FI 3221 Electromagnetic Interactions in matterfismots.fi.itb.ac.id/FMF/wp-content/uploads/2016/01/Lecture03... · Dressel and G. Gruner : Section 6.1 and 5.1 Supplementary Jai Singh

10/01/2018

1

Alexander A. Iskandar

Physics of Magnetism and Photonics

FI 3221 ELECTROMAGNETIC

INTERACTIONS IN MATTER

• Lorentz

Model

• Drude

Model

CLASSICAL MODEL OF

PERMITTIVITY

Alexander A. Iskandar Electromagnetic Interactions in Matter 2

10/01/2018

2

Main

▪ A.M. Fox : Section 2.1.1 and 2.1.2, 7.1 – 7.3

▪ M. Dressel and G. Gruner : Section 6.1 and 5.1

Supplementary

▪ Jai Singh : Section 2.3

▪ S.A. Maier : Section 1.2


REFERENCES


)2()1( ~~ i

~ ~ N

N

~

~

)1()2(

)2()1(

~1~

~~

n

n

2~

~

)2(

22)1(

22)2(

)1(

1~

2~

n

n)2()1( ~~~ i

)1()2(

)2()1(

~~

~1~

)1(2)2(2)1(2

)1(2)2(2)1(2

~~~

2

1

~~~

2

1

nin

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Harmonic oscillation model can be used to

approximately modelled the frequency dependence

of susceptibility – the Lorentz model.

Behaviour of bound electrons in an electromagnetic

field.

Charges in a material are treated as harmonic

oscillators.

Optical properties of insulators are determined by

bound charges,


LINEAR DIELECTRIC RESPONSE OF

MATTER

fieldapplieddampingspringdt

dm FFF

r

2

2

Harmonic oscillation model can

be used to approximately

modelled the frequency

dependence of susceptibility –

the Lorentz model.

Consider a harmonic applied

field :


LINEAR DIELECTRIC RESPONSE OF

MATTER

+

-e, m

r

rp

e

fieldappliedeCdt

dm

dt

dm Er

rr

2

2

ti

fieldapplied e 0EE

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Solve for the dipole moment from the

following

yields the following steady-state the solution

or


ATOMIC POLARIZABILITY

rp

e

tieem

C

dt

d

dt

d 0

2

2

2

Eppp

tie 0pp

0

2

0

2

000

2Eppp

m

ei

022

0

2

0

1Ep

im

e

Recall the definition of atomic polarizability

hence


ATOMIC POLARIZABILITY

000 Ep

im

e

22

00

2 1

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Polarization is defined as the dipole moment per unit

volume

N is the atomic density per unit volume.

Thus, from the previous results, we obtain

where is defined as the plasma frequency.


SUSCEPTIBILITY AND PERMITTIVITY

0000001

EEEpP

NVV j

j

j

j

iim

NeN

p

22

0

2

22

00

2 1

m

Nep

0

22

Recall the relation between susceptibility and

permittivity


SUSCEPTIBILITY AND PERMITTIVITY

ii rrr 11

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From the last relation of susceptibility and

permittivity, we obtain


FREQUENCY DEPENDENCE OF

PERMITTIVITY

22222

0

2

22222

0

22

0

2

1

p

r

p

r

2

0

2

p

1

p

0

<< 0 : high n ’ → low vphase

0 : strong dependence

vphase , large absorpt ion ( n” )

>> 0 : n ’ =1 → vphase = c


REFRACTIVE INDEX

2

2

22

22

rrr

rrr

n

n

rn

nnnn rr 2,

22

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For the range higher than the frequency of largest

absorption, 0, we can approximate

On the other hand, from KK theory

We can make the following identification


KRAMERS – KRONIG SUM RULE

2

2

22222

0

22

0

2

)1( 11~

pp

0

)2(

2

0

22

)2()1( )(~2

1)(~2

1)(~

ddP

2

0

2

0

22

)()(also2

)( pp dnd

Electrons in metal are free ( free electron gas model ,

Drude model), however in its motion there are

collisions, hence its equation of motion is given as

The mean free path of the electron is characterized

by its relaxation time t, hence we can write

Assume a time harmonic applied

field.


OPTICAL PROPERTIES OF METAL

fieldappliededt

dm

dt

dm E

rr

2

2

fieldappliedemdt

dm E

vv

t

ti

fieldapplied e 0EE

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We are looking for a solution in the form of

Substituting yields


OPTICAL PROPERTIES OF METAL

tie vv

t

01

Ev

im

e

Recall the current density in metal is given in terms

of the drift velocity, the charge and its volume

density, hence

Comparing with the Ohm’s law , we

deduce

Where 0 is called the DC conductivity.


CONDUCTIVITY

t

0

2

1EvJ

im

NeNe

0EJ

m

Ne

ii

mNe t

t

t

2

00

2

,11

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Separating the real and imaginary parts of the

conductivity, yield

Recall the definition of plasma frequency

then


CONDUCTIVITY

22

0

22

021

11 t

t

t

ii

m

Nep

0

22

22

2

022

01

1

1

1 tt

t

p

22

2

022

02

11 t

tt

t

t

p

m

Ne t

2

0

Some values of are


CONDUCTIVITY

m

Nep

0

22

Metal p (eV)

Al 15.1

Cu 8.8

Ag 9.2

Au 9.1

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Recall that there are actually two kinds of electrons

in a metal, namely the bound electrons and

conduction electrons. Both of these electrons

contributes to the permittivity.

Consider the Ampere-Maxwell equation,

with time-varying electric field and the current

density as follows


BOUND AND CONDUCTION ELECTRONS

JD

H

t

ti

fieldapplied e 0EE

0EJ

We have

The effective permittivity consist of contribution from

the bound charges B() and the conduction electrons


BOUND AND CONDUCTION ELECTRONS

000 EEJD

H

Bi

t

00

0

0

0

0

0

0

E

E

EJD

H

eff

B

B

i

ii

ii

t

0

iBeff

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Using the expression of conductivity

We have

i.e.


DIELECTRIC CONSTANT OF METAL

22

0

0

22

0

00 1

1

1 t

t

t

ii BBeff

22

0

22

021

11 t

t

t

ii

22

0

0

11

1

t

t

B 22

0

0

21

1

t

At frequencies visible since visiblet >> 1, then

Hence, effective permittivity become



t

t

t

0

22

00

1i

i

t

t

2

0

0

23

0

0

0

ii BBeff

22

0

0

11

1

t

t

B 22

0

0

21

1

t

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Define

Then

can be written as



m

Nep

0

2

0

02

t

t

3

2

2

2

pp

Beff iFree electrons

Bounds electrons

t

t

2

0

0

23

0

0

0

ii BBeff


EXAMPLE : ALUMINIUM

t

3

2

2

2

pp

Beff i

At >> 0, similar

behaviour as dielectric

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Dielectric function () of the free electron gas (solid

line) fitted to the literature values of the dielectric

data for gold [Johnson and Christy, 1972] (dots).

Interband transitions limit the validity of this model

at visible and higher frequencies.


COMPARISON WITH EXPERIMENTAL DATA


COMPARISON WITH EXPERIMENTAL DATA

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The discrepancies between the experimental data

and the theoretical model can be reconcile by

considering interband transition of the electrons.

To this end, we add extra terms in the permittivity

expression that correspond to this interband

transition.


CONTRIBUTION FROM INTERBAND

TRANSITION

pp

i

pp

i

ppp

p

p

p

i

e

i

eA

i

pp

2

12

2

)(


CONTRIBUTION FROM INTERBAND

TRANSITION

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The Clausius-Mossotti equation

relates the dielectric constant of a material to the

polarisability of its atom.

Derive this relationship by carefully assuming that

the field at point in the dielectric can be written as

the sum of the “local” field (that consist of the

external field and the field generated by all other

molecule outside of the spherical exclusion) and the

field induced by the dipole in the spherical exclusion.


HOMEWORK

2

13 0

r

r

N

V

r

Recall that the Polarization vector is defined as


HOMEWORK

r

extE

localE

dipoleinducedE

dipoleinducedlocaltotal EErE

)(

localV

Nfieldlocallitypolarizabiatomicdensityatomic EP

))()((

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The following graphs show the real and imaginary

part of the permittivity function of an unknown

dielectric material


HOMEWORK

From the previous graphs, estimate the resonant

frequency of the Lorentz oscillator model.

Estimate the plasma frequency of the model.

With the electron mass value of me = 9.1 10-31 kg,

and , estimate the valence

electron density N that contribute to this Lorentz

oscillator model.

Estimate the damping constant of the material.


HOMEWORK

p

2212 mNC1085.8

Documents

FI 3221 Electromagnetic Interactions in matterfismots.fi.itb.ac.id/FMF/wp-content/uploads/2016/01/Lecture03... · Dressel and G. Gruner : Section 6.1 and 5.1 Supplementary Jai Singh