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25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties...

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25. Optical properties of materials-Metal Drude Model Conduction Current in Metals EM Wave Propagation in Metals Skin Depth Plasma Frequency
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Page 1: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

25. Optical properties of materials-Metal

Drude Model

• Conduction Current in Metals• EM Wave Propagation in Metals• Skin Depth• Plasma Frequency

Page 2: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Drude model

Drude model : Lorenz model (Harmonic oscillator model) without restoration force(that is, free electrons which are not bound to a particular nucleus)

Remind! Linear Dielectric Response of Matter

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Conduction Current in MetalsConduction Current in Metals

2 2

2

:

v

:

e

The current density is definedA CJ N e

m s m

Substituting in the equation of motion we obtain

dJ N eJ Edt m

γ

⎡ ⎤= − =⎢ ⎥⎣ ⎦

⎛ ⎞+ = ⎜ ⎟

⎝ ⎠

i

The equation of motion of a free electron (not bound to a particular nucleus; ),

( : relaxation time )2

142

0

1 10ee e e

C

md r d r dvm eE m m v eE sdt dtdt

Cr γ ττ γ

=

= − − ⇒ − = ≈− + =

Lorentz model(Harmonic oscillator model)

Drude model(free-electron model)

IfC = 0

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( ) ( )

( )( ) ( ) ( )

0 0

00 0 0

:

exp exp

:

expexp exp exp

Assume that the applied electric field and the conduction current density are given by

E E i t J J i t

Substituting into the equation of motion we obtain

d J i tJ i t i J i t J i t

dt

ω ω

ωγ ω ω ω γ ω

= − = −

⎡ ⎤−⎣ ⎦ + − = − − + −

( )

( )

( )

2

0

2

0 0

exp

exp :

,

e

e

N e E i tm

Multiplying through by i t

N ei J E or equivalentlym

ω

ω

ω γ

⎛ ⎞= −⎜ ⎟⎝ ⎠

+

⎛ ⎞− + = ⎜ ⎟

⎝ ⎠

Local approximation to the current-field relation

2

e

dJ N eJ Edt m

γ⎛ ⎞

+ = ⎜ ⎟⎝ ⎠

( )2

e

N ei J Em

ω γ⎛ ⎞

− + = ⎜ ⎟⎝ ⎠

Page 5: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

( )

( ) ( )

2 2

2

,

, :

:

/ ,

0

1 //

1:

e e

e

N e N eJ E E where static conductivitym m

For the general case of a

For static field

J Ei

n oscillating applied field

N e mE where dynamic conduci

s

iω ω

σ σγ γ

σσ σω γ γ ω

σω γ

ω

⎡ ⎤= ⎢ ⎥−⎣

⎛ ⎞= = =⎜ ⎟⎝ ⎠

= = =− −⎦

=

( ) ,.

, 1

,

tivity

the dynamic conductivity is purely real and the electrons follow the electric fFor very low frequencies

As the frequency of the applied field inc

ield

the inertia of electrons introducesa phase lag in th

ree electron

ases

ω γ <<

( ) ( )2

, .

,, 1

90

i

response to the field and the dynamic conductivity is complex

J i E e E

the dynamic conductivity is purely imaginaryand the electron oscillations are out of phase with the a

For very high frequ

pplied

encies

fi

π

σ σω γ ≈ =

°

>>

.eld

( )2

e

N ei J Em

ω γ⎛ ⎞

− + = ⎜ ⎟⎝ ⎠

Page 6: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Propagation of EM Waves in MetalsPropagation of EM Waves in Metals

( )

( )

22

2 2 20

22

2 2 20

' :

1 1

,1 /

:

1 11 /

Maxwell s relations give us the following wave equation for metals

E JEc t c t

But J Ei

Substituting in the wave equation we obtain

E EEc t c i t

The wave equation is

ε

σω γ

σε ω γ

∂ ∂∇ = +

∂ ∂

⎡ ⎤= ⎢ ⎥−⎣ ⎦

⎡ ⎤∂ ∂∇ = + ⎢ ⎥∂ − ∂⎣ ⎦

( )

( )

0

22

22

0 0

0

:

ex

1

p

1 , /

satisfied by electric fields of the form

E E i k

k i

r t

whei

e cc

r

ω

εω μω γ μ

σω

⎡ ⎤= ⋅ −⎣ ⎦

⎡ ⎤= + ⎢⇒ ⎥− ⎦

=⎣

0 ,0 ≠= JP

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Skin Depth at low frequencySkin Depth at low frequency

( )

( )

2

22 0

0 02

00 0 0

:

exp1 / 2

, exp exp cos sin 12 4 4 4 2

Consider the case where is small enough that k is given by

k i i ic i

Then k i i i i

ω

σ ω μω πσ ω μ σ ω μω γ

σ ω μπ π π πσ ω μ σ ω μ σ ω μ

⎡ ⎤ ⎛ ⎞= + ≅ =⎢ ⎥ ⎜ ⎟− ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞≅ = = + = +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

( ) ( ) ( ) ( )

20 0

,0

0 0 0

2 2 2

, :

exp exp exp exp exp

R I R I R I

I R R

cck k n n k

In the metal for a wave propagating in the z directionzE E ikz E k z i k z t E i k z t

σ ω μ σ μ σω ω ωε

ω ωδ

⎛ ⎞= = ⇒ = = = =⎜ ⎟⎝ ⎠

⎛ ⎞= = − − = − −⎡ ⎤ ⎡ ⎤⎜ ⎟⎣ ⎦ ⎣ ⎦⎝ ⎠

20

0

27 1 1 7

21 2:

5.76 10 5.76 10 0.66

I

cThe skin depth is given byk

C sFor copper the static conductivity m mJ m

εδ δσ ω μ σ ω

σ δ μ− −

= = =

−= × Ω = × → =

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Refractive Index of a metalRefractive Index of a metal

( )

( ) ( )

22 0

2

2 222 2 0 0

2

22 0

2

22 2

0

, 1 /

1 11 / 1 /

1

, pe

Now consider again the general case k ic i

c cc in k i iii i

cni

N eThe plasma frequency is defined cm

σ ω μωω γ

σ μ σ μγω γω ω γ ω ω γ

γ σ μω ωγ

ω γ σ μ γγ

⎡ ⎤= + ⎢ ⎥−⎣ ⎦

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪= = + = +⎨ ⎬ ⎨ ⎬− −⎡ ⎤ ⎡ ⎤⎪ ⎪ ⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭ ⎩ ⎭

⇒ = −+

⎛= =

22

00

2 22

20

,

1 ,

e

pp

e

N ecm

The refractive index of the conductive medium is given by

N en wherei m

με

ωω

ω ωγ ε

⎞=⎜ ⎟

⎝ ⎠

= − =+

: Plasma frequency

Page 9: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

If the electrons in a plasma are displaced from a uniform background of ions, electric fields will be built up in such a direction as to restore the neutrality of the plasma 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.

/ ( ) / : electrostatic field by small charge separation

exp( ) : small-amplitude oscillation

( ) ( ) 2 2 2

2 22

2

s o o o

o p

s p po

p

o

o

E Ne x x

x x i t

d x Ne Nem e E

m

mt

e

m

N

d

σ ε δ ε δ

δ δ ω

δ ω ωε

ωε

ε

= =

= −

= − ⇒ − = − ⇒ =

=

What is the plasma Frequency? What is the plasma Frequency?

Page 10: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Critical wavelength (or, plasma wavelength) Critical wavelength (or, plasma wavelength)

Born and Wolf, Optics, page 627.

ppc

cωπλλ 2

==

Page 11: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

For a high frequency ( ),

by neglecting .2

221 pn

ω γ

ωγ

ω

>>

≈ −

22

2 1 pni

ωω ωγ

= −+

Refractive Index of a metalRefractive Index of a metal

: is complex

: is real

and radiation is attenuated.

and radiation is not attenuated(transparent).

p

p

n

n

ω ω

ω ω

<

>

EM waves with lower frequencies are reflected/absorbed at metal surfaces.

EM waves with higher frequencies can propagate through metals.

Page 12: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Dispersion of Refractive Index for copperDispersion of Refractive Index for copper

Page 13: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

2

22

2

2 2

2 2

2 2 3 2

( )

( ) 1

( ) 2

1

R I

pR I

R I R I

p p

i n

n ini

n n i n n

i

ε ω ε ε

ωω ωγ

ω ω γω γ ω ωγ

= + =

= + = −+

= − +

⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

Dielectric constant of metal given by Drude model

τγω 1=>>

2 2

2 3( ) 1/

p piω ω

ε ωω ω γ

⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Page 14: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Ideal case : metal as a free-electron gas

• no decay (infinite relaxation time)• no interband transitions

2

0 2( ) ( ) 1 pτγ

ωε ω ε ω

ω→∞→

⎛ ⎞⎯⎯⎯→ = −⎜ ⎟⎜ ⎟

⎝ ⎠

2

21 pr

ωε

ω= − 0

Page 15: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Plasma wave (oscillation) = density fluctuation of charged particles

Plasmon = plasma wave with well defined oscillation frequency (energy)

Plasmon in metals = collective oscillation of free electrons with well defined energy

Surface plasmons = Plasmons on metal surfaces

An application of Drude model: Surface plasmons

An application of Drude model: Surface plasmons

Page 16: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Plasmons propagating through bulk media with a resonance at ωp

Bulk Plasmons

Plasmons confined but propagating on surfaces

Surface Plasmons (SP)

Plasmons confined at nanoparticles with a resonance at

Localized Surface Plasmons

+ + +

- - -

+ - +

k

Plasma oscillation = density fluctuation of free electrons

0

2

εω

mNedrude

p =

0

2

31

εω

mNedrude

particle =

Plasma waves (plasmons)

Page 17: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Dispersion relation for EM waves in electron gas (bulk plasmons)

( )kω ω=

• Dispersion relation:

Page 18: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Dispersion relation for surface plasmons

TM waveεm

εd

xm xdE E= ym ydH H= m zm d zdE Eε ε=• At the boundary (continuity of the tangential Ex, Hy, and the normal Dz):

Z > 0

Z < 0

Page 19: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Dispersion relation for surface plasmons

zm zd

m d

k kε ε

=

xm xdE E=

ym ydH H=

xmmymzm EHk ωε=

ydd

zdym

m

zm HkHkεε

=

),0,( ziixii EiEi ωεωε −−),0,( yixiyizi HikHik−

xiiyizi EHk ωε=

xddydzd EHk ωε=

Page 20: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Dispersion relation for surface plasmons

• For any EM wave:2

2 2 2i x zi x xm xdk k k , where k k k

cωε ⎛ ⎞= = + ≡ =⎜ ⎟⎝ ⎠

m dx

m d

kc

ε εωε ε

=+

SP Dispersion Relation

kkx

kzi

Page 21: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

For a bound SP mode:

kzi must be imaginary: εm + εd < 0

k’x must be real: εm < 0

So,

Dispersion relation:Dispersion relation for surface plasmons

'm dε ε< −

2 22 2 zi i x x i x ik k i k k

c c cω ω ωε ε ε⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ± − = ± − ⇒ >⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

1/ 22

' izi zi zi

m d

k k ikc

εωε ε

⎛ ⎞= + = ± ⎜ ⎟+⎝ ⎠

z-direction:2

2 2zi i xk k

cωε ⎛ ⎞= −⎜ ⎟⎝ ⎠

+ for z < 0- for z > 0

1/ 2

' " m dx x x

m d

k k ikc

ε εωε ε

⎛ ⎞= + = ⎜ ⎟+⎝ ⎠

x-direction: ' "m m miε ε ε= +

Page 22: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Plot of the dispersion relation

2

2

1)(ωω

ωε pm −=

m dx

m d

kc

ε εωε ε

=+

• Plot of the dielectric constants:

• Plot of the dispersion relation:

d

psp

dm

ωωε

ωεε

+=≡∞→⇒

−→•

1 ,k

, When

x

22

22

)1()(

pd

dpspx c

kkωωεεωωω

−+

−==

Page 23: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Surface plasmon dispersion relation:

2/1

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=dm

dmx c

kεεεεω

ω

ωp

d

p

ε

ω

+1

Re kx

real kxreal kz

imaginary kxreal kz

real kximaginary kz

d

xckε

Bound modes

Radiative modes

Quasi-bound modes

Dielectric: εd

Metal: εm = εm' +

εm"

xz

(ε'm > 0)

(−εd < ε'm < 0)

(ε'm < −εd)

2 2 2 2p xc kω ω= +

Surface plasmon dispersion relation 1/ 22

izi

m d

kc

εωε ε

⎛ ⎞= ⎜ ⎟+⎝ ⎠

λπ /2=λx~λ Λx<<λ

Page 24: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Barnes et al., Nature (2003)

Surface plasmon 응용

Page 25: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Several 1 cm long, 15 nm thin and 8 micron wide gold stripes guiding LRSPPs3-6 mm long control electrodes low driving powers (approx. 100 mW) and high extinction ratios (approx. 30 dB) response times (approx. 0.5 ms)total (fiber-to-fiber) insertion loss of approx. 8 dB when using single-mode fibers

Surface plasmonicwaveguides

금속선

전자신호

광 신호광 신호

전자신호

Metal waveguides

Page 26: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Asymmetric mode : field enhancement at a metallic tip

Er EzEr

Ez

M. I. Stockman, “Nanofocusing of Optical Energy in Tapered Plasmonic Waveguides,” Phys. Rev. Lett. 93, 137404 (2004)]

Page 27: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Nano-scale light guiding

Page 28: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Metal Nanoparticle WaveguidesMetal Metal NanoparticleNanoparticle WaveguidesWaveguides

Maier et al., Adv. Mater. 13, 1501 (2001)

Page 29: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Nano PlasmonicsNanoNano PlasmonicsPlasmonics

Page 30: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

Nano-Photonics Based on PlasmonicsNanoNano--Photonics Based on Photonics Based on PlasmonicsPlasmonics

By Prof. M. Brongersma

Nanoscale integrated circuits with the operating speed of photonics can be possible!

Page 31: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr
Page 32: 25-Optical properties of materials-metal - Hanyangoptics.hanyang.ac.kr/~shsong/25-Optical properties of materials...Optical properties of materials-Metal ... G JJGGJJG e eee C dr dr

A World of NanoPlasmonics

Could such an Architecture be Realized with Metal rather than Dielectric Waveguide Technology?

Harry Atwater, California Institute of Technology

On-chip light source Short-range(~ nm)

waveguides

Nano-photonics

~ cm

Long-range(~ cm) waveguides

Nano-electronics

Photonic integrated circuit


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