Surface Plasmon Polaritons (SPPs) - Introduction and basic properties

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)

Elementary 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
Volume plasmon polaritons
propagate through the volume for frequencies
Surface plasmon polaritons
Plasmons
h ω = h
p
4πne
m
0
2
n
≈
23
10 cm
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 ω = ω 2
Particle (localized) plasmon polaritons
p
ω > ω p
ω
p
−3

Bulk
metal
Metal
surface
Metal sphere
localized SPPs
Plasmon resonance positions in vacuum
+ + + +
- - - -
++ -- ++ -- ++ --
+
-
+ +
-
-
ω p
ε =
ε =
ε =
0
−1
−2
drude
model
drude
model
ω p
ω p
2
3

Optical technology using
- propagating surface plasmon polaritons
- localized plasmon polaritons
Topics include:
Surface Plasmon Photonics
Localized resonances/ - nanoscopic particles
local field enhancement - near-field tips
Propagation and guiding - photonic devices
- near-field probes
Enhanced transmission - aperture probes
- filters
Negative index of refraction - perfect lens
and metamaterials
SERS/TERS - surface/tip enhanced Raman scattering
Molecules and - enhanced fluoresence
quantum dots
Also called:
• Plasmonics
• Plasmon photonics
• Plasmon optics

Nanophotonics using plasmonic circuits
Atwater et.al., MRS Bulletin 30, No. 5 (2005)
Nanoscale plasmon waveguides
• Proposal by Takahara et. al. 1997
Metal nanowire
Diameter

Subwavelength-scale plasmon waveguides
Krenn, Aussenegg, Physik Journal 1 (2002) Nr. 3

Some applications of plasmon resonant nanoparticles
• SNOM probes
• Sensors
• Surface enhanced
Raman scattering (SERS)
• Nanoscopic waveguides for light
S.A. Maier et.al., Nature Materials 2, 229 (2003)
T. Kalkbrenner et.al., J. Microsc. 202, 72 (2001)

Literature:
Light-matter interactions in solids
EM waves in matter
Lorenz oscillator
Isolators, Phonon polaritons
Metals, Plasmon polaritons
- 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) - definitions
Polarization
P = ε χ with suszeptibility
χ
0 E
Electric displacement
( + χ ) E ε E
D = ε 0E + P = ε 0 1 = 0ε
Complex refractive index
N = n + iκ
=
ε = ε′
+ iε
′
ε ′ = n
2
ε ′
= 2nκ
= 1+
χ
2
+ κ
ε
Complex dielectric function
Relationship between N and ε

EM-waves in matter (linear media) - dispersion
E = E 0e
i( kr −ωt)
i( kr −ωt)
B = B0e c
ω = k =
ε
c
n
k
2 ω
( k⋅ k)=
c 2 ε ε(ω) and thus k = k´ + ik´´ are complex numbers!
( kr−ωt
) i(
k′
r−
t ) −k′
′ r
i ω
E = E e = E e e
0
wavevector k = 2π
λ
frequency ω = 2πf
Dispersion in transparent media without absorption:
Dispersion generally:
k and ε are real
ε >0
0
propagating wave exponential decay of amplitude

+
2
ω0 = Km
γ = bm
-
ω 0
Harmonic oscillator (Lorentz) model
m&x
& + bx&
+ Kx
= eE
= eE
p = ex
= ε αE
P = Np
= ε χE
0
0
0
e
iωt
e m
iΘ eE
x =
E = Ae
2 2
ω −ω
− iγω
m
0
2
ω p
ε = 1+
χ = 1+
2 2
ω −ω
− iγω
2 2 ωp = Ne / mε0 plasma frequency
0
ω 0

One-oscillator (Lorentz) model
from Bohren/Huffman
N =
n + iκ
=
R =
n
ε
2 2 ( n −1)
+ κ
2 2 ( + 1)
+ κ

2
1,6
1,2
0,8
0,4
0
-0,4
-0,8
eps'
Weak and strong molecular vibrations
weak oscillator: ε‘ > 1
wavenumber / cm -1
eps''
860 880 900 920 940
caused by: molecular vibrations
examples: PMMA, PS, proteins
1
0,8
0,6
0,4
0,2
0
-0,2
-0,4
25
20
15
10
5
0
-5
-10
strong oscillator: ε‘ < 0
eps' eps''
860 880 900 920 940
wavenumber / cm -1
crystal lattice vibrations
SiC, Xonotlit, Calcite, Si 3 N 4
25
20
15
10
5
0
-5
-10

Optical properties of polar crystals
note:
lattice has transversal T and longitudinal L
oscillations but only transversal phonons
can be excited by light
N = n + iκ
=
2
ω
2
( k ⋅k ) = ε
E = E0e
i
c
ε
( kr−ωt
)
A n = 0,
κ =
B
A
B
n = ε , κ =
ω
k = i
c
k =
ω
c
ε
ε
ε
A: total reflection
B: transmission and reflection
0
R =
n
B A B
γ = 0
2 2 ( n −1)
+ κ
2 2 ( + 1)
+ κ

γ > 0
SiC - single oscillator model

General dispersion for nonconductor
ω 0 = 0
permanent dipoles
e.g. water
phonons
molecular vibrations
resonances
restoring forces
ω0 > 0

Dispersion in polar crystals - phonon polaritons
Dispersion relation for
k
2
2 2
ω ω
= ε = ε
2 2
c c
photon like
γ = 0
2 2
ω −ω
L ( ωs
) 2 2
ω −ω
T
strong coupling of mechanical and
electromagnetic waves, polariton like
γ = 0
between TO and LO
there is no solution for
real values ω and k
frequency gap:
E ∝ e
ω
−i
t −kr
e
no propagation
reflection

- Intraband transitions
- longitudinal plasma oscillations
- ω 0 = 0 i.e. no restoring force
- ω P = plasma frequency
ε
2
ω p
1+
2
ω −ω
− iγω
= 2
0
ω 0 = 0
ε
2
ω p
1−
ω − iγω
= 2
2
ω p
ε′
= 1−
2 2
ω + γ
ε′
′ =
ω
ω
2
p
γ
2
ω p
≈1
− 2
ω
2
ω pγ
≈
2 2 3
( ω + γ ) ω
Metals - drude model
γ = 0
E = E0e
i
( kr−ωt
)
ω
k = i
c
n = 0,
κ =
ω >> γ = 1/τ (1/ collision time)
collisions usually by electron-phonon scattering
ε
ε
k =
ω
c
ε
n = ε , κ =
total reflection transmission
generally:
γ > 0 leads to damping of transmitted wave n > 0, κ > 0
0

ω T = 0
Aluminium
ω L = ω P

Free and bound electrons in metals
ε bound
ε drude
Bound electrons contribute like a Lorenz oscillator
where
ε = ε + ε
ε
metal
drude
drude
2
ω p,
d
1−
ω − iγ
ω
= 2
d
bound
2
ω p,
j
∑ 2
ω −ω
− iγ ω
ε bound = 2
j 0
j

ω
ω p
0
Metals - plasmon polaritons
Plasmon polariton dispersion (γ = 0)
2
ω
= ω
2
p +
ω =
c
2
k
ck
2
k
E ∝ e
ω
−i
t −kr
e
no propagation
reflection

Dielectric function of metals and polar crystals
Metal
collective free electron oscillations (plasmons)
no restoring force
plasma frequency
(longitudinal oscillation)
Polar crystal
strong lattice vibrations (phonons)
transversal optical
phonon frequency, TO
Reststrahlenband
longitudinal optical
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)

z
x
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.
dielectric ( ω)
> 0
surface wave
ε d
++ -- ++ -- ++ --
ε
( ω)
< 0
z
E
( z)
∝
e
E
− Im
SP
E(
z)
k z
z
±
= E0
e
( k x±
k z−ωt
)
i x z
k = k′
+ ik
′′
x
k′
x
=
x
2π
λ
SP
x

z
Ez
H y⊗
x
dielectric:
E
E
⎛ E
⎜
= ⎜ 0
⎜
⎝ E
metal/polar crystal:
x
+ + - - x
x
z
⎞
⎟
⎟e
⎟
⎠
e
e
i
( k x+
k z−ωt
)
x
z
( k x+
k z−ωt
)
i xd zd
( k x−k
z−ωt
)
i xm zm
SP fields
Ez y H
E
H
++ -- ++ -- ++ --
++ -- ++ -- ++ --
++ -- ++ -- ++ --
++ -- ++ -- ++ --
kx
⎛ 0
⎜
= ⎜ H
⎜
⎝ 0
y
⎞
⎟
⎟e
⎟
⎠
i
( k x+
k z−ωt
)
x
z

z
z
x
x
Derivation of SPP dispersion – boundary conditions
H
Ez
y
dielectric
z > 0
z
< 0
metal
⊗
E
x
+ + - -
longitudinal surface wave
++ -- ++ -- ++ --
εd ( ω)
εm( ω)
H
E
H
E
d
d
m
m
Ey
H
x
= 0
( 0,
H , )
= 0
( 0,
H , )
= 0
( E , 0 E )
= ,
xm
yd
ym
e
( E , 0 E )
= ,
xd
H
= z
= 0
zd
e
zm
i
( ) t z k x k + −ω
e
i
i
( ) t z k x k − −ω
e
xd
xm
i
( ) t z k x k − −ω
xm
zd
( ) t z k x k + −ω
xd
zm
zd
zm
xm
Boundary
bonditions (z=0)
ε E = ε E
m
E =
xm
H =
ym
xd
zm
E
xd
H
k = k =
d
yd
k
x
zd

Derivation of SPP dispersion
Derivation of SPP dispersion
E
H
t
c ∂
∂
= 1
curl ε
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
∂
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
×
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
∂
∂
∂
z
x
t
y
z
y
x
E
E
c
H 0
1
0
0
ε
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
∂
∂
−
z
x
y
x
y
z
E
E
c
H
H
0
0
ω
ε
Maxwell eq.:
z-component:
Diel.:
Metal:
zm
m
y
x
E
c
H
k
ω
ε
−
=
+
zd
d
y
x
E
c
H
k
ω
ε
−
=
+
x-component:
Diel.:
Metal: ym
zm
ym
z
H
k
H +
=
∂
−
yd
zd
yd
z
H
k
H −
=
∂
−
x
m
y
zm
E
c
H
k
ω
ε
−
=
+
x
d
y
zd
E
c
H
k
ω
ε
+
=
+
( )
t
z
k
x
k
i zd
x
e
ω
−
+
( )
t
z
k
x
k
i zm
x
e
ω
−
−
Diel.:
Metal:

x-component:
Metal
Diel.
I:
II:
I / II:
Derivation of SPP dispersion
k
k
zm
zd
H
H
k
k
y
y
zm
zd
ω
= −ε
m E
c
ω
= ε d Ex
c
ε m = −
ε
d
x
k
ε
zd
d
+
k
ε
zm
m
=
0

generally:
2 2 2
kx + k y + kz
at interface metal/dielectric:
kzd
ε
d
0
+
k
ε
zm
m
=
= 0
k
2
2
ω
ε
2
c
Dispersion relation of SPPs
k
k
k
2
x
2
zd
k =
2
zm
xd
k
xm
2
⎛ ω ⎞
= ⎜ ⎟
⎝ c ⎠
2
⎛ ω ⎞
= ⎜ ⎟
⎝ c ⎠
ε mε
d
ε + ε
2
⎛ ω ⎞
= ⎜ ⎟
⎝ c ⎠
m
d
( ε + ε )
m
ε
2
d
d
( ε + ε )
m
ε
2
m
d
dielectric:
metal:
2 2
kx + kzd
= ε d
2 2
kx + kzm
= ε m
2
ω
2
c
2
ω
2
c
( ε m ) < 0 and ε m > ε d real kx
Re →
k zd and k zm are imaginary

E
k
k
k
2
x
2
zd
2
zm
SP
2
⎛ ω ⎞
= ⎜ ⎟
⎝ c ⎠
±
= E0
2
⎛ ω ⎞
= ⎜ ⎟
⎝ c ⎠
⎛ ω ⎞
= ⎜ ⎟
⎝ c ⎠
ε mε
d
ε + ε
2
m
e
d
( ε + ε )
m
ε
2
d
( ε m ) < 0 and ε m > ε d real kx
Re →
k zd and k zm are imaginary
d
( ε + ε )
m
ε
( k x±
k z−ωt
)
i x z
2
m
d
Dispersion relation of SPPs
ω
SP
= ω
p
ω p
1
1+
ε
d
ω
photon
in air
ω =
ckx
ε ′ → −
ε d
surface plasmon polariton
ω =
ε + ε
ε
m d ckx
mε
d
k x

ω
SP
= ω
p
ω p
1
1+
ε
d
ω
Volume vs. surface plasmon polariton
2 2
ω = ω p + c
plasmon
polariton
2
k
2
x
ω = ck
photon
in air
with damping
x
ε ′ → −
ε d
surface plasmon polariton
ω =
ε + ε
ε
m d ckx
mε
d
k x
volume plasmon
surface plasmons
non-propagating
collective oscillations
of electron plasma
near the surface

ω p
ω
Plasmon polaritons:
photon
in air
Light-electron coupling in
• metals
• semiconductors
SP dispersion - plasmon vs. phonon
ω
ε ′ → −1
surface plasmon
polariton
kx
ck
= x
ε + 1
ε
ω LO
ω TO
ω
Phonon polaritons:
photon
in air
ε ′ → −1
surface phonon
polariton
kx
Light-phonon coupling in polar crystals
• SiC, SiO 2
• III-V, II-VI-semiconductors

metal/air
interface
E
( x)
k
x
ω
= k′
+ ′
x ikx
=
c
ik xx ik′
x x −k
x′
′ x
= E0e
= E0e
e
propagating term exponential decay
in x-direction
L
x
1
=
2k′
′
x
SP propagation length
ε m
ε + 1
m
propagation
length
intensity !
ε d
= 1
2
ω p
ε m = 1−
2
ω + iγω
γ = 0.
2
10
7.5
5
2.5
-2.5
-5
-7.5
-10
Lx( λvac)
ε ′′ m
ε ′ m
Drude
model
0.4 0.6 0.8 1
ω p
ωSP
=
2
200 ε = −1
= 514. 5 nm : Lx
= 1060 nm : Lx 22 μ
500 μm
20
10
5
2
1
0.2 0.4 0.6 0.8 ω
ω 1 p
Example silver: λ = m
λ =
100
50
ω
ω p

z
x
dielectric
()
ε d ω
metal
()
ε m ω
k
x
z
SPP field perpendicular to surface
E z
Examples:
silver:
gold:
E
( z)
= E
0
e
− Im
λ = 600 nm : Lz,
m = 390 nm and Lz,
m
λ = 600 nm : Lz,
m = 280 nm and Lz,
m
k z
z
L
z
=
1
Im k
z-decay length
(skin depth):
z
= 24 nm
=
31nm

SPPs have transversal and longitudinal el. fields
z
Ez
H
x
y⊗
El. field
E
E
x
=
zd i
k
E z = i
k
ε ′
E
+ + - -
At large values,
m
x
− ε
ε
d
x
z
the el. field in air/diel. has a strong
transvers E z component compared to the
longitudinal component E x
E
In the metal E z is small against E x
m
x
Themag. fieldH is
parallel to surface
and perpendicular to
propagation
E
E
zm i
x
= −
ε d
− ε
m
−1
ε ′ m
At large k x , i.e. close to ε = - ε d ,
both components become equal
E ± iE
z
= (air: +i, metal: -i)
x
ωSP
ω p
ω

ω
photon in air
Dispersion and excitation of SPPs
photon in
dielectric
SPP dispersion
k of photon in air is always < k of SPP
k x
no excitation of SPP is possible
in a dielectric k of the photon is increased
k of photon in dielectric can equal k of SPP
SPP can be excited by p-polarized light
(SPP has longitudinal component)
thin metal film
dielectric
Kretschmann configuration
E
θ0
k x
R
z
x

Methods of SPP excitation
n prism > n L !!

E
ε d
ε m
ε
0
θ0
k x
R
z
Excitation by ATR
Kretschmann configuration Otto configuration
total reflection at prism/metal interface
-> evanescent field in metal
-> excites surface plasmon polariton at
interface metal/dielectric medium
metal thickness < skin depth
ATR: Attenuated Total Reflection
x
E
ε m
ε d
ε
0
θ0
total reflection at prism/dielectric medium
-> evanescent field excites surface plasmon
at interface dielectric medium/metal
usful for surfaces that should not be damaged
or for surface phonon polaritons on thick crystals
k x
distance metal – prism of about λ
R
z
x

z
x
SPP excitation requires kphoton,x = kSP,x ε d
ε m
ε 0
k photon,x
θ
k SP,x
R
< k SP,x
k photon,x
no SPP excitation
Excitation by ATR
θ
k photon,x
k SP,x
= k SP,x
R
SPP excitation
k photon,x

z
ε d
ε m
ε
0
x
k
c
ω
=
k =
ω
c
ε 0
Excitation by Kretschmann configuration
θ0
k x =
ω
c
0
( )
sin θ ε
0
m
0
kx ε m
0
ω0
ω
ω ε + 1 c
= c =
ε sin
0
ω = ck
photon
in air
( θ )
0
0
kx
Resonance
condition
ω x
photon in
dielectric
0 sin
kx
( )
= c / k ε θ
SPP dispersion
ω
ε m
kx
=
c ε + 1
m
0

ω
ω0
θ0
photon
in air
Kretschmann configuration – angle scan
θ0
R
k x
illumination freq. ω 0 = const.
R
s-polarized
-> no excitation of SPPs
θ0
p-polarized