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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 (spatially local response of media) : relative dielectric constant = relative permittivity = dielectric function

Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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Page 1: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 2: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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)

Page 3: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

Ampere-Maxwell’s Law

Stokes theorem (very general)

Faraday’s Law (Faraday 1775-1836)

Page 4: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

EM wave Equation in bulk media ()

: EM wave equation in bulk media without net charge ( and =0)

Page 5: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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.

Page 6: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 7: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 8: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 9: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

Now, how to determine and b() ?

Microscopic origin of -response of matterProfessor Vladimir M. Shalaev, Univ of Purdue

Page 10: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 11: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

A. of insulating media (dielectric : P, J = 0)A. Insulator

Page 12: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

A. Insulator

Page 13: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

A. Insulator

Page 14: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

Assume j ‘s are the same for all atoms

( ) jN

A. Insulator (single atomic gas)

Page 15: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

A. Insulator (single atomic gas)

Page 16: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

1

A. Insulator (single atomic gas)

Page 17: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

A. Insulator (realistic gas with different atoms)

Page 18: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

A. Insulator (solids)

Page 19: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

Local field at an atom

A. Insulator (solids)

유도해보시오!

Page 20: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 21: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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.

Page 22: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 23: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 24: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 25: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 26: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 27: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 28: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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.

Page 29: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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)

Page 30: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 31: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 32: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 33: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 34: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 35: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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

Page 36: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

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 :

Page 37: Optical properties of materialsoptics.hanyang.ac.kr/~shsong/5-Optical properties of...dr mdr dv meEmmveE s d dt dt Cr t Lorentz model (Harmonic oscillator model) ... exp( ) : small-amplitude

Also, keep in mind…..