Gothic Cathedrals and Solar Cells (and maybe a Grail?)

A short introduction to the phenomenon of

Surface Plasmonsand their role in the scattering of light

Martin Kirkengen, Fysisik Institutt, Universitetet i Oslo

Electromagnetic wave in homogeneous medium

• Maxwell equations, wave solution:

• Wave with frequency and wavenumber k

2 2

2 2

2 2

0

0

E k E

H k H

k

( )

0

( )

0

x y z

x y z

i k x k y k z t

i k x k y k z t

E E e

H H e

Stationary solution at boundary:

• Flat boundary at z=0

• Wave propagating in one direction only (no reflection)

• Ey = 0

y

z

xH Ex

Ez

2

1

Plane wave at boundary (at z=0)(no reflection)

• Fields

• Boundary conditions:

1 2 1 2

1 2 1 2

1 1 2 2

, 0

, 0

x x y y

y y x x

z z

E E E E

H H H H

E E

1 1

1 1

2 2

2 2

( )1 1 1

( )1 1

( )2 2 2

( )2 2

( ,0, )

(0, ,0)

( ,0, )

(0, ,0)

x z

x z

x z

x z

i k x k z tx z

i k x k z ty

i k x k z tx z

i k x k z ty

E E E e

H H e

E E E e

H H e

y

z

xH Ex

Ez

21

H

Solution at boundary

Maxwell equations give (Raether 1988)

And unless 1=2

Plasmon solution for 2 =1, 1<-1, imaginary kz

1 2

1 2

22 2

1

21 2

1 21 2

0z z

x zi i

x x x

k k

k kc

k k kc

What? < 0?!Lorentz model – electron on spring:

20 local

ex x x E

m

0

220

2 2 2 20 0

/1 1 1

el

pel

P N ex E

N e m

i i

2 2

2 21 1p p

i

Polarizability and dielectric constant:

Drude model – cut the spring:

<0 , imaginary k, what about c?

• Wavenumber:2

,

k k

k N N cc

• Imaginary -> absorption

• Real , imaginary N-> no penetration, no absorption

Im( ) ( )0

z xk z i k x tE E e

Back to the Surface Plasmons...

• Visible light, metal/air interface• Real wave number along x• Imaginary wave number along y• Longitudinal charge fluctuations at surface• Evanescent waves – no propagated power

+++_ _ _

+++_ _ _ x

z 1

2

Some Plasmon Geometries

• Multiple interfaces give new possibilities

• More stable modes(up to 1 cm propagation)

Coupling to light

• Plasmon:

• Light: kc

1

21

1 1xk c c

• Momentum mismatch – requires help- Grating (discreet, periodic)- Roughness (periodic Fourier components)- Periodic due to curved surface - spheres

The Cathedral Bit...

Red color due to embedded gold particles spreading light

An early application:

• The Lycurgus Cup (British museum 400 A.D.)• When illuminated from within, it glows red. Again

due to gold particles embedded in the glass, with an absorption peak at around 520 nm

Illustrations stolen from:David G. Stroud, David G. Stroud, Ohio State Ohio State University University Columbus OHColumbus OH

Light scattering from spheresMie Theory

• Solve Wave equations in spherical coordinates

• Expand a plane wave in spherical harmonics

• Determine coefficients for the scattered wave and for the internal field of the sphere

• Try to extract some physical meaning...

Bohren&Huffman, Absorption &Scattering of Light by Small Particles (Wiley 1983)

Jumping to the Coefficients...

• m=N1/Nair

• Resonance (in the limit of small x) at

21

2 1 11

( )[ ( )]' ( )[ ( )]'

( )[ ( )]' ( )[ ( )]'n n n n

nn n n n

m j mx xj x j x mxj mxa

m j mx xh x h x mxj mx

2 1nm

n

• Lowest mode for metal particle in air (Nair=1):

N12 = m2 = -2, = -2

Frequency Dependence of Resonance

Ag

Free Electrons(Drude model)

Bound electrons(Lorentz model)

as function of frequency for silverExtinction (absorption+scattering) for gold particles in fluid

Kreibig and Vollmer , Optical Properties of Metal Clusters , Springer-Verlag:Berlin, 1995

Elghanian et al, Science 277, 1078 (1997)Storhoff et al, JACS 120, 1959 (1998)Park and Stroud, PRB 68, 224201 (2003)]

The Scattered Fieldcomponents of the scattered field

r=0.01 r=0.2 r=

Smallest particles – dipole field

Larger particles – multipole contributions

Destructive interference with incoming wave

Constructive interference with incoming wave

Changing the resonance

• SizeLarger particles, higher modes contribute, each mode red-shifted

• ShapeElliptic shape, flatter particles have red-shifted resonance

• Coating/substrateResonance is given as a relative refraction index, changing surroundings changes resonance

• Arrays/clustersLoads of opportunities...

Applications in biophysics

• Gold Nanoparticles as markers

• Shift in resonance gives dielectric constant of medium

Forschungszentrum Jülich

Applications in Solar Cells• Placing small particles of a reflective medium at surface reduces

reflection...• Strong coupling between plasmons and waveguides – increased

fields in the active part of the solar cell.

• The reverse process (LED) has been shown to gain a factor 8 from plasmon coupling

• No Solar Cell demonstrated – yet...

Catchpole & Pillai

School of Photovoltaic and Renewable Energy Engeneering UNSW, Australia

JoAP 100, 044504 (2006)