4-Optical Properties of Materials-dielectrics and Metals
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Transcript of 4-Optical Properties of Materials-dielectrics and Metals
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4. Optical properties of materials
Insulating media (dielectric) : Lorentz model
Conducting media (Metal, in free-electron region) : Drude model
Conducting media (Metal, in bound-electron region) : Drude-Sommerfeld model
Extended Drude model (Lorentz-Drude model)
0( ) ( ) ( )
( )
r
r
D E ω ε ε ω ω
ε ω
= (spatially local response of media)
: relative dielectric constant= relative permittivity
= dielectric function
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Let’s start from Maxwell equations
Flux densities in a medium
Total electric flux density = Flux from external field + flux due to material polarization
in most of optical materials.
ε = permittivity of material
(external)
(external)
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Ampere-Maxwell’s Law
Stokes theorem (very general)
Faraday’s Law (Faraday 1775-1836)
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EM wave Equation in bulk media (ε, μ)
: EM wave equation in bulk media without net charge
( μ=μ0
and ρ=0)
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EM waves in dispersive media : ε=(ω), μ=μ(ω)
Relation between E and P (J) is dynamic:
In ideal case, assuming an instantaneous response,
But, in real life,
P(r,t) results from response to E over some characteristic time τ : Function x(t) is a scalar function lasting a characteristic time τ
Convolution theorem
Response to E is DISPERSIVE
Therefore, the temporally dispersive response of a medium induces a frequency dependence in permittivity.
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A. Propagation in insulating media (dielectric : P, J = 0)
Consider a linear, homogeneous, isotropic medium .
: all the materials properties resulting from P
: EM wave equation in insulating media (dielectric)
( linear, homogeneous, isotropic, J=0, μ=μ0 , and ρ=0)
Note 3 : In inhomogeneous media:
22
2dispersion rel( ), whereation 1 ( )r r k
c
ω ε ε χ ω = = +
2 22
0 0 2 2 2
1r r
E E E
t c t
μ ε ε ε ∂ ∂
∇ = =
∂ ∂
The wave equation is satisfied by an electric field of the form of:
( )0exp E E i k r t ω ⎡ ⎤= ⋅ −
⎣ ⎦
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B. Propagation in conducting media (free-electron region: J, P = 0)
Consider an ideally free electron (not bounded to a particular nucleus) Drude model
J linearly proportional to E: J = σ E
σ is the conductivity
The wave equation is satisfied by an electric field of the form of:
( )0exp E E i k r t ω ⎡ ⎤= ⋅ −⎣ ⎦
2 22
2 2
0
0
dispersion r1 ( ) ( )
( )( )
elati
1
onr
r
k ic c
i
σ ω ω ε ω
ε ω
σ ω ε ω
ε ω
⎡ ⎤= + =⎢ ⎥
⎣ ⎦
= +
2
22 2 2
0
1 E E E c t c t
σ ε
∂ ∂∇ = +∂ ∂
: EM wave equation in conducting media (free-electron region)( linear, homogeneous, isotropic, P=0, μ=μ0 , and ρ=0)
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Consider a bounded electron (not ideal free-electron due to interband transition ) Modified Drude model
P b induces internal current density :
The wave equation is satisfied by an electric field of the form of:
( )0exp E E i k r t ω ⎡ ⎤= ⋅ −⎣ ⎦
[ ]
2 22
2 2
0
0
1 ( )
( )( ) 1 ( )
b r
r b
k ic c
i
σ ω ω χ ε ω
ε ω
σ ω ε ω ε χ ω ε
ε ω ∞ ∞
⎡ ⎤= + + =⎢ ⎥
⎣ ⎦
= + = +
( )22 2
2
0 0 02 2 2 2 2
1 1bb
P E J E E J J
c t t t c t t μ μ μ
∂∂ ∂ ∂ ∂∇ = + + = + +
∂ ∂ ∂ ∂ ∂
: EM wave equation in conducting media (bound-electron region)
( linear, homogeneous, isotropic, μ=μ0
, and ρ=0)
C. Propagation in conducting media (bound-electron region: J, P = 0)
For the noble metals* (e.g. Au, Ag, Cu), the filled d-band close to Fermi surfacecauses a residual polarization, P b , due to the positive back ground of the ion cores.
Ag 47
: 1s 2
… 4d 10
5s1
Au79
: 1s 2
… 5d 10
6s1
Cu 29
: 1s 2
… 3d 10
4s1
Al 13
: 1s 2
… 3s 2
3p1
(different)
*In physics the definition of a noble metal is strict. It is required that the d-bands of the electronic structure are filled.Taking this into account, only copper, silver and gold are noble metals, as all d-like band are filled and don't cross the Fermi level.
P b also linearly proportional to E:0b bP E ε χ =
0b
b b
P E J
t t
ε χ ∂ ∂
= =
∂ ∂
2 22
2 2 2 2 2
0
1 b E E E E
c t c t c t
χ σ
ε
∂ ∂ ∂∇ = + +
∂ ∂ ∂
*Interband transition : excitation of electrons from deeper bands into the conduction band
Quasi-free-electron model Drude-Sommerfeld model
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Now, how to determineNow, how to determine χ(ω), σ(ω),(ω), σ(ω), andand χbb((ω)) ??
Microscopic origin of ω-response of matterProfessor Vladimir M. Shalaev, Univ of Purdue
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The equation of motion of the oscillating electron,
: of each electron
1= : dampling constant (or, collision freq
( )
effect
( )
uency
ive ma
)
( is kno
( )
ss
2
2
E r x F v d r dr
m Cr m eE dt dt
m
F F E r γ γ
γτ
τ
= + + = − − −
14
wn as the relaxation time of the free electron gas)
( is typically 100 THz = 10 Hz)
+
No external fieldApplied external electric field
+-r(t)
Ex
Electron cloud
-r(t)Fr
Ex FE
Fγ
Classical Electron Oscillator (CEO) Model = Lorentz model
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A. χ(ω) of insulating media (dielectric : P, J = 0)A. Insulator
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A. Insulator
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A. Insulator
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Assume α j ‘s are the same for all atoms
( ) j
N χ ω α ∴ =
A. Insulator (single atomic gas)
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A. Insulator (single atomic gas)
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1
A. Insulator (single atomic gas)
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A. Insulator (realistic gas with different atoms)
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A. Insulator (solids)
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Local field at an atom
A. Insulator (solids)
숙제 - 유도해보시오!
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γω ω ω ε α
im
e
jo
j −−⎟⎟ ⎠
⎞⎜⎜⎝
⎛ =
22
2 1
A. Insulator (solids)
( )
0
3
2 2
4 .2
embed embed
r r solid j j embed embed
j r r
embed
embed
V N V V
a if the medium is a sphere with radius a
ε ε ε ε α α ε
ε ε ε ε
ε ε π
ε ε
⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠⎝ ⎠
⎛ ⎞−= ⎜ ⎟+⎝ ⎠
∑
The Polarizability of a solid with volume V given by the Clausius-Mossotti relation is
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γω ω ε
ε α α χ
im
Ne
m
C m Ne
N N
o −−⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
⎟⎟ ⎠ ⎞⎜⎜
⎝ ⎛ =−=
2
0
2
2
31
3
11
For a simple case when Cj = C for all atoms in solid,
γω ω ω ε α
im
e
jo
j −−⎟⎟ ⎠
⎞⎜⎜⎝
⎛ =
22
21
∑
∑−
=
j
j j
j
j j
N
N
α
α
χ 311
where and 2 j
j
C C
m mω = =
2
2 2
0
( )p
i
ω χ ω
ω ω γω
=
− −
2o, where
2
3 o
C Ne
m m
ω
ε
≡ −
A. Insulator (solids)
This is the same form as the single atomic gasses, except the different definition of ω 0 .
In general when Cj
are not identical,
( )
( )
2
2 2
p j
j j j
f
i
ωχ ω
ω ω ωγ=
− −∑
f j is the oscillation strength (the fraction of dipoles having ω j .
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B. Drude model
B. σ(ω) in free-electron region: Drude model = Free-electron model
Drude model : Lorenz model (Harmonic oscillator model) without restoration force (that is, free electrons which are not bound to a particular nucleus)
2 2: v
A C The current density is defined J N e
m s m
⎡ ⎤= − =⎢ ⎥⎣ ⎦
i
The equation of motion of a free electron (not bound to a particular nucleus; ),
====> ( : relaxation time )2
14
2
0
110
C
d r m d r dv m eE m m v eE s
dt dt d Cr
tγ τ
τ γ
−
=
= − − + = − = ≈
Lorentz model
(Harmonic oscillator model)
Drude model
(free-electron model)
If C = 0
dv m m v eE
dtγ = −
2dJ N e
J E dt m
γ ⎛ ⎞
+ = ⎜ ⎟⎝ ⎠
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( ) ( )
( )
( )
0 0
0
0 0
:
( ) ( ) exp ( ) ( ) exp
:
exp
exp exp
Assume that the applied electric field and the conduction current density are given by
E r E r i t J r J r i t
Substituting into the equation of motion we obtain
d J i t
J i t i J dt
ω ω
ω
γ ω ω
= − = −
⎡ ⎤−⎣ ⎦
+ − = −
( ) ( )
( )
( )
( )
0
2
0
2
0 0
exp
exp
exp :
,
e
e
i t J i t
N e E i t
m
Multiplying through by i t
N ei J E or equivalently
m
ω γ ω
ω
ω
ω γ
− + −⎛ ⎞
= −⎜ ⎟⎝ ⎠
+
⎛ ⎞− + = ⎜ ⎟
⎝ ⎠
Local approximation
to the current-field relation
2
e
dJ N e J E
dt mγ
⎛ ⎞+ =
⎜ ⎟⎝ ⎠
( )2
e
N ei J E
mω γ
⎛ ⎞− + = ⎜ ⎟
⎝ ⎠
B. Drude model
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( )
( )
( )
2 2
0 0
0
0
2
, , :
:
( ) ,
1 /
/ ( )
1:
0
/
N e N e J E E where static conductivity
m m
of an oscillating applied field
J
For static fields
For t
E E
i
dynamic N e m
i i
he general cas
conduc
e
σ σ γ γ
σ σ
σ σ ω
ω γ γ ω
ω γ
ω
ω
⎛ ⎞= = =⎜ ⎟⎝ ⎠
⎡ ⎤= =⎢ ⎥
= =−
−⎣
−
=
⎦
tivity
( )2
e
N ei J E
mω γ
⎛ ⎞− + = ⎜ ⎟
⎝ ⎠
0
( )( ) 1r i σ ω ε ω ε ω
= +
( ) ( )
2 2 2
0 0 0 0 0 0
2
2 22 2 2
0 0 0
0
2 22
2
0
( ) 1 1 11 / 1 /
,
( ) 1 ,
r
p
p
r p
c c cii i
i ii i
N e N eThe plasma frequency is defined c c
m m
N e
i m
σ μ σ μ γ σ μ γ ε ω
γ ω ω γ ω ω γ ω ω γ
ω γ σ μ γ μ γ ε
ω ε ω ω
ω ω γ ε
⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪= + = + = −⎨ ⎬ ⎨ ⎬
+− −⎡ ⎤ ⎡ ⎤⎪ ⎪ ⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭ ⎩ ⎭
⎛ ⎞= = =⎜ ⎟
⎝ ⎠
= − =
+
B. Drude model
B D d d l
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( )
( )
0
0
( )1 /
,
( )~ :
,1
,
.
i
For very low frequencies
As the frequency of the applied field increase
and the electrons follow the electric field
the inertia of electrons introduces a phase l
purely r
ag in the electron response t
eal
s
σ σ ω ω γ
σ ω σ
ω γ
= −
<<
( )
( ) ( )( )2
0 0 0
,
.
,
,
( )
( ) ~ / /
90
/ :
.
, 1
i
is complex
i purely imag
o the field
and
J i E e E
and the electron oscillations are out of phase with
For very hi
the appli
gh freq
ed field
uenc s
inary
ie
π
σ
σ ω
σ ω σ γ ω γ ω σ γ ω
ω γ >>
≈ =
°
Note : Dynamic conductivity σ(ω)
0
( )( ) 1r
iσ ω
ε ω ε ω
= +
B. Drude model
B D d d l
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2 2 2
2 2 2 3 2( ) 1 1 p p pr i
iω ω ω γ ε ω
ω ωγ ω γ ω ωγ ⎛ ⎞ ⎛ ⎞= − = − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ + +⎝ ⎠ ⎝ ⎠
2 2 2 2
2 3 2 3( ) 1 1
/
p p p p
r i iω ω ω ω
ε ω ω ω γ ω ω τ
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞≈ − + = − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
(1) For an optical frequency, ωvisible >> γ
Dielectric constant of free-electron plasma (Drude model)
(2) Ideal case : metal as an undamped free-electron gas
• no decay (infinite relaxation time)• no interband transitions
0
( )r
τ
γ
ε ω →∞
→
⎯⎯→2
2
( ) 1p
r
ω ε ω
ω = −
B. Drude model
2
0 : N e
static conductivitym
σ γ
=
22 2
0 0
0
p
N ec
m
ω γ σ μ
ε
= = =
B Drude model
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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 22 2
2
s o o o
o p
s p p o o
E Ne x x
x x i t
d x Ne Ne m e E m
m dt
σ ε δ ε δ
δ δ ω
δω ω
ε ε
= =
= −
= − ⇒ − = − ⇒ =
Note : What is the actual meaning of ω p
Note : plasma wavelength , λ p
2 p
p
cπ
λ ω =
B. Drude model
B Drude model
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Note : What happens in ε(ω) when ω∼ ω p ?
B. Drude model
22
0 2
22 ( )
02
( ) ( 0)
( ) ( , ) : j k r t
D E E E assume J
t
k k E k E k E E E ec
ω
μ
ω ε ω ⋅ −
∂∇×∇× =∇ ∇ ⋅ − ∇ = − =∂
⋅ − = − =
(1) For transverse waves, 0k E ⋅ =2
2
2( , )k k
cω ε ω =
(2) For longitudinal waves, ( , ) 0k ε ω =2( ) 0k k E k E ⋅ − =
00 D E Pε = = +
Therefore, at the plasma frequency ω = ω p,
E is a pure depolarization field (No transverse field strength in media).
( , ) 0 pk ε ω ω = =
The quanta of these longitudinal charge oscillations are called plasmons(or, volume plasmons)
Volume plasmons do not couple to transverse EM waves
(can be only excited by particle impact)
B Drude model
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B. Drude model
( ) 0, inc If K V ε ω ≤ ≤
Note : What happens in ε(ω) when ω∼ ω p ?
optical potential
Incident energy 20incK k =
, the incident wave is completely reflected.
0ε ≤
(N. Garcia, et. Al, “Zero permittivity materials”, APL, 80, 1120 (2002))
Wave propagation in this material can happen only with phase velocity being infinitely large satisfying the “static-like” equation
(A. Mario, et. Al, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation
phase pattern”, PR B, 75, 155410, 2007)
C Drude-Sommerfeld model
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C. χb(ω) in bound-electron region : Drude-Sommerfeld model
= Quasi-free-electron model
[ ]0
1 1( )
( ) 1 ( ), usually 0r bi
σ ω ε ω ε ε χ ω ε
ε ω ∞∞ ∞= + = + ≤ ≤
2 2 2
2 2 2 3 2( )p p p
r ii
ω ω ω γ
ε ω ε ε ω ωγ ω γ ω ωγ ∞ ∞
⎛ ⎞ ⎛ ⎞
= − = − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ + +⎝ ⎠ ⎝ ⎠
C. Drude Sommerfeld model
Plasmons in metal nanostructures,Dissertation, University of Munichby Carsten Sonnichsen, 2001
silver
gold
C. Drude-Sommerfeld model
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Measured data and modified Drude model for Ag:
γ
ω
ω ε
ω
ω ε
3
2
2
2
",1'p p =−=
γ ω
ω
ε ω
ω
ε ε 3
2
2
2
",'p p
=−= ∞
Drude model:
Modified Drude model:
Contribution of
bound electrons
3.7
9.1 p
eV
ε
ω
∞ =
=
200 400 600 800 1000 1200 1400 1600 1800-150
-100
-50
0
50
Measured data:
ε' ε"
Drude model: ε' ε"
Modified Drude model:
ε'
ε"
ε
Wavelength (nm)
ε'
-εd
Bound SP mode : ε’m < -εd
ε
Wavelength (nm)
2.48 eV
500 nm
1.24 eV
1000 nm
0.62 eV
1500 nm
For Ag, good matched bellow 2.5 eV
For Ag,
C. Drude Sommerfeld model
C. Drude-Sommerfeld model
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Let’s compare the modified Drude model with experimental measurement
P. B. Johnson and R. W. Christy, “Optical constants of the noble metals”, Phys. Rev. B, 6, pp. 4370-4379 (1972).
32
2
2
2
2( )
p
r
pi
ω ε
ω γ ε ω
ω γ
ω ωγ ∞
⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ + ⎠−
+
Regions of interband transitions
Not good matched in interband regions!
Need something more.
C. Drude-Sommerfeld model
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P. B. Johnson and R. W. Christy, “Optical constants of the noble metals”, Phys. Rev. B, 6, pp. 4370-4379 (1972).
D. Extended Drude model
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D. bound-electron region : Extended Drude (Drude-Lorentz) model
( )22
2
2
2
( ) p
L
r L
Li i
ω ε ω ε
ω ω
ε
ω γ ω ∞
⎛ ⎞Δ Ω⎜ ⎟−⎜ ⎟− Ω + Γ⎝
⎛ ⎞= − ⎜ ⎟⎜ ⎟+⎝ ⎠
( )
2
2 2
j p
j j j
f
i
ωχ ω
ω ω ωγ=
− −Since the Lorentz terms of insulators have a general form of
The Drude-Lorentz model consists in addition of one Lorentz term to the modified Drude model.
For gold (Au),
Alexandre Vial, et.al, “Improved analytical fit of gold dispersion: Application to the modeling of extinction
spectra with a finite-difference time-domain method”, Phys. Rev. B, 71, 085416 (2005).
Excellent agreementwithin 500 nm ~ 1 μmfor gold (Au)
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Extended Drude (Drude-Lorentz) model
( )
2 2
2 2 2
( )
p Lr
L Li i
ω ε ε ω ε
ω ωγ ω ω ∞
⎛ ⎞⎛ ⎞ Δ Ω⎜ ⎟= − −⎜ ⎟
⎜ ⎟ ⎜ ⎟+ − Ω + Γ⎝ ⎠ ⎝ ⎠
Modified Drude model for metal in bound-electron region
2 2 2
2 2 2 3 2( )
p p p
r i
i
ω ω ω γ ε ω ε ε
ω ωγ ω γ ω ωγ ∞ ∞
⎛ ⎞ ⎛ ⎞= − = − +⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟+ + +⎝ ⎠ ⎝ ⎠
Drude model for metal in free-electron region
2 2 2
2 2 2 3 2( ) 1 1
p p p
r i
i
ω ω ω γ ε ω
ω ωγ ω γ ω ωγ
⎛ ⎞ ⎛ ⎞= − = − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ + +
⎝ ⎠ ⎝ ⎠
2
2 2
/ ,
p j
j
j
N
i
ω α
ω ω γω =
− −13
( ) 1 ,1
j j
j
r
j j
j
N
N
α
ε ω α
= +−
∑
∑2 j
j
C
mω =
Lorentz model for dielectric (insulator)
In summary :