Dielectric and Optical Properties
• As with conductivity, we will start with macroscopic property and
connect to the microscopic
• All aspects of free electrons have been covered: only bound electrons left
• Capacitance, Optical properties --> ε,n --> molecules and atoms
Review of capacitance and connection to dielectric constant
First, no material in capacitor
to
Vo
Q = CV
dq
dV
=i=C
dt
dt
V
C
i
Vo
V
©1999 E.A. Fitzgerald
t
1
The Capacitor
+
+
+
+
+
+
-- I=0 always
- in capacitor
-t
d/2
d/2
+V
ρ
∇⋅E =
E=
E
ρ
εo
−d
2
∫
−d
−t
2
V=
ρ
ρt
Q
=
dx =
εo
ε o Aε o
d
2
∫ Edx =
−d
2
V
Qd
Aε o
Q
Qd
=V =
C
Aε o
©1999 E.A. Fitzgerald
C=
εo A
d
2
The Capacitor
•
•
•
The air-gap can store energy!
If we can move charge temporarily without current flow, can store even more
Bound charge around ion cores in a material can lead to dielectric properties
•Two kinds of charge can create plate
charge:
•surface charge
•dipole polarization in the volume
•Gauss’ law can not tell the difference
(only depends on charge per unit area)
Image from Wikimedia Commons, http://commons.wikimedia.org
3
©1999 E.A. Fitzgerald
Material Polarization
+ + -+++ E’
E' = P
E
D = ε o E + P = εE
ε = ε rε o
ε r = 1+
P
= 1+ χ
εoE
+
++
++
+
+
- - - C=
ε rε o A
d
P is the Polarization
D is the Electric flux density or the Dielectric
displacement
χ is the dielectric or electric susceptibility
All detail of material response is in εr and therefore P
©1999 E.A. Fitzgerald
4
Origin of Polarization
• We are interested in the true dipoles creating polarization in materials (not
surface effect)
• As with the free electrons, what is the response of these various dipole
mechanisms to various E-field frequencies?
• When do we have to worry about controlling
– molecular polarization (molecule may have non-uniform electron density)
– ionic polarization (E-field may distort ion positions and temporarily create dipoles)
– electronic polarization (bound electrons around ion cores could distort and lead to
polarization)
• Except for the electronic polarization, we might expect the other mechanisms
to operate at lower frequencies, since the units are much more massive
• What are the applications that use waves in materials for frequencies below the
visible?
5
©1999 E.A. Fitzgerald
Application for different E-M Frequencies
Courtesy of the Opensource Handbook of Nanoscience and Nanotechnology,
http://en.wikibooks.org/wiki/Nanotechnology
In communications, many E-M waves travel in insulating materials:
What is the response of the material (εr) to these waves?
6
©1999 E.A. Fitzgerald
Wave Eqn. With Insulating Material and Polarization
r
r
∂B
∇xE = −
∂t
r r
r
r
r r ∂D nonmag
r r ∂ ε o E + P insulating
r
∂E
∇xH = J +
⎯⎯⎯→ ∇xB = J +
⎯⎯ ⎯
⎯→ ∇xB = ε
∂t
∂t
∂t
r
r r
r
(D = ε 0 E + P = εE)
(
)
∂2E εr ∂2E
∇ E = μ 0ε 0ε r 2 = 2 2
∂t
c ∂t
2
E = E0 e i(k •r −ϖt ) = E0 eik •r e −iϖt = E(r)e −iϖt
∇ E(r) = −
2
ω =
2
ω=
c2
εr
c
ω 2ε r E(r)
c2
k2
c
⎯⎯→ k
k ⎯optical
n
εr
So polarization slows down the
velocity of the wave in the
material
7
©1999 E.A. Fitzgerald
Compare Optical (index of refraction) and Electrical Measurements of ε
Material
Optical, n2
Electrical, ε
diamond
5.66
5.68
Only electrical polariztion
NaCl
2.25
5.9
Electrical and ionic polariztion
H 2O
1.77
80.4
Electrical, ionic, and
molecular polariztion
Polarization that is active depends on material and frequency
8
©1999 E.A. Fitzgerald
Microscopic Frequency Response of Materials
• Bound charge can create dipole through charge displacement
• Hydrodynamic equation (Newtonian representation) will now have a
restoring force
• Review of dipole physics:
-q
r
d
-
+
+q
r
r
Dipole moment: p = qd
r
p
Applied E-field rotates dipole to align with field:
r r
Torque τ = pxE
r
r r r
r
Potential Energy U = − p ⋅ E = p E cos θ
9
©1999 E.A. Fitzgerald
Microscopic Frequency Response of Materials
•
For a material with many dipoles:
r
r r
r
r
P = Np = NαE ( p = αE )
(polarization=(#/vol)*dipole polarization)
α=polarizability
χ=
r
r
p = αE
r
P
r , so χ = Nα
εo E
Actually works well only for low density of dipoles, i.e. gases: little screening
For solids where there can be a high density: local field
Eext
Eloc
For a spherical volume inside (theory of local field),
r
r
r
P
Eloc = E ext +
3ε o
10
©1999 E.A. Fitzgerald
Microscopic Frequency Response of Materials
• We now need to derive a new relationship between the dielectric constant and
the polarizability
D = ε r ε o Eext = ε o Eext + P
P = ε r ε o Eext − ε o Eext
⎛ 2 + εr ⎞
Eloc = Eext ⎜
⎟
⎝ 3 ⎠
Plugging into P=NαEloc:
ε r ε o Eext − ε o Eext = N α
(ε r + 2 ) E
3
ext
(ε r − 1)ε o = Nα (ε r + 2 )
3
Clausius-Mosotti Relation:
ε r − 1 Nα
α
=
=
ε r + 2 3ε o 3νε o
Macro
©1999 E.A. Fitzgerald
Micro
Where v is the volume per dipole (1/N)
11
Different Types of Polarizability
Highest natural frequency
•
•
•
Lowest natural frequency
Atomic or electronic,αe
Displacement or ionic, αi
Orientational or dipolar, αo
Lightest mass
Heaviest mass
α = αe + αi + αo
As with free e-, we want to look at the time dependence of the E-field:
E = Eo e −iωt
∂ 2 x m ∂x
m 2 =
− eE − Kx
∂t
τ ∂t
Restoring Force
Response
Driving
Force
Drag
m&x& = −eE − Kx
x = xo e −iωt
m(−ω 2 )xo = −eEo − Kxo
xo =
ωo =
©1999 E.A. Fitzgerald
eEo
K⎞
⎛
m⎜ ω 2 − ⎟
m ⎠
⎝
K
m
=
eEo
m ω 2 − ω o2
(
)
So lighter mass will have a
higher critical frequency
12
Classical Model for Electronic Polarizability
•
Electron shell around atom is attached to nucleus via springs
+
+
r
E
K
K
Zi electrons,
mass Zim
r
E
r
p
r
Z i m&r& = −Kr − Z i eEloc , assume r = ro e −iωt
13
©1999 E.A. Fitzgerald
Electronic Polarizability
ro =
ro =
eEo
K
ω
;
=
oe
m ω 2 − ω oe2
mZ i
(
)
p = qd = −Z i er; p = po e
po =
2
)
Zie2
αe =
m ω 2 − ω oe2
)
(
ω >> ω oe , α e = 0
ω << ω oe , α e =
2
Zie
mω oe2
⎛
K ⎞
⎟⎟
m⎜⎜ ω 2 −
mZ
i ⎠
⎝
If no Clausius-Mosotti,
ε r = 1+
−iωt
Zie
E o = α e Eo
m ω 2 − ω o2
(
eEo
Nα e
εo
NZ i e 2
= 1+
= n 2
2
2
ε o m ω − ω oe
(
)
εr
NZ i e 2
1+
ε o m ω oe2
( )
1
ωoe
ω
14
©1999 E.A. Fitzgerald
QM Electronic Polarizability
•
•
•
At the atomic electron level, QM expected: electron waves
QM gives same answer qualitatively
QM exact answer very difficult: many-bodied problem
E1
E0
e2
f10
E1 − E0
=
α e (ω ) =
;
ω
10
h
m ω102 − ω 2
f10 is the oscillator strength of the transition (ψ1 couples to ψo by E-field)
For an atom with multiple electrons in multiple levels:
f
E − E0
e2
α e (ω ) = ∑ j≠0 2 j 0 2 ; ω j 0 = j
h
m
j10 − ω
15
©1999 E.A. Fitzgerald
Ionic Polarizability
•
•
•
Problem reduces to one similar to the electronic polarizability
Critical frequency will be less than electronic since ions are more massive
The restoring force between ion positions is the interatomic potential
E(R)
Parabolic at bottom near Ro
Nucleii repulsion
k(R − Ro ) 2
E=
2
∂E
F=
= k(R − Ro )
∂R
Ro
-
+
R
Electron bonding in between ions
F = kx ⇒ σ ij = Cijkl ε kl
16
©1999 E.A. Fitzgerald
Ionic Polarizability
Eloc
•2 coupled differential eqn’s
•1 for + ions
•1 for - ions
+
-
+
-
&& = u&&+ − u&&−
w = u+ − u− , w
M=
p
u-
u+
Ionic materials always have ionic and
electronic polarization, so:
α tot
e2
= αi + αe = α + + α − +
M ω oi2 − ω 2
(
1
1
1
+
M+ M−
&& = −2Kw + eEloc
Mw
Eloc = Eo e −iωt , w = wo e −iωt
wo =
)
eEo
2K
ω
=
,
oi
M ω oi2 − ω 2
M
(
)
po = ewo = α i Eo
e2
αi =
M ω oi2 − ω 2
(
)
17
©1999 E.A. Fitzgerald
Ionic Polarizability
•
Usually Clausius-Mosotti necessary due to high density of dipoles
⎤
ε r −1 Nα tot
1 ⎡
e2
=
=
+
+
α
α
−
⎢ +
⎥
ε r + 2 3ε o
3ε o v ⎣
M (ω oi2 − ω 2 )⎦
By convention, things are abbreviated by using εs and ε∞:
ε s −1
1 ⎡
e2 ⎤
α+ +α− +
=
ω << ω oi ,
⎥
ε s + 2 3ε o v ⎢⎣
M (ω oi2 )⎦
ε ∞ −1 n 2 −1
1
ω >> ω oi ,
= 2
=
[α + + α − ]
ε ∞ + 2 n + 2 3ε o v
εr
εs
n2=ε∞
∴ εr = ε∞ +
ε∞ − εs
2
2 ⎛ ε∞ + 2 ⎞
⎜⎜
⎟
,
ω
ω
=
T
oi
⎟
ω2
ε
2
+
s
⎝
⎠
−1
2
ωT
ωT
ω
18
©1999 E.A. Fitzgerald
Orientational Polarizability
• No restoring force: analogous to conductivity
O
- p
H
+
θ
O
O
p=0
Analogous to conductivity, the
molecules collide after a certain
time t, giving:
α
CD
= oα
τωi−1
+q
C
H
-q
For a group of many molecules at some temperature:
f =e
−U
k bT
pE cosθ
k bT
= e
After averaging over the polarization of the
ensemble molecules (valid for low E-fields):
α DC
p2
~
3kbT
19
©1999 E.A. Fitzgerald