Upload
hoangcong
View
213
Download
0
Embed Size (px)
Citation preview
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
Alexander A. Iskandar Electromagnetic Interactions in Matter 3
REFERENCES
Alexander A. Iskandar Electromagnetic Interactions in Matter 5
)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
10/01/2018
3
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,
Alexander A. Iskandar Electromagnetic Interactions in Matter 6
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 :
Alexander A. Iskandar Electromagnetic Interactions in Matter 7
LINEAR DIELECTRIC RESPONSE OF
MATTER
+
-e, m
r
rp
e
fieldappliedeCdt
dm
dt
dm Er
rr
2
2
ti
fieldapplied e 0EE
10/01/2018
4
Solve for the dipole moment from the
following
yields the following steady-state the solution
or
Alexander A. Iskandar Electromagnetic Interactions in Matter 8
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
Alexander A. Iskandar Electromagnetic Interactions in Matter 9
ATOMIC POLARIZABILITY
000 Ep
im
e
22
00
2 1
10/01/2018
5
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.
Alexander A. Iskandar Electromagnetic Interactions in Matter 10
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
Alexander A. Iskandar Electromagnetic Interactions in Matter 11
SUSCEPTIBILITY AND PERMITTIVITY
ii rrr 11
10/01/2018
6
From the last relation of susceptibility and
permittivity, we obtain
Alexander A. Iskandar Electromagnetic Interactions in Matter 12
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
Alexander A. Iskandar Electromagnetic Interactions in Matter 13
REFRACTIVE INDEX
2
2
22
22
rrr
rrr
n
n
rn
nnnn rr 2,
22
10/01/2018
7
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
Alexander A. Iskandar Electromagnetic Interactions in Matter 14
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.
Alexander A. Iskandar Electromagnetic Interactions in Matter 16
OPTICAL PROPERTIES OF METAL
fieldappliededt
dm
dt
dm E
rr
2
2
fieldappliedemdt
dm E
vv
t
ti
fieldapplied e 0EE
10/01/2018
8
We are looking for a solution in the form of
Substituting yields
Alexander A. Iskandar Electromagnetic Interactions in Matter 17
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.
Alexander A. Iskandar Electromagnetic Interactions in Matter 18
CONDUCTIVITY
t
0
2
1EvJ
im
NeNe
0EJ
m
Ne
ii
mNe t
t
t
2
00
2
,11
10/01/2018
9
Separating the real and imaginary parts of the
conductivity, yield
Recall the definition of plasma frequency
then
Alexander A. Iskandar Electromagnetic Interactions in Matter 19
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
Alexander A. Iskandar Electromagnetic Interactions in Matter 20
CONDUCTIVITY
m
Nep
0
22
Metal p (eV)
Al 15.1
Cu 8.8
Ag 9.2
Au 9.1
10/01/2018
10
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
Alexander A. Iskandar Electromagnetic Interactions in Matter 23
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
Alexander A. Iskandar Electromagnetic Interactions in Matter 24
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
10/01/2018
11
Using the expression of conductivity
We have
i.e.
Alexander A. Iskandar Electromagnetic Interactions in Matter 25
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
Alexander A. Iskandar Electromagnetic Interactions in Matter 26
DIELECTRIC CONSTANT OF METAL
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
10/01/2018
12
Define
Then
can be written as
Alexander A. Iskandar Electromagnetic Interactions in Matter 27
DIELECTRIC CONSTANT OF METAL
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
Alexander A. Iskandar Electromagnetic Interactions in Matter 28
EXAMPLE : ALUMINIUM
t
3
2
2
2
pp
Beff i
At >> 0, similar
behaviour as dielectric
10/01/2018
13
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.
Alexander A. Iskandar Electromagnetic Interactions in Matter 29
COMPARISON WITH EXPERIMENTAL DATA
Alexander A. Iskandar Electromagnetic Interactions in Matter 30
COMPARISON WITH EXPERIMENTAL DATA
10/01/2018
14
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.
Alexander A. Iskandar Electromagnetic Interactions in Matter 31
CONTRIBUTION FROM INTERBAND
TRANSITION
pp
i
pp
i
ppp
p
p
p
i
e
i
eA
i
pp
2
12
2
)(
Alexander A. Iskandar Electromagnetic Interactions in Matter 32
CONTRIBUTION FROM INTERBAND
TRANSITION
10/01/2018
15
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.
Alexander A. Iskandar Electromagnetic Interactions in Matter 33
HOMEWORK
2
13 0
r
r
N
V
r
Recall that the Polarization vector is defined as
Alexander A. Iskandar Electromagnetic Interactions in Matter 34
HOMEWORK
r
extE
localE
dipoleinducedE
dipoleinducedlocaltotal EErE
)(
localV
Nfieldlocallitypolarizabiatomicdensityatomic EP
))()((
10/01/2018
16
The following graphs show the real and imaginary
part of the permittivity function of an unknown
dielectric material
Alexander A. Iskandar Electromagnetic Interactions in Matter 35
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.
Alexander A. Iskandar Electromagnetic Interactions in Matter 36
HOMEWORK
p
2212 mNC1085.8