chapter 6. optics of solids part 2 - plasmonicsocw.snu.ac.kr/sites/default/files/note/lecture...

Changhee Lee, SNU, Korea

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Changhee Lee

School of Electrical and Computer Engineering

Seoul National Univ.

[email protected]

Chapter 6. Optics of Solids

Part 2 - Plasmonics


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https://en.wikipedia.org/wiki/Plasmon

Plasmon

A plasmon is a quantum of plasma oscillation.

Just as light [optical oscillation] consists of photons,

the plasma oscillation consists of plasmons. The

plasmon can be considered as a quasiparticle since it

arises from the quantization of plasma oscillations,

just like phonons are quantizations of mechanical

vibrations. Thus, plasmons are collective (a discrete

number) oscillations of the free electron gas density.

For example, at optical frequencies, plasmons can

couple with a photon to create another quasiparticle

called a plasmon polariton.

Gothic stained glass rose window of Notre-Dame de Paris. The colors were

achieved by colloids of gold nano-particles.Lycurgus cup (Roman empire, 4th century AD), which has a green color when

observing in reflecting light, while it shines in red in transmitting light conditions

http://www.thebritishmuseum.ac.uk/ science/lycurguscup/sr-lycugus-p1.html


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Plasma frequency

Plasmons can be described in the classical picture as an oscillation of free electron density with

respect to the fixed positive ions in a metal. If the electrons in a metal are displaced from a uniform

background of ions, electric fields will be built up in such a direction as to restore the neutrality by

pulling the electrons back to their original positions. Because of their inertia, the electrons will

overshoot and oscillate around their equilibrium positions with a characteristic frequency known as

the plasma frequency.

o

p

o m

Nex

NeEe

dt

xdm

22

2

2

2

+++++++

-------

x

Surface charge density = N e x

electric field

o

xNeE

Surface charge density = -N e x

Free electrons in metal: eVm

NeN

o

p 10~ cm102

3-23


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Role of plasmons

Plasmons play a large role in the optical properties of metals and semiconductors.

• Light of frequencies below the plasma frequency is reflected by a material because the electrons in

the material screen the electric field of the light.

• Light of frequencies above the plasma frequency is transmitted by a material because the electrons

in the material cannot respond fast enough to screen it.

• In most metals, the plasma frequency is in the ultraviolet, making them shiny (reflective) in the

visible range. Some metals, such as copper and gold have electronic interband transitions in the

visible range, whereby specific light energies (colors) are absorbed, yielding their distinct color.

• In semiconductors, the valence electron plasmon frequency is usually in the deep ultraviolet, while

their electronic interband transitions are in the visible range, whereby specific light energies

(colors) are absorbed, yielding their distinct color. It has been shown that the plasmon frequency

may occur in the mid-infrared and near-infrared region when semiconductors are heavily doped.

• Surface plasmons have been used to control colors of materials. Controlling the particle's shape

and size determines the types of surface plasmons that can couple to it and propagate across it. This

in turn controls the interaction of light with the surface. In the stained glass, the color is given by

metal nanoparticles of a fixed size which interact with the optical field.

• Surface plasmons can confine light below the diffraction limit of light (near-field optics).

https://en.wikipedia.org/wiki/Plasmon


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Operating speed of data transporting and processing systems• The ever-increasing need for faster information processing and transport is undeniable.

• As data rates and component packing densities increase, electrical interconnects become

progressively limited by RC-delay.

• Photonics is diffraction-limited in size.

Why nanophotonics needs plasmons?


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Why nanophotonics needs plasmons?


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Summary of relations for EM waves in matter

• Polarization

EEPED

EEP

rooo

o

oo

)1(

litysusceptibi electric ,1

)(

• Complex refractive index

1"' i

o

r• Complex dielectric function

cn

ck

nn

nnN

ikKNc

KinN

r

,

2

" ),"''(

2

1

2" ,' ,~

~

,~~

,~

222

222


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EM waves in matter

• Dispersion relation

ztkzi

o

tzKi

o eeEeEE )()~

(

)~

( tzKi

oeHH

kn

cn

ck

Ncc

K r

,

~

)()(~ 2222

2

2

22

2

2

2

22

2

2

2

2

2

2

2

2

2

2

1)1(

11

1)(

t

E

ct

E

t

E

ct

P

t

E

cE

t

J

t

P

t

E

cE

rrooo

oo

2

2

2

2 1

t

E

cE r

Dielectric material


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When isolated atoms condense to form a metal, core electrons remain bound to the nucleus to form

the metallic ion, but valence electrons can wander about and are called conduction electrons

Optical Properties of Metals: Drude-Sommerfeld Model

Eedt

rdm

dt

rdm

2

2

frequency plasma volume ,11

)(11

)1()( ,)(

2

2

2

2

2

2

22

o

p

p

oo

r

ro

m

Ne

iim

Ne

E

P

EEmim

NereNpNPEermim

Equation of motion for electron in the metal under the influence of an electric field:


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Since collisions occur usually by electron-phonon

scattering, t>>1,

12

2

2

2

11)(

t

ii

pp

r

t

tt

t

t

3

2

22

2

2

2

22

2

12

2

)()("

11)('

1)(")(')(

pp

pp

p

ri

i

2222

2

2

2 )1( kcp

p

Total reflection

For most metals, plasma frequency p

is in the UV range ~1015 Hz.


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e-

EE-E

1

2

34

5

bulk

surface

E =

Observation of Plasmons


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Optical Properties of Metals: Interband transition

• The Drude-Sommerfeld model gives accurate results for optical properties of metals in infrared

regime, but need to be supplemented in the visible range by the response of bound electrons

(lower lying shells). e.g., gold and silver, the most important metals for plasmonic studies in the

visible and near-infrared.

• Equation of motion has to include the restoring force: EerKdt

rdm

dt

rdm

2

2

EKmim

NePEerKmim

2

22 ,)(

frequency resonance ,1122

2

m

K

io

o

p

r

0

j jj

jp

bound

Drude

Drudep

Drude

boundDrudemetal

i

i

22

2

,

2

2

,1


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Energy-Saving Window Coatings

• The reflectivity cutoff at the plasmon energy can be used for energy-saving window coatings

which transmit visible sunlight (photon energy above Ep), but reflect thermal IR radiation back

into a heated room.

• To get a reflectivity cutoff in the infrared one needs a smaller electron density than in a metal. A

highly-doped semiconductor fits just right, such as indium-tin-oxide (ITO). This material is also

widely used as transparent front electrode for solar cells, LEDs, and liquid crystal displays.

An ITO film transmits visible light and reflects thermal infrared radiation, keeping the heat

inside a building.

ReflectivityTransmission


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Surface plasmon polaritons (SPPs)

Surface plasmon polaritons (SPPs): Excitaton of a coupled state between photons and plasma

oscillations at the interface between a metal and a dielectric.

• They are electromagnetic excitations propagating at the interface between a dielectric (exhibiting

positive real part of dielectric constant) and a conductor (exhibiting negative real part of dielectric

constant), evanescently confined in the perpendicular direction.

• Surface plasmon polaritons only exist for TM (p-)polarization.

• Surface plasmon, plasmon that is confined to the surface, is the limiting form of a SPP as kx→ ∞

plasmon waveguiding dielectric waveguiding


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yH xE

zE

for the region z > 0; dielectric for the region z < 0; metal

)(

)(

)(

tzkxki

ydy

tzkxki

zdz

tzkxki

xdx

zdxd

zdxd

zdxd

eHH

eEE

eEE

)(

)(

)(

tzkxki

ymy

tzkxki

zmz

tzkxki

xmx

zmxm

zmxm

zmxm

eHH

eEE

eEE

TM (p-)polarization

0

0

zx

y

HH

E


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Boundary conditions at z=0:

• Tangential components of E and H are continuous.

xxdxmydymxdxm kkkHHEE

0

m

zm

d

zd

d

m

zd

zm

kk

k

k


• From Maxwell’s equationt

DH

xmmymxmzddydxdzxy

xmmymzmxddydzdxyz

EHkEHkt

E

y

H

x

H

EHkEHkt

E

z

H

y

H

,

,

222222 )( ,)(c

kkc

kk mzmxdzdx

• Dispersion relation


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2/1)(dm

dmx

ck


])([)()()()()( ,)( 2222222222222

xm

m

ddzm

m

ddzddxdzdx k

cck

ck

ck

ckk

dm

md

dm

dmmd

dm

dmmd

m

d

m

dmd

x

m

dmdx

m

d

cccck

ck

2

22

2

22

22

2

2

2

22

2222

)()(

)()(

)(1

)(

)(

])([)(])(1[

zm

m

dzd kk

222 )(

ckk mzmx

dm

m

dm

mddxmzm

dm

d

dm

mddxdzd

ccck

ck

ccck

ck

2

222222

2

222222

)()()()(

)()()()(

imaginary are and , real ,0)Re( zmzdxdmm kkk


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real)( 2/1

dm

dmx

ck

imaginary are and zmzd kk

)(

)(

)(

zktxki

ydy

zktxki

zdz

zktxki

xdx

zdx

zdx

zdx

eeHH

eeEE

eeEE

)(

)(

)(

zktxki

ymy

zktxki

zmz

zktxki

xmx

zmx

zmx

zmx

eeHH

eeEE

eeEE

for the region z > 0; dielectric for the region z < 0; metal


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2/1)(dm

dmx

ck

• At low ,

d

dm

dmx

cck

m

2/1)(

• At sp (when m = d)

x

dm

dmx k

ck ;)( 2/1

• Surface plasmon frequency sp, (at sp, m = d)

d

p

spd

sp

p

m

1 ;1

2

2


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Dispersion relation of surface-plasmon polaritons


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Dispersion relation of surface-plasmon polaritons

Higher index dielectric on metal results in lower sp.


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Excitation of SPPs with light

Problem: SPP modes lie below the light line

• k of photon in air is always less than k of SPP No coupling of SPP modes

• Need a “trick” to excite modes below the light line

• k of the photon is increased in a dielectric possible to match with k of SPP mode

• SPP can be excited by p-polarized light from a high index medium


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Excitation of SPPs by Kretschmann configuration

condition resonance ;sin

1

oom

m

o

o cc

k

Kretschmann configuration

Thin metal film

dielectric


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Excitation of SPPs by Kretschmann configurationExcitation of SPPs by Kretschmann configuration

condition resonance ;sin

1

oom

m

o

o cc

k

angle scan


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Applications: Determination film thickness of deposited films

Langmuir-Blodget-Kuhn (LBK) films


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Surface Plasmon Sensors

• Evanescent field interacts with adsorbed molecules only

• Coupling angle strongly depends on εd

• Use of well-established surface chemistry for Au (thiol chemistry)


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Nanoscopic waveguides for light

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

M.L. Brongersma, et al., Phys. Rev. B 62, R16356 (2000)

S.A. Maier et al., Advanced materials 13, 1501 (2001)

• Guides electromagnetic energy at optical frequency below the diffraction limit

• Enables communication between nanoscale devices

• Information transport at speeds and densities exceeding current electronics


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Cloaking: Making an Object Invisible

Surrounding an object with a material having the right kind of dielectric properties

(negative refractive index - metamaterials) can make the object invisible.

Cloaking simulation in two dimensions:

A. The black disc blocks the light coming from the left and reflects it back, leaving a shadow towards the right

(shown in light green + yellow).

B. The surrounding ring of cloaking material guides the light around the disc. Reflection and shadow are

avoided, thereby avoiding any trace of the object.


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Perfect Lens

A medium with refractive index n = -1 acts as perfect lens.

For n = 𝜀 𝜇= -1 both 𝜀 and 𝜇 need to be negative.

Negative n refracts light towards the same side of the normal.

chapter 6. optics of solids part 2 - plasmonicsocw.snu.ac.kr/sites/default/files/note/lecture...

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