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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
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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.
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• 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
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2pω
pω
3pω
++ -- ++ -- ++ --
+ + + +
+++
---
Bulkmetal
Metalsurface
Metal spherelocalized SPPs
Plasmon resonance positions in vacuumPlasmon resonance positions in vacuum
0=ε
1−=ε
2−=εdrudemodel
- - - -
drudemodel
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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
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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
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Subwavelength-scale plasmon waveguidesSubwavelength-scale plasmon waveguides
Krenn, Aussenegg, Physik Journal 1 (2002) Nr. 3
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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)
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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
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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
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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
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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
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One-oscillator (Lorentz) modelOne-oscillator (Lorentz) model
from Bohren/Huffman
( )( ) 22
22
11
κκ
+++−
=nnR
εκ =inN +=
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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
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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
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SiC - single oscillator modelSiC - single oscillator model
γ > 0
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permanent dipolese.g. water
phononsmolecular vibrations
ω0 = 0
ω0 > 0
resonancesrestoring forces
General dispersion for nonconductorGeneral dispersion for nonconductor
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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:
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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
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ωT = 0 ωL = ωP
AluminiumAluminium
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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,
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Metals - plasmon polaritonsMetals - plasmon polaritons
Plasmon polariton dispersion (γ = 0)
ωp
2222 kcp += ωω
0
ck=ωkrti eeE −−∝ ω
no propagation
reflection
k
ω
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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
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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
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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
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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
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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
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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:
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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
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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
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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 +
=
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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
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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
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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
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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
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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
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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
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Methods of SPP excitationMethods of SPP excitation
nprism > nL !!
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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
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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
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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
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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