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Optical properties of materialsInsulating 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
(spatially local response of media)
: relative dielectric constant= relative permittivity= dielectric function
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.
= permittivity of material
(external)(external)
Ampere-Maxwell’s Law
Stokes theorem (very general)
Faraday’s Law (Faraday 1775-1836)
EM wave Equation in bulk media ()
: EM wave equation in bulk media without net charge ( and =0)
EM waves in dispersive media : )
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.
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, , and =0)
Note 3 : In inhomogeneous media:
22
2 dispersion rel ( ), whereation 1 ( )r rkc
2 22
0 0 2 2 2
1r r
E EEt c t
The wave equation is satisfied by an electric field of the form of:
0 expE E i k r t
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 = E
is the conductivity
The wave equation is satisfied by an electric field of the form of:
0 expE E i k r t 2 2
22 2
0
0
dispersion r1 ( ) ( )
( )( )
elati
1
onr
r
k ic c
i
22
2 2 20
1 E EEc t c t
: EM wave equation in conducting media (free-electron region)( linear, homogeneous, isotropic, P=0, , and =0)
1 ( ) for dielectricr
Consider a bounded electron (not ideal free-electron due to interband transition) Modified Drude model
Pb induces internal current density :
The wave equation is satisfied by an electric field of the form of:
0 expE E i k r t
2 22
2 20
0
1 ( )
( )( ) 1 ( )
b r
r b
k ic c
i
22 2
20 0 02 2 2 2 2
1 1bb
PE J EE J Jc t t t c t t
: EM wave equation in conducting media (bound-electron region)( linear, homogeneous, isotropic, , and =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 surface causes a residual polarization, Pb, due to the positive back ground of the ion cores.
Ag47 : 1s2 … 4d10 5s1 Au79 : 1s2 … 5d10 6s1Cu29 : 1s2 … 3d10 4s1 Al13 : 1s2 … 3s2 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.
Pb also linearly proportional to E: 0b bP E
0b
b bP EJt t
2 22
2 2 2 2 20
1 bE E EEc t c t c t
*Interband transition : excitation of electrons from deeper bands into the conduction band
Quasi-free-electron model Drude-Sommerfeld model
Now, how to determine and b() ?
Microscopic origin of -response of matterProfessor Vladimir M. Shalaev, Univ of Purdue
The equation of motion of the oscillating electron,
: of each electron1 = : dampling constant (or, collision freq
( )
effect
( )
uency
ive ma
)
( is kno
( )
ss
2
2
Er xF vd r drm Cr m eE
dtdt
m
F F Er
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
F
Classical Electron Oscillator (CEO) Model = Lorentz model
A. of insulating media (dielectric : P, J = 0)A. Insulator
A. Insulator
A. Insulator
Assume j ‘s are the same for all atoms
( ) jN
A. Insulator (single atomic gas)
A. Insulator (single atomic gas)
1
A. Insulator (single atomic gas)
A. Insulator (realistic gas with different atoms)
A. Insulator (solids)
Local field at an atom
A. Insulator (solids)
유도해보시오!
ime
joj
22
2 1
A. Insulator (solids)
0
3
3 32 2
4 .2
embed embedr r
solid j j embed embedj r r
embed
embed
V N V V
a if the medium is a sphere with radius a
The Polarizability of a solid with volume V given by the Clausius-Mossotti relation is
imNe
mCm
NeN
N
o
2
0
2
2
31
3
11
For a simple case when Cj = C for all atoms in solid,
ime
joj
22
2 1
jjj
jjj
N
N
311 where and 2 j
j
C Cm m
2
2 20
( ) p
i
2o, where
2
3 o
C Nem m
A. Insulator (solids)
This is the same form as the single atomic gasses, except the different definition of 0.
In general when Cj are not identical,
( )2
2 2p j
j j j
f
i
fj is the oscillation strength (the fraction of dipoles having j.
B. Drude model
B. 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 CThe current density is defined J N em s m
The equation of motion of a free electron (not bound to a particular nucleus; ),
====> ( : relaxation time )2
142
0
1 10
C
d r m d r dvm eE m m v eE sdt dtd
Crt
Lorentz model(Harmonic oscillator model)
Drude model(free-electron model)
IfC = 0
dvm m v eE
dt
2dJ N eJ Edt m
0 0
00 0
:
( ) ( ) exp ( ) ( ) exp
:
expexp 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 tJ i t i J
dt
0
2
0
2
0 0
exp
exp
exp :
,
e
e
i t J i t
N e E i tm
Multiplying through by i t
N ei J E or equivalentlym
Local approximation to the current-field relation
2
e
dJ N eJ Edt m
2
e
N ei J Em
B. Drude model
2 2
0 0
0
0
2
, , :
:
( ) ,1 /
/( ) 1
:
0
/
N e N eJ E E where static conductivitym m
of an oscillating applied field
J
For static fields
For t
E Ei
dynamicN e mi i
he general cas
conduc
e
tivity
2
e
N ei J Em
0
( )( ) 1r i
2 2 2
0 0 0 0 0 02
2 22 2 2
0 0 00
2 22
20
( ) 1 1 11 / 1 /
,
( ) 1 ,
r
p
pr p
c c cii ii ii i
N e N eThe plasma frequency is defined c cm m
N ei m
B. Drude model
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
2
0 0 0
, .
,
,
( )
( ) ~ / /
90
/ :
.
, 1i
is complex
i purely imag
o the fieldand
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
Note : Dynamic conductivity
0
( )( ) 1r i
B. Drude model
2 2 2
2 2 2 3 2( ) 1 1p p pr i
i
2 2 2 2
2 3 2 3( ) 1 1/
p p p pr i i
(1) For an optical frequency, visible
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
2
2( ) 1 pr
B. Drude model
2
0 :N e static conductivitym
22 2
0 00
pN ecm
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
s o o o
o p
s p po o
E Ne x xx x i t
d x Ne Nem e E mmdt
Note : What is the actual meaning of p
Note : plasma wavelength p2
pp
c
B. Drude model
Note : What happens in when p ? B. Drude model
22
0 2
22 ( )
02
( ) ( 0)
( ) ( , ) : j k r t
DE E E assume Jt
k k E k E k E E E ec
(1) For transverse waves, 0k E 2
22( , )k k
c
(2) For longitudinal waves, ( , ) 0k 2( ) 0k k E k E
00D E P
Therefore, at the plasma frequency p,
E is a pure depolarization field (No transverse field strength in media).
( , ) 0pk
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)
means that at the plasma frequency only a longitudinal wave can propagate through.
B. Drude model
( ) 0, incIf K V
Note : What happens in when p ?
Optical potential
Incident energy20incK k
, the incident wave is completely reflected.
0
(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 radiationphase pattern”, PR B, 75, 155410, 2007)
C. b in bound-electron region : Drude-Sommerfeld model = Quasi-free-electron model
0
1 1( )( ) 1 ( ), usually 0 r bi
2 2 2
2 2 2 3 2( ) p p pr i
i
C. Drude-Sommerfeld model
Plasmons in metal nanostructures, Dissertation, University of Munich by Carsten Sonnichsen, 2001silver
gold
Measured data and modified Drude model for Ag:
3
2
2
2
",1' pp
3
2
2
2
",' pp
Drude model:
Modified Drude model:
Contribution of bound electrons
3.79.1 p eV
200 400 600 800 1000 1200 1400 1600 1800-150
-100
-50
0
50
Measured data: ' "
Drude model: ' "
Modified Drude model: '
"
Wavelength (nm)
'
-d
Bound SP mode : ’m < -d
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
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( ) pr
p i
Regions of interband transitions
Not good matched in interband regions! Need something more.
C. Drude-Sommerfeld model
P. B. Johnson and R. W. Christy, “Optical constants of the noble metals”, Phys. Rev. B, 6, pp. 4370-4379 (1972).
C. Drude-Sommerfeld model
D. bound-electron region : Extended Drude (Drude-Lorentz) model
22
2
2
2
( )
p
Lr
L
Li i
( )
2
2 2j p
j j j
fi
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 agreement within 500 nm ~ 1 mfor gold (Au)
D. Extended Drude model
1.4 1.6 1.8 2.0 2.2 2.40.51.01.52.02.53.03.54.01000900 800 700 600 500
Im P
erm
ittiv
ityE [eV]
J & C Exp Drude DrudeLorentz
[nm]
1.4 1.6 1.8 2.0 2.2 2.4-80-70-60-50-40-30-20-10
010
Rel
ativ
e er
ror i
n %
E [eV]
Drude DrudeLorentz
1.4 1.6 1.8 2.0 2.2 2.4-40-35-30-25-20-15-10-51000900 800 700 600 500
Re
Perm
ittiv
ity
E [eV]
J & C Exp Drude DrudeLorentz
[nm]
1.4 1.6 1.8 2.0 2.2 2.4-202468
10
Rel
ativ
e er
ror i
n %
E [eV]
Drude DrudeLorentz
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0-40-35-30-25-20-15-10-505180015001200900 600 300
Re
Perm
ittiv
ity
E [eV]
J & C Exp Drude DrudeLorentz
[nm]
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0012345678180015001200900 600 300
Im P
erm
ittiv
ity
E [eV]
J & C Exp Drude DrudeLorentz
[nm]
But, not correct at higher energy
Extended Drude (Drude-Lorentz) model
2 2
2 2 2
( )
p Lr
L Li i
Modified Drude model for metal in bound-electron region 2 2 2
2 2 2 3 2( ) p p pr i
i
Drude model for metal in free-electron region2 2 2
2 2 2 3 2( ) 1 1p p pr i
i
2
2 2
/,p j
jj
Ni
13
( ) 1 ,1
j jj
rj j
j
N
N
2 jj
Cm
Lorentz model for dielectric (insulator)
In summary :
Also, keep in mind…..