Dielectric Materials

• Pre-requisite
• Capacitive behavior
• Polarization
• Dielectric Loss
• Insulating behavior
• Dielectric Breakdown
• Refractive Index
• Piezoelectricity & Ferroelectricity

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 Pre-requisite

Gauss’s Law:
Qtotal
 En dA 
surface
 r 0

Where:
the integral refers to that over the whole
of the surface enclosing the charge Qtotal
and En is the electric field normal to a
small area dA on the closed surface.

 DdA  Q
surface
total

D is the displacement (surface charge density).

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 Polarization

• Microscopic View
• Macroscopic View
• Clausius-Mossotti Equation:
The link between Microscopic and Macroscopic
Views
• Types of Polarizations

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 Polarization vs Charge Flow

• Dielectric materials typically have an energy gap of

greater than 2.5 eV.
• Insulating at room temperature
• No need to be concerned with the transport of
charge carriers.
• Need to consider how the bound charges are
polarized in the presence of an applied electric field

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 Polarization: Microscopic View
“polarization” process
• Consider a negative and a positive charge separated by a distance a,
as shown below.

• We define the electric dipole moment, p, of a pair of equal charges

as the product of the charge and the separation. Direction of vector
p points from the negative to the positive charge.

p=Qa

• The separation of negative and positive charges resulting in an

induced dipole is termed polarization.

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 “Polarization” in Dielectric Slab (Dipole Distribution)

- In the presence of an applied electric

field, atoms and molecules become
polarized  distribution of dipole
moments

- Dipoles aligned head to tail  every

+ve charge has a neighboring –ve
charge to neutralize it  no net
charge in bulk

Note: the free and bound charges are different. +Q and –Q are free
charges that arrive on the plates of the battery whereas +Qp and –Qp
are polarization charges bound to the molecules.
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“Polarization” in Dielectric Slab (polarization “P”)

- However, there is a net +Qp on RHS

and –Qp on LHS face. These bound
charges are a result of polarization and
termed as surface polarization charges.

- The polarization P is equal to the dipole

moment per unit volume, given by

where p1, p2, …, pN are the average

dipole moments induced at N
(b) (c) molecules in the volume.

Notes: - If pav is the average dipole moment per

- Polarization “P” is a macroscopic parameter. molecules, then an equivalent definition
- dipole moments pi (i=1, 2…, N), pav are of P is
microscopic parameters. P=Npav
- The equations link “macro-” and “micro-”
parameters. where N is number of molecules per
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“Polarization” in Dielectric Slab (Surface polarization charge density)

- A large dipole between –Qp and

+Qp separated with d, its total
dipole moment ptotal is

ptotal=Qpd

- From the definition of the

polarization “P”, we have

P=(Qpd)/(Ad)=Qp/A

(b) (c) - Introducing σp as surface

(polarization) charge density, we
Notes: have
- Polarization “P” (magnitude) equals “surface
charge density” (charge per unit area) P=σp
appearing on the surface of a dielectric slab

- Direction of “P” is normal to the

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 Summary

Direction of P is normal to surface.

For +ve surface charge, P points
outward from surface. For -ve surface
charge, P points into surface.

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“Polarization” in Dielectric Slab (“micro-view” picture)

“polarization”
process No
Nonet
netbulk
bulk
charge
charge

Electric Dipole
Dipole
Electric
distribution
distribution
Field
Field Surface
Surface
(polarization)
(polarization)
charge
charge

Introduce Treat as a
polarization large dipole
parameter “P”
(definition) P=Np
P=Npavav
P=σ
P=σpp

Macro-parameter Micro-parameter

Polarization “P” is a macro-parameter, but …

And pav is a micro-parameter, but …
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 Polarization: Macroscopic View
Dielectric displacement (or surface charge density)

- When vacuum is present between two parallel pates, the surface

charge density (or dielectric displacement) D is given by
D=0E

- Hence at the macroscopic level, a dielectric is characterized by its

permittivity  which relates the surface charge density (or dielectric
displacement D’) to the electric field via
D’=E=0rE

 is the product of the permittivity of free space 0 and the relative

dielectric constant r.

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Dielectric constant r (macro parameter)

- D’ can also be expressed as

D’ = 0rE
=0E+P
= 0E+ 0(r-1)E

here, P is the increase in charge density above that for a vacuum,

and its value equals to 0(r-1)E

- The meaning of “P” here is “additional” charge density in dielectric

compared to a vacuum case, thus has the same meaning as the
polarization “P” (surface charge density).

- To express the dependence of P on E, we define the dielectric

susceptibility χ, by
P=0χE

- Thus, we have
χ=r-1 or r=χ+1

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 Polarization (macro view)

In vacuum:

Displacement (surface
Electric
Electric charge density)
Field
Field D=0E

In dielectric: Item 0E: is the same

as the case in vacuum
Displacement (surface
Electric
Electric charge density)
Field
Field D=0E+P Item “P=0(r-1)E”: is
the polarization (or
surface polarization
charge density), due to
contribution of dipole
formation in dielectric
Polarization “P” is a function of “E”, how about “pav” … under E field
(Recall: P=Npav from micro-view)
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 Clausius-Mossotti Equation

• Local (or Internal) Field: A re-visit

of micro-view
• Link between “macro-” and “micro-”
• Clausius-Mossotti Equation

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 Local (or Internal) Field: A Revisit of “Micro-” View

Electric field experienced by - In general, the induced polarization

molecules is not just “E”… depends on the actual field, or local
field, experienced by the molecule.

- This includes field due to free

charges (resulting from the applied
field) on the plates AND the field
arising from the dipoles surrounding
the molecule.

- Local field is actual field that acts on

a molecule. It can be calculated by
removing molecule and evaluating
field at that point from charges on
capacitor plate and dipoles
surrounding that point

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Local (or Internal) Field: Some comments

- For solids and many liquids where the number of atoms or

molecules per unit volume is large, the local field is greater than
the applied field.

- The greater the polarization, the greater the local field because
there are bigger dipoles around the point of interest. Thus Elocal
depends on the arrangement of polarized molecules around the
point and hence the crystal structure.

- On the other hand, with dilute gases where the concentration of

molecules or atoms is very low (surrounding dipoles are very
sparse and far away), then the influence of other dipoles on the
local electric field is negligible, and the local field is effectively the
same as the applied field.

How does the “local” field relate to applied field “E”?

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Relation of “Elocal” to “E”

- The relation between the local field and applied field:

“P” including contribution of “dipoles” in

dielectric to local field

where E is the applied field, P is the polarization induced by the applied

field.
(Note: the equation is derived by considering the simplest cubic crystal
structure or a liquid).

- Polarizability
At low electric fields, we assume that the dipole moment p is
proportional to the local electric field Elocal
p=Elocal
where  is a constant called polarizability which depends on the
polarization mechanism of the material concerned.

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Link between micro- and macro-view
An inherent micro-
parameter of a dielectric;
An inherent
different materials have
macro-parameter P=Npav Dipole different types of
of a dielectric;
Polarization
P pav polarization mechanisms
values depends on
(different type of )
material types
Polarizability
Dielectric constant
p=Elocal 
r P=0(r-1)E

E Elocal

Macro-view Micro-view

How to link “dielectric constant” (r) and “polarizability” ()?

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Clausius-Mossotti Equation

 r  1 N - CM equation relates the dielectric


 r  2 3 0 constant (r) and polarizability ()
directly

DIY - A higher  will cause a higher

dielectric constant.

- Four different types of polarization

mechanisms exist in dielectric
+
P=Npav=NElocal  Results in different dielectric
constant

 Results in different capacitive

+ behavior in response to frequency
P=0(r-1)E and temperature

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 Types of Polarization

- Four main mechanisms for polarization

(i) Electronic polarization
(ii) Ionic polarization
(iii) Orientational/dipolar polarization
(iv) Interfacial polarization

- Need to understand:
(i) origin of polarization
(ii) magnitude of polarization in various materials
(iii) speed with which dipoles can align

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 Electronic Polarization
Dipole moment defined by
pe=eElocal

and polarization
Pe=Npe=NeElocal

where
e is the electronic polarizability
and
N is the number of molecules
per unit volume.
Mechanism:
Displacement of electron cloud in a neutral
atom by an electric field, thereby inducing a
dipole moment on the atom.

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Properties of Electronic Polarization

- Clausius Mossotti (CM) Equation of Electronic polarization

 r  1 N e

 r  2 3 0

- Type of Materials
Since this form of polarization is atomic in nature, it is present in all
materials regardless of type of bonding. When field is removed,
polarization vanishes.

- Temperature dependence of e
Electronic structure of an atom is independent of the temperature  e
has no dependence on temperature.

- Operational frequency
Light electron cloud  the response speed to the electric field is very
fast. Typically up to ~1015-1016 Hz. (optical frequency)

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 Ionic Polarization

Dipole moment defined by

pi=iElocal

and polarization
Pi=Nipi=NiiElocal

where
i is the ionic polarizability
and
Mechanism: Ni is the number of ion pairs per
Displacement of anions and cations in unit volume.
crystals relative to their normal positions
by an electric field, resulting in net dipole
moment.

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Properties of Ionic Polarization

- Type of Materials
Occurs predominantly in ionic materials. Like NaCl, CsCl, …

- Temperature dependence of i
None.

- Operational frequency
Since entire ions (much heavier than electrons) are being
displaced, the mechanism only operates up to ~1012 – 1013 Hz
(infra-red frequencies).

- Clausius Mossotti equation also valids for (i+e) and r.

 r  1 N e  N i i

r  2 3 0

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 Orientational/Dipolar Polarization
Example: HCl molecules
Mechanism:
In the presence of an applied field,
(1) When E=0
these molecules with permanent dipole
moments orientate themselves to try to
- Each molecule has a
align with the applied field.
permanent dipole moment po

- Due to random thermal

motion, the alignment is upset

 thermal energy randomizes

orientations of all dipole
moments

 the dipole moment per

molecule po,av averaged over
Certain molecules have permanent dipoles. We call the whole system is zero.
the permanent dipole moment of one molecule po
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Example of Orientational/Dipolar Polarization

(2) When E≠0

- H+ and Cl- experience forces in opposite direction under applied field

 torque rotates molecule to align with E
- If all molecules would to simply rotate and align with the applied field,
the dipole moment of the solid would be Ps=Npo (N is # of molecules per
unit of volume).
- Thermal energy will try to randomize the orientations due to collisions of
molecules which destroy dipole moments.
- There is nevertheless still a net average polarization directed along the
field with average dipole moment per molecule po,av.
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Types of Materials for Orientational/Dipolar Polarization

Type of Material:
Molecules that possess permanent dipole moments and are free to rotate to
align with the electric field direction. Common in silicates, polar polymers, polar
liquids (water, alcohol, acetone), polar gases (gaseous HCl, steam).

Dipole moment or not?

Key factor: geometry

Ex1:
A linear molecule like carbon dioxide
has two oxygen atoms placed
symmetrically around the carbon 
net dipole moment of zero.

Ex2:
Water molecule is bent (has a δ+ and
δ- region) and has a dipole moment.
(left figure)
E3406/NUS 27
Properties of Orientational/Dipolar Polarization

Temperature dependence of o:

- While dipole moments are trying to line up with the applied field, they
are jostled by their thermal motion and not all of them succeed in
lining up properly.

- Assuming Boltzmann statistics, the average dipole moment per

molecule is given (without proof)

- Thus, the orientational polarizability given by

- Hence the polarization due to orientational polarization

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Properties of Orientational/Dipolar Polarization

- We see that polarization due to orientational polarization

Po is inversely proportional to the absolute temperature.

(1) Unlike electronic and ionic polarization, orientational

polarization is strongly dependent on temperature
(2) Orientational polarization can be retained after the field
is removed.
(3) o decreases with increasing temperature  r also
decreases with increasing temperature.

Operational frequency:

Involves of rotation of whole molecules (or molecular groups) to

align with the electric field, therefore only occurs at lower
frequencies of ~1011-1012 Hz (sub-infrared frequencies)

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 Interfacial Polarization
Mechanism:
Accumulation of charge at (i) an interface between two materials or
(ii) between two regions in a material

Example 1: Interfacial polarization

With an electric field:

 mobile +ve ions pile up at

interface between dielectric and
–ve electrode

 increased polarization

 increased dielectric constant

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Another Example of Interfacial Polarization

Example 2: interfacial polarization due to grain

boundaries

Trapping of electrons by
dangling bonds at grain
boundaries causing polarization

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Properties of Interfacial Polarization

Type of Materials:
Occurs predominantly at interfaces and in materials with
defects, grain or phase boundaries, impurities.

Operational frequency
Occurs at even lower frequencies of ~10-3 to 103 Hz.

CM Equation is always not valid for interfacial

polarization
Interfacial polarization cannot be considered as a
straightforward contribution because it occurs at
interfaces (locally) and cannot be put into an average
polarization per molecule in the bulk.

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 Combined Polarization
- The total polarization of a material is the sum of contributions from all
components.

- In general, the average polarization due to the electronic, ionic and

orientational/dipolar components is

- The dielectric constant under electronic, ionic and orientational

polarization is given by the combined Clausius Mossotti equation.

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Some discussions on “Combined Polarization”

- Not all atoms or molecules posses each of the above types of

polarizability
(i) All display electronic polarizability, since this aries from polarization of
atoms.
(ii) Many materials have ionic polarizations (exceptions are elements, eg.
Ar, Ne, He, and molecules of the same atomic species, eg. H2, O2, Si).
(iii) Those with permanent dipole moments have orientational/dipolar
polarization.

- Several factors determine the extent with which each

dielectric is affected by each of the polarization mechanism:
(1) Atomic scale structure
(2) Type of bonding
(3) Type of atoms
(4) Frequency of applied electromagnetic field.

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Examples of polarization types for different materials

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