, === mzmz LLλ

nm 31 and nm 280 :nm 600 ,, === mzmz LLλ

Ez

zdielectric

metal

εd ω( )

εm ω( )zx

xk

SPPs have transversal and longitudinal el. fieldsSPPs have transversal and longitudinal el. fields

xz

xz E

kkiE =

At large values,

the el. field in air/diel. has a strongtransvers Ez component compared to thelongitudinal component Ex

mε ′

In the metal Ez is small against Ex

At large kx, i.e. close to ε = - εd, both components become equal

xz iEE ±= (air: +i, metal: -i)

m

d

x

zm iEE

εε

−−=

pω

ω

mε ′

SPω

1−

The mag. field H isparallel to surfaceand perpendicular to propagation

d

m

x

zd iEE

εε−

=

+ + - -xE

zE

zx

⊗yH

El. field

Dispersion and excitation of SPPsDispersion and excitation of SPPs

thin metal filmdielectric

zx

Kretschmann configuration

photon indielectric

k of photon in air is always < k of SPP

photon in air

kx

ω

SPP dispersion

no excitation of SPP is possible

in a dielectric k of the photon is increased

SPP can be excited by p-polarized light (SPP has longitudinal component)

k of photon in dielectric can equal k of SPP

E0θ

R

kx

Methods of SPP excitationMethods of SPP excitation

nprism > nL !!

zx

E0θ

R

kxdε

mε

0ε

Excitation by ATRExcitation by ATR

Kretschmann configuration Otto configuration

zx

E0θ

R

kx

dεmε

0ε

total reflection at prism/metal interface-> evanescent field in metal-> excites surface plasmon polariton at

interface metal/dielectric medium

metal thickness < skin depth

total reflection at prism/dielectric medium-> evanescent field excites surface plasmon

at interface dielectric medium/metal

usful for surfaces that should not be damagedor for surface phonon polaritons on thick crystals

distance metal – prism of about λ

ATR: Attenuated Total Reflection

zx

θ

R

kSP,xdε

mε

0ε kphoton,x

< kSP,xkphoton,x

θ

R

kSP,x

kphoton,x

= kSP,xkphoton,x

no SPP excitation SPP excitation

Excitation by ATRExcitation by ATR

SPP excitation requires = kSP,xkphoton,x

Excitation by Kretschmann configurationExcitation by Kretschmann configuration

photon indielectric

photonin air

xk

ω

SPP dispersion

ck=ω

z

x

0θ

dε

mε0ε

0εωc

k =

ck ω

=

( )00 sin θεωc

kx =

( )00 sin/ θεω xkc=

0ω

( )0000

sin1

θεεεω cc

k m

m

x

=+

= Resonancecondition

0xk

1+=

m

mx c

kε

εω

Kretschmann configuration – angle scanKretschmann configuration – angle scan

0θ R

0θ

p-polarized

s-polarized-> no excitation of SPPs

illumination freq. ω0= const.

photonin air

kx

ω

0ω

R

0θ