• Conduction Current in Metals

25. Optical properties of materials-Metal

Drude Model

• Conduction Current in Metals

• EM Wave Propagation in Metals

• Skin Depth

• Plasma Frequency

Drude model

z

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

Conduction Current in Metals Conduction Current in Metals

2 2

2

: v

:

e

The current density is defined

A C

J N e

m s m

Substituting in the equation of motion we obtain

dJ N e

J E

dt γ m

⎡ ⎤

= − ⎢ ⎣ = ⎥ ⎦

⎛ ⎞

+ = ⎜ ⎟

⎝ ⎠

i

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

( : relaxation time )

2

14 2

0

1 10

e

e e e

C m

d r d r d v

m e E m m v e E s

dt dt

dt C r γ τ

τ γ

=

= − − − ⇒ + = − = ≈

Lorentz model

(Harmonic oscillator model)

Drude model (free-electron model) If

C = 0

( ) ( )

( )

( ) ( ) ( )

0 0

0

0 0 0

:

exp exp

:

exp exp 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 t

J i t i J i t J i t

dt

ω ω

ω γ ω ω ω γ ω

= − = −

⎡ − ⎤

⎣ ⎦ + − = − − + −

( )

( )

( )

2 0

2

0 0

exp

exp :

,

e

e

N e E i t

m

Multiplying through by i t

i J N e E or equivalently

m

ω ω ω γ

⎛ ⎞

= ⎜ ⎟ −

⎝ ⎠

+

⎛ ⎞

− + = ⎜ ⎟

⎝ ⎠

Local approximation to the current-field relation

2

e

dJ N e

J E

dt + γ = ⎜ ^⎛ m ^⎞ ⎟

⎝ ⎠

( )

e

i J N e E

ω γ ^⎛ m ^⎞

− + = ⎜ ⎟

⎝ ⎠

( )

( ) ( )

2 2

2

,

, :

:

, /

0

1 /

/

1 :

e e

e

N e N e

J E E where static conductivity

m m

For the general case of a For static field

J E

i

n oscillating applied field

N e m

E where dynamic conduc

i s

ω ω i

σ σ

γ γ

σ σ σ

ω γ γ ω

σ ω γ

ω

⎡ ⎤

= ⎢⎣ − ⎥

⎛ ⎞

= ⎜ ⎟ = =

⎝ ⎠

= = =

− −

⎦

=

( )

.

, 1

,

tivity

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

As the frequency of the applied field inc

ield

the inertia of electrons introduces a phase lag in th

re e electron

ases ω γ <<

( ) ( )

, .

,

, 1

90

i

response to the field and the dynamic conductivity is complex

J i E e E

the dynamic conductivity is purely imaginary

and the electron oscillations are out of phase with the a For very high frequ

pplied encies

fi

σ π σ

ω γ ≈ =

°

>>

eld.

( )

e

i J N e E

ω γ ^⎛ m ^⎞

− + = ⎜ ⎟

⎝ ⎠

Propagation of EM Waves in Metals Propagation of EM Waves in Metals

( )

( )

2 2

2 2 2

0

2 2

2 2 2

0

' :

1 1

, 1 /

:

1 1

1 /

Maxwell s relations give us the following wave equation for metals

E J

E c t c t

But J E

i

Substituting in the wave equation we obtain

E E

E c t c i t

The wave equation is ε

σ ω γ

σ

ε ω γ

∂ ∂

∇ = +

∂ ∂

⎡ ⎤

= ⎢ ⎣ − ⎥ ⎦

⎡ ⎤

∂ ∂

∇ = ∂ + ⎢ ⎣ − ⎥ ⎦ ∂

( )

( )

0

2 2

2

2

0 0 0

: ex

1

p

, 1

/

satisfied by electric fields of the form

E E i k

k i

r t

i

whe e c

c

r

ω

ε ω μ

ω γ μ

ω σ

⎡ ⎤

= ⎣ ⋅ − ⎦

⎡ ⎤

= + ⎢

⇒

=

⎣

0

,

0 ≠

= J

P

Skin Depth at low frequency Skin Depth at low frequency

( )

( )

2

2

2 0

0 0

2

0

0 0 0

:

1 / exp 2

, exp exp cos sin 1

2 4 4 4 2

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

k i i i

c i

Then k i i i i

ω σ ω μ

ω σ ω μ π σ ω μ

ω γ

σ ω μ

π σ ω μ π σ ω μ π π σ ω μ

⎡ ⎤ ⎛ ⎞

= + ⎢ ⎣ − ⎥ ⎦ ≅ = ⎜ ⎝ ⎟ ⎠

⎡ ⎤

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

≅ ⎜ ⎝ ⎟ ⎠ = ⎜ ⎝ ⎟ ⎠ = ⎢ ⎣ ⎜ ⎟ ⎝ ⎠ + ⎜ ⎟ ⎝ ⎠ ⎥ ⎦ = +

( ) ^{( ) ( ) ( )}

2

0 0

,

0

0 0 0

2 2 2

, :

exp exp exp exp exp

R I R I R I

I R R

c

k k n n c k

In the metal for a wave propagating in the z direction

E E ikz E k z i k z t E z i k z t

σ ω μ σ μ σ

ω ω ωε

ω ω

δ

= = ⇒ = = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = =

−

⎛ ⎞

= = − ⎡ ⎣ − ⎤ ⎦ = ⎜ ⎝ − ⎟ ⎠ ⎡ ⎣ − ⎤ ⎦

2 0

0

2

7 1 1 7

1 2 2

:

5.76 10 5.76 10 0.66

I

The skin depth is given by c

k

C s

For copper the static conductivity m m

J m

δ δ ε

σ ω μ σ ω

σ

δ μ

= = =

= × Ω = × − → =

−

Refractive Index of a metal Refractive Index of a metal

( )

( ) ( )

2

2 0

2

2 2

2

2 2 0 0

2

2

2 0

2

2

2 2

0

,

1 /

1 1

1 / 1 /

1

,

e

Now consider again the general case k i

c i

c c

c i

n k i i

i i i

n c

i

The plasma frequency is defined c N e

m σ ω μ ω

ω γ

σ μ γ σ μ

ω ω ω γ γ ω ω γ

γ σ μ ω ω γ

ω γ σ μ γ

γ

⎡ ⎤

= + ⎢ ⎣ − ⎥ ⎦

⎧ ⎫ ⎧ ⎫

⎪ ⎪ ⎪ ⎪

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

⇒ = −

+

= = ⎛

0

2 2

2

2

0

,

1 ,

e

p

p

e

c N e

m

The refractive index of the conductive medium is given by

n where N e

i m

μ ε

ω ω

ω ω γ ε

⎞ =

⎜ ⎟

⎝ ⎠

= − =

+

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 2

2

2

s o o o

o p

s p p

o

p

o

o

E Ne x x

x x i t

d x Ne Ne

m e E

m t m

e

m

N d

σ ε δ ε δ

δ δ ω

δ ω ω

ε

ω ε

ε

= =

= −

= − ⇒ − = − ⇒ =

=

What is the plasma Frequency?

What is the plasma Frequency?

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

Born and Wolf, Optics, page 627.

p p

c

c ω λ π

λ = = ²

For a high frequency ( ),

by neglecting .

2 2

1

n

ω γ

ω γ

ω

>>

≈ −

2 2

n 1 2 ^p

i

ω

ω ω γ

= − +

Refractive Index of a metal Refractive 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

<

>

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

ÆEM waves with higher frequencies can propagate through metals.

Dispersion of Refractive Index for copper

Dispersion of Refractive Index for copper

2

2 2

2

2 2

2 2

2 2 3 2

( )

( ) 1

( ) 2

1

R I

p

R I

R I R I

p p

i n

n in

i

n n i n n

i

ε ω ε ε

ω

ω ωγ

ω ω γ

ω γ ω ωγ

= + =

= + = −

+

= − +

⎛ ⎞ ⎛ ⎞

= − ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠ + ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠

Dielectric constant of metal given by Drude model

γ τ

ω >> = ^{1 2 2}

2 3

( ) 1

/

p p

ω i ω

ε ω ω ω γ

⎛ ⎞ ⎛ ⎞

= − ⎜ ⎜ ⎟ ⎟ + ⎜ ⎜ ⎟ ⎟

⎝ ⎠ ⎝ ⎠

Ideal case : metal as a free-electron gas

• no decay (infinite relaxation time)

• no interband transitions

2

0 2

( ) ^τ _γ ( ) 1 ω ^p

ε ω ε ω

ω

→∞

→

⎛ ⎞

⎯⎯⎯ → = − ⎜ ⎜ ⎟ ⎟

⎝ ⎠

2

1 ^p 2 r

ε ω

= − ω ⁰

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

An application of Drude model : Surface plasmons

An application of Drude model

: Surface plasmons

Plasmons propagating through bulk mediawith a resonance at ω_p

Æ Bulk Plasmons

Plasmons confined but propagating on surfaces

Æ Surface Plasmons (SP)

Plasmons confined at nanoparticles with a resonance at

Æ Localized Surface Plasmons

+ + +

- - -

+ - +

k

Plasma oscillation = density fluctuation of free electrons

0 2

ω ε

m Ne

drude

p

=

0 2

3 1 ω ε

m Ne

drude

particle

=

Plasma waves (plasmons)

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

( ) k

ω ω =

• Dispersion relation:

Dispersion relation for surface plasmons

TM wave

ε

ε

xm xd

E = E H

= H

ε

E

= ε

E

• At the boundary (continuity of the tangential E

, H

, and the normal D

):

Z > 0

Z < 0

Dispersion relation for surface plasmons

zm zd

m d

k k

ε ⁼ ε

xm xd

E = E

ym yd

H = H

xm m ym

zm

H E

k = ωε

yd d

zd ym

m

zm

k H

k H

ε

ε ⁼

) ,

0 ,

( − i ωε

E

− i ωε

E

)

, 0 ,

( − ik

H

ik

H

xi i yi

zi

H E

k = ωε

xd d yd

zd

H E

k = ωε

Dispersion relation for surface plasmons

• For any EM wave:

2

2 2 2

i x zi x xm xd

k k k , where k k k

c ε ^{⎛ ⎞} ω

= ⎜ ⎟ ⎝ ⎠ = + ≡ =

m d x

m d

k c

ε ε ω

ε ε

= +

SP Dispersion Relation

k

k

k

For a bound SP mode:

k

must be imaginary: ε

+ ε

< 0

k’

must be real: ε

< 0 So,

Dispersion relation:

Dispersion relation for surface plasmons

'

m d

ε ^{< −} ε

2 2

2 2

zi i x x i x i

k k i k k

c c c

ω ω ω

ε ^{⎛ ⎞} ε ^{⎛ ⎞} ε ^{⎛ ⎞}

= ± ⎜ ⎟ ⎝ ⎠ − = ± − ⎜ ⎟ ⎝ ⎠ ⇒ > ⎜ ⎟ ⎝ ⎠

2 1/ 2

'

zi zi zi

m d

k k ik

c

ε ω

ε ε

⎛ ⎞

= + = ± ⎜ ⎝ + ⎟ ⎠

z-direction:

2

2 2

zi i x

k k

c ε ^{⎛ ⎞} ω

= ⎜ ⎟ −

⎝ ⎠

+ for z < 0 - for z > 0

1/ 2

' "

x x x

m d

k k ik

c

ε ε ω

ε ε

⎛ ⎞

= + = ⎜ ⎝ + ⎟ ⎠

x-direction: ε

= ε

+ i ε

Plot of the dispersion relation

2 2

1 )

( ω

ω ω

ε

= −

m d x

m d

k c

ω ε ε

ε ε

= +

• Plot of the dielectric constants:

• Plot of the dispersion relation:

d p sp

d m

ω

ω ε

ω ε

ε

= +

≡

∞

→

⇒

−

→

•

1 , k

, When

x

2 2 2 2

) 1

(

) (

p d

d p sp

x

k c

k ε ω ω

ε ω ω ω

− +

= −

=

Surface plasmon dispersion relation:

2 / 1

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

= +

d m

d m

x

c

k ε ε

ε ε ω

ω

ω

d p

ε ω

+ 1

Re k

real k

real k

imaginary k

real k

real k

imaginary k

d

ck

ε

Bound modes Radiative modes

Quasi-bound modes

Dielectric:

ε_d

Metal: ε_m= ε_m^'+ ε_m^"

x z

(ε'

> 0)

(−ε

< ε'

< 0)

(ε'

< −ε

)

2 2 2 2

p c kx

ω =ω +

Surface plasmon dispersion relation

2 1/ 2 i zi

m d

k c

ε ω

ε ε

⎛ ⎞

= ⎜ ⎝ + ⎟ ⎠

λ π /

= 2

λ_x~λ Λ_x<<λ

Barnes et al., Nature (2003)

Surface plasmon 응용

Several 1 cm long, 15 nm thin and 8 micron wide gold stripes guiding LRSPPs 3-6 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 plasmonic waveguides

금속선 전자신호

광 신호 광 신호

전자신호

Metal waveguides

Asymmetric mode : field enhancement at a metallic tip

E _r ^E

E _z

E

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

Nano-scale light guiding

Metal Nanoparticle Waveguides Metal

Metal Nanoparticle Nanoparticle Waveguides Waveguides

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

Nano Plasmonics

Nano Nano Plasmonics Plasmonics

Nano-Photonics Based on Plasmonics Nano Nano - - Photonics Based on Photonics Based on Plasmonics Plasmonics

By Prof. M. Brongersma

Nanoscale integrated circuits with the operating

speed of photonics can be possible!

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