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
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
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
( ) ( )
( )( ) ( ) ( )
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⎛ ⎞
− + = ⎜ ⎟⎝ ⎠
( )
( ) ( )
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⎛ ⎞
− + = ⎜ ⎟⎝ ⎠
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
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 = × → =
−
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
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?
Critical wavelength (or, plasma wavelength) Critical wavelength (or, plasma wavelength)
Born and Wolf, Optics, page 627.
ppc
cωπλλ 2
==
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.
Dispersion of Refractive Index for copperDispersion of Refractive Index for copper
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
⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
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
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
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)
Dispersion relation for EM waves in electron gas (bulk plasmons)
( )kω ω=
• Dispersion relation:
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
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=
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
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= +
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
−+
−==
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
Barnes et al., Nature (2003)
Surface plasmon 응용
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
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)]
Nano-scale light guiding
Metal Nanoparticle WaveguidesMetal Metal NanoparticleNanoparticle WaveguidesWaveguides
Maier et al., Adv. Mater. 13, 1501 (2001)
Nano PlasmonicsNanoNano PlasmonicsPlasmonics
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!
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