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Title: ELECTROCERAMICS
Description: This coursework consist of 41 lectures , Suitable for material science , mechanical,electrical students , chemistry. 1: Structure of Ceramic Materials 5 2: Defect Chemistry and Equilibria 7 3: Diffusion and Conduction in Ceramics 7 4: Linear Dielectric Ceramics 8 5: Nonlinear Dielectric Ceramics 6 6: Magnetism and Magnetic Ceramics 5 7: Superconducting Ceramics 1 8: Multiferroic and Magnetoelectric Ceramics 1 9: Synthesis Methods 1 Total number of equivalent lectures 41 total lectures. Total no of pages : 376.
Description: This coursework consist of 41 lectures , Suitable for material science , mechanical,electrical students , chemistry. 1: Structure of Ceramic Materials 5 2: Defect Chemistry and Equilibria 7 3: Diffusion and Conduction in Ceramics 7 4: Linear Dielectric Ceramics 8 5: Nonlinear Dielectric Ceramics 6 6: Magnetism and Magnetic Ceramics 5 7: Superconducting Ceramics 1 8: Multiferroic and Magnetoelectric Ceramics 1 9: Synthesis Methods 1 Total number of equivalent lectures 41 total lectures. Total no of pages : 376.
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Electroceramics
Overview
Electro Ceramics Web Course (NPTEL)
Contact information of the course instructor:
Ashish Garg, Associate Professor
Department of Materials science and Engineering
Indian Institute of Technology Kanpur
Kanpur 208016, India
Telephone: 0512-259-7904
Email: ashishg@iitk
...
in
Web: http://home
...
ac
...
Some of the examples are
Dielectric materials such as SiO2 are used as data storage elements in random access
memories or RAMs
Ferroelectrics such as BaTiO3 and PbTiO3 are used as sensors and actuators
Magnetic oxides such as iron oxides are used for data storage in magnetic heads
ZnO is used circuit protection materials in devices named as varistors
ZrO 2 stabilized with other oxides is used in fuel cells and batteries
Hence, to understand these materials better and to engineer them as per our needs, we need to
understand their science viz
...
A sound understanding of these would (hopefully)
enable us tailor the structure and properties of these material with good degree of control
...
List of Topics
Module Topics
Equivalent
each)
1: Structure of Ceramic Materials
5
2: Defect Chemistry and Equilibria
7
3: Diffusion and Conduction in Ceramics
7
4: Linear Dielectric Ceramics
8
5: Nonlinear Dielectric Ceramics
6
6: Magnetism and Magnetic Ceramics
Lectures
(50-60
m
5
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Electroceramics
Table of Contents
Table of Contents
1
...
1 Brief Review of Structure of Materials
1
...
3 Packing of atoms in metals
1
...
5 Structure of Covalent Ceramics
1
...
7 Compounds based on FCC Packing of ions
1
...
9 Orthogonal Structures
1
...
11 Summary
2
...
1 Point Defects
2
...
3 Defect Reactions
2
...
5 Defect Structures in Non-stoichiometric Oxides:
2
...
7 Concentration of Intrinsic Defects
2
...
9 Units for defect Concentration
2
...
11 Defect Equilibria in Stoichiometric Oxides
2
...
13 Defect Structures involving Oxygen vacancies and interstitials:
2
...
15 A Simple General Procedure for constructing at Brouwer’s Diagram
2
...
17 Example: Comparative behaviour of TiO2 and MgO vis-à-vis oxygen pressure
2
...
19 Examples
2
...
Defects, Diffusion and Conduction in Ceramics
3
...
2
3
...
4
Diffusion
Diffusion Kinetics
Examples of Diffusion in Ceramics
Mobility and Diffusivity
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5 Analogue to the electrical properties
3
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7 Ionic Conduction: Basic Facts
3
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9 Characteristics of Ionic Conduction
3
...
11 Conduction in Glasses
3
...
13 Examples of Ionic Conduction
3
...
15 Nernst Equation and Application of Ionic Conductors
3
...
17 Summary
4
...
1 Basic Properties: Dielectrics in DC electric field
4
...
3 Microscopic Approach
4
...
5 Analytical treatment of Polarizability
4
...
7 Frequency dependence of dielectric properties: Resonance
4
...
e
...
9 Circuit Representation of a Dielectric and Impedance Analysis
4
...
11Dielectric Breakdown
4
...
Nonlinear Dielectrics
5
...
2 Classification based on Crystal Classes
5
...
3
...
3
...
3
...
3
...
3
...
3
...
3
...
3
...
3
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3
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3
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3
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4 Piezoelectric Ceramics
5
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1 Direct Piezoelectric Effect
5
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2 Reverse or Converse Piezoelectric Effect
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4
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4
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4
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4
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4
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5 Pyroelectric Ceramics
5
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1 Difference between and pyroelectric and ferroelectric material
5
...
2 Theory of Pyroelectric Materials
5
...
3 Measurement of Pyroelectric coefficient
5
...
4 Direct and Indirect effect
5
...
5 Common Pyroelectric Materials
5
...
6 Common Applications
5
...
Magnetic Ceramics
6
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2 Macroscopic view of Magnetization
6
...
4 Diamagnetism
6
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6 Ferromagnetism
6
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8 Ferrimagnetism
6
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10 Magnetic Losses and Frequency Dependence
6
...
12 Summary
7
...
1
7
...
3
7
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5
7
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7
7
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Multiferroic and Magnetoelectric Ceramics
8
...
2
8
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4
8
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6
8
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8
Introduction
Historical Perspective
Requirements of a magnetoelectric and multiferroic material
Magnetoelectric Coupling
Type I Multiferroics
Type II Multiferroics
Two Phase Materials
Summary
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Synthesis Methods
9
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2
9
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4
Bulk Preparation Methods
Thin Film Preparation Methods
Thin film deposition: Issues
Summary
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More references are provided
in each module
...
Physical Ceramics: Principles for Ceramic Science and Engineering, Y
...
Chiang, D
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Birnie, and W
...
Kingery, Wiley-VCH
2
...
D
...
K
...
R
...
Principles of Electronic Ceramics, by L
...
Hench and J
...
West, Wiley
4
...
J
...
M
...
Nonstoichiometry, Diffusion and Electrical Conductivity in Binary Metal Oxides (Science &
Technology of Materials), P
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Kofstad, John Wiley and Sons Inc
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7
...
9
...
11
...
13
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15
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Introduction to Solid State Physics, C
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Solymer and D
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W
...
D
...
J
...
A
...
R
...
Toft Sørensen, Academic Press
Dielectrics and Waves, A
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von Hippel, John Wiley and Sons
Feynman Lectures on Physics, Volume 1-3, R
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Feynman, Addison Wesley Longman
Materials Science and Engineering: A first course, V
...
D
...
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Module 1: Structure of Ceramic Materials
Introduction
In this module, we will first review the structure and bonding in the materials in general followed by a
brief discussion on how atoms pack together in the solids and what are the types of interstices
present in various structures
...
Together, this information will form the basis for structures in ceramic
materials which are typically bonded with a mix of ionic and covalent bonding
...
These are essentially based on packing of
anions closed packed forms where cations fill the interstices
...
D
...
, Wiley
Physical Ceramics: Principles for Ceramic Science and Engineering, Y
...
Chiang, D
...
Birnie, and W
...
Kingery, Wiley-VCH
Introduction to Ceramics, W
...
Kingery, H
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Bowen, D
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Uhlmann, Wiley
Fundamentals of Ceramics, Michael Barsoum, McGraw Hill
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1 Brief Review of Structure of Materials
In the following sections, we will quickly look at the concepts of lattice, unit-cell and crystal
structures which will be useful to understand the crystal structures of common ceramic compounds
...
1
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(Figure 1
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(Figure 1
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1(a) Unit-cell
representation
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1(b) Schematics of arrangement of points in
space
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1
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2)
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Figure 1
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Module 1: Structure of Ceramics
Brief Review of Structure in Materials
1
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3 Motif and Crystal Structure
Crystal structure: a combination of motif and point lattice
Motif is defined as a unit or a pattern
...
Often it is also called as Basis
...
Figure 1
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Module 1: Structure of Ceramics
Brief Review of Structure in Materials
1
...
4 Types of Lattice
Lattice can further be classfied into two types
Primitive lattice having one formula unit or one lattice point or one unit of motif per
unit cell, and
Non-primitive lattices having more than one lattice points or more than one unit
of motif per unit cell
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4 Primitive and Non-primitive
lattices
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1
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Symmetry elements underlying a point lattice (see the figure)
Reflection: reflection across a mirror plane
Rotation: rotation around a crystallographic axis by an angle θ such as 360°/θ is
an integer of value 1, 2, 3, 4 and 6 and is referred to as n -fold rotation
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5 Basic symmetry operations in crystals
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1
...
However, as one starts defining various shapes, we come up with seven categories, called as
crystal systems, in which all possible unit cells shapes would fit provided space filling criteria
is fulfilled
...
Crystal system and lattice
parameters
Cubic
a = b =
c,
Minimum
symmetry
elements
Four 3-fold
rotation axes
Tetragonal
One 4-fold
rotation (or
rotation-inversion)
axis
Orthorhombic
Three
perpendicular 2fold rotation (or
rotation-inversion)
axis
Rhombohedral
One 3-fold
rotation (or
rotation-inversion)
axis
a=b=c
Hexagonal
One
6-fold
rotation
(or
rotation-inversion)
axis
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6 Seven crystal systems
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1
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Cubic
a=b=c,
Tetragonal
Orthorhombic Rhombohedral Hexagonal
Monoclinic
Triclinic
a=b=c
Figure 1
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Module 1: Structure of Ceramics
Brief Review of Structure in Materials
1
...
8 Planes and Directions
Faces and directions joining atoms in crystals can be best described by Miller Indices (in the
names of W
...
Miller ) ascribed to determine various planes and directions
...
1
...
8
...
k
...
k
...
1)
h/a, k/b, and c/l are the intercepts of the plane on x, y, and z axes
...
1
...
8
...
Miller indices are [ u , v , w ] for a direction or < u , v , w > for identical set of directions
where u , v , w are integers
Vector components of the direction resolved along each of the crystal axis reduced to smallest
set of integers
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8(a) Planes and Directions in Crystals
Crystal Directions
How to locate a direction :
Example: [231] direction would be
1/3 intercept on cell a-length
1/2 intercept on cell b-length and
1/6 intercept on cell c-length
Directions are always denoted with [
uvw ] with square brackets and family
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8(b) Planes and Directions in Crystals
We will not go into too much details of this assuming that you would know about planes and
directions in a crystal
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2 A Brief Review of Bonding in Materials
Bonding in materials is a very important criterion and determines many of the physical properties of
the materials
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e
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Bonding in materials can be divided in two categories:
Primary bonding
Secondary bonding
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2
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Figure 1
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2
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1 Metallic bonding :
This kind of bonding is characterized by presence of a sea of electrons around atoms in metal
giving rise to flexible bonds, good malleability, high electrical and thermal conductivity
...
1
...
1
...
Examples of materials showing this bonding are mainly group IV elements and compounds
such as Si, C, Ge, and SiC and gases like methane
...
10 Schematic of covalent bonding
1
...
1
...
This type of bonding typically leads to high bond energies, high bond strength, high modulus,
brittle nature, generally low thermal and electrical conductivities making them excellent
insulators
...
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Figure 1
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Module 1: Structure of Ceramics
A Brief Review of Bonding in Materials
1
...
2 Secondary Bonding :
I t arises from the interaction between charge dipoles
...
2
...
1 Fluctuating Dipoles:
Observed in gases like hydrogen
...
12 Secondary bonding due to fluctuating dipoles
1
...
2
...
13 Secondary bonding due to permanent
dipoles
Examples are materials like polymers
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14 Secondary bonding in
polymers
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2
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Module 1: Structure of Ceramics
Packing of Atoms in Metals
1
...
First we will see how atoms pack in metals
...
Some metals are a little loosely packed in
the form of body-centered cubic structure
...
1
...
1 Simple Cubic Structure
Simplest structure crystallographically but in the entire periodic table only polonium (Po)
possesses this structure
...
Figure 1
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3
...
Contains 2 atoms per unit-cell
Figure 1
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Packing Factor: Volume of all atoms in one unit cell divided by Volume of
one unit-cell
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3
...
Figure 1
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stacking leads to the formation of cubic closed packed (CCP) or face centered
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The closed packed A, B, C
planes are (111) planes in the structure
...
stacking leads to hexagonal closed packed (HCP) structure
...
Figure 1
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74
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Module 1: Structure of Ceramics
Interstices in Structures
1
...
These interstices are very useful because there can contain smaller atoms which modify the
properties of materials tremendously, such as Carbon (C) in Iron (Fe) makes steel and makes iron
stronger
...
4
...
19 Interstices in a FCC
structure
So, by simple geometry, you can also estimate the size of the largest interstitial atom that would fit in these
interstices without distorting them
...
225* r
r tet = 0
...
4
...
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Figure 1
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Module 1: Structure of Ceramics
Structure of Covalent Ceramics
1
...
In most ionically
bonded materials, there is a significant level of covalency which is decreases as the difference
between the electronegativities of cations and anions increases
...
Covalent
bonding, as we saw in preceding sections, arises from the sharing of orbitals and as a result
materials with this type of bonding are characterized by significant hybridization of orbitals and
directionality of the bonds which play a crucial role in determining the crystal structure
...
In this section, we will understand the structures of a few covalently bonded materials with emphasis
on the Diamond structure
...
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Module 1: Structure of Ceramics
Structure of Covalent Ceramics
1
...
1 Diamond Cubic Structure
Typical and well known purely covalent bonded materials are carbon (Diamond), Si, Ge and
SiC
...
Thus, the unit cell of diamond contains a total of 8 atoms
...
Orbital hybridization of C atoms (sp 3 ) requires that the atoms are tetrahedrally co-ordinated
and thus the structure has high degree of directionality
...
What it means is that there are two FCC unit-cells of C intermingled into each other,
with origin of one at (0,0,0) and another at (¼,¼,¼)
...
Figure 1
...
Figure 1
...
You can also work out the packing factor of this unit which is lower than the typical FCC unit
cell
...
225*r while in
this structure, the size of the atom sitting at the site is much larger i
...
same size as the
base lattice atom
...
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Module 1: Structure of Ceramics
Structure of Covalent Ceramics
1
...
2 Structure of Graphite
Other forms of Carbon such as graphite and fullerene are also covalent bonded but the
structures are entirely different
...
However, the bonding between individual layers is Van der
Walls type of bonding
...
Figure 1
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Module 1: Structure of Ceramics
Ionically Bonded Ceramic Structures
1
...
Hence, when these ions are brought together, they form a very strong ionic
bond
...
...
As there is an involvement of two different types of ions to form
a crystal structure, there are certain rules or say guidelines which need to be followed to give
rise to a stable crystal structure
...
Based on these rules, typically ceramic structures are based on anions forming the base
lattice and cations occupying the interstices in them
...
As a result, we can categorize the structures of most ceramic materials into
following categories
Compounds based on cubic closed packing (CCP or FCC) of ions
Compounds based on hexagonal closed packing (HCP) of ions
Other structures with some deviations from above two
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Module 1: Structure of Ceramics
Ionically Bonded Ceramic Structures
1
...
1 Pauling’s Rules
Anions being the larger ions form the base lattice and lead to the formation of coordinated
polyhedrons around cations
...
e
...
Also, another point to note is that the ionic radius of each ion is also
dependent on its co-ordination
...
0-0
...
155-0
...
225-0
...
414-0
...
732-1
...
0 or above
FCC or HCP
The structure will be stable when it preserves the charge neutrality (Electrostatic valence
rule)
...
This is especially true for solids with smaller cations and cations
with bigger charges e
...
Ti4+ and Zr 4+
...
In a crystal containing different cations, those of high valence and small coordination number
tend not to share the polyhedron elements with one another such as in materials like BaTiO3
...
The
repeating units will tend to be identical because each atom in the structure is most stable in a
specific environment
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6
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Bond strength of an ion is
defined as the ratio of the valence of an ion to its co-ordination, i
...
In a stoichiometric and charge neutral solid, the bond strengths of cations must be equal to
those of anions
...
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7 Compounds based on FCC Packing of ions
In these structures, typically anions form the FCC lattice and cations fill tetrahedral or octahedral
sites in most cases, although in some cases, some other co-ordination may be preferred
...
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Module 1: Structure of Ceramics
Compounds based on FCC Packing of Ions
1
...
1 Rocksalt Structure
MX Type Compounds Based on NaCl or Rocksalt Structure
Anions (X) form the cation sub lattice with FCC structure
...
100% occupancy of sites according to the stoichiometry since there will be one octahedral site
per anion
...
414 - 0
...
Examples of ceramic materials with such structure as NaCl, MgO, NiO, FeO etc
...
23 Schematic of structure of Rocksalt structured
compounds
Lattice type: FCC and motif will be M at 0 0 0 and X at ½ 0 0
Four formula units per unit cell
...
102
0
...
564
MgO
0
...
140
0
...
118
0
...
842
NiO
0
...
140
0
...
078
0
...
557
MnO
0
...
140
0
...
119
0
...
85
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e
...
The octahedrons shares at the edges
...
Schematic representation of atomic arrangement on (110) plane of rock-salt structure
compounds shows the empty rows of tetrahedral voids along [001]-direction
...
24 (110) plane of a Rocksalt structured
compound
You can see that if both octahedral and tetrahedral voids were filled, this would bring cations
quite close to each other, leading to large electrostatic repulsion as like neighbours do not
make an energetically preferred configuration
...
The complete structure can be viewed as two FCC lattices, one of Na and another of Cl,
interpenetrating into each other
...
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Module 1: Structure of Ceramics
Compounds based on FCC Packing of Ions
1
...
2 Antifluorite (A 2 X) and Fluorite (AX 2 ) Structures
1
...
2
...
225-0
...
059 nm, r(Na + ) : 0
...
14 nm
Figure 1
...
In turn, anions
are cubic coordinated by cations (CN: 8)
The structure shows corner sharing of tetrahedra
...
7
...
2 Fluorite Structure (CaF 2 Structure)
Slightly bigger cations in comparison to other structures
Example:UO2 , ZrO 2 , CaF 2 , CeO2
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Figure 1
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Figure 1
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1 nm, Zr 4+ : 0
...
097 nm, O 2- : 0
...
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The structure as you can also see has a large void in the center of unit cell made by cations
...
g
...
For having some fun with the structure, we can also draw as projection of this material on
(110) plane
...
Figure 1
...
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Module 1: Structure of Ceramics
Compounds based on FCC Packing of Ions
1
...
3 Zinc Blende (MX) Structure
MX type compounds, also called as sphalerite structured compounds based on a mineral
name of sphalerite
...
Examples are ZnO, ZnS, BeO etc
...
You can also visualize diamond also having similar structure with both anion and cation
being of same type
...
In this structure anions form FCC lattice and cations occupy the tetrahedral interstices
...
Compounds with radius ratio
: 0
...
414 follow this structure with a few exceptions
where bonding favours a tetrahedral coordination despite unfavourable radius ratio, especially
covalently bonded compounds
...
06 nm,Be 2+ - 0
...
14 nm, S2-- 0
...
29 Zinc Blende or Sphalerite
structure
Coordination numbers: M - 4, X - 4
Lattice type: FCC
Motif: M – 0 0 0; X – ¼ ¼ ¼
4 formula units per unit cell
...
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7
...
B 2 O 3
FCC Packing of anions
Partial occupancy of both tetrahedral and octahedral sites i
...
1/8 th of tetrahedral and ½ of the
octahedral sites are occupied
...
Depending on how cations occupy different interstices, spinel structure can be Normal or
Inverse
...
7
...
1 Normal Spinel
Chemical formula: (A2+ )(B 3+ )O 4
Examples are many aluminates such as MgAl2 O 4 , FeAl2 O 4 , CoAl 2 O 4 and a few ferrites such
as ZnFe 2 O 4 and CdFe2 O 4
...
Apply bond strength rule to verify the stoichiometry
Cations: - A2+ - 2 /4 ; B3+ - 3 /6
Oxygen valence = ( 2 /4 x1)+ ( 3 /6 x3) = 2
Figure 1
...
7
...
2 Inverse Spinel B(AB)O 4
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Most ferrite follow this structure such as Fe 3 O 4 (or FeO
...
In this structure, ½ of the B3+ ions occupy the tetrahedral sites and remaining ½ B3+ and all
A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in the
previous line)
...
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Module 1: Structure of Ceramics
Other Cubic Structures
1
...
However the actual structure is rather different and many of these structures are merely
based on
the cubic packing of anions
...
ABO 3 structure,
1
...
1 Perovskite (ABO3 ) Structure
ABO 3 type compounds
Examples are many titanates like BaTiO3 , SrTiO3 , PbTiO3 etc
...
In ABO 3 structured compounds, A ion is twelve fold coordinated by oxygen (like a
dodecahedra) and B ion is octahedrally coordinated by oxygen ions
...
Bond strength check:
Cation: Ba: 2 /12 = 1 /6 and Ti: 4 /6 = 2 /3
Oxygen valence = 1 /6 x Coordination number by Ba + 2 /3 x coordination number by Ti
...
31 Perovskite structure
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32 Polyhedra model of perovskite
structure
Lattice type: Primitive Cubic (NOT FCC!)
Motif: A ion - 0 0 0, B ion – ½ ½ ½, O ion - ½ ½ 0, 0 ½ ½, ½ 0 ½
One Formula unit per unit cell
Coordination
B cation is surrounded by oxygen octahedra which share corners
...
An important parameters about perovskites is the their “Tolerance Factor (t)” which is defined
as
This is derived from the geometry of a cube in which the atoms are of such sizes that they
touch each other and hence, the face diagonal of the unit cell would be
times the unit-cell
length, as result t = 1 for a perfect cubic perovskite
...
Deviations from t = 1 signify the level of
lattice distortion
...
Perovskites can also have various combinations of ionic valence such as
e
...
A2+ B4+ O 4 , BaTiO3, PbTiO3 , CaTiO 3 , SrTiO3 etc
...
g
...
Mixed Perovskites:
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Pb(Mg 1/3 Nb2/3 )O 3
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...
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Module 1: Structure of Ceramics
Other Cubic Structures
1
...
2 ReO3 Structure
Stoichiometry : MX 3
Lattice type: Primitive cubic
Atomic Positions: M- 0 0 0; X - ½ 0 0, 0 ½ 0, 0 0 ½
Coordination Numbers
M
CN = 6
Octahedral coordination
X
CN = 2
Linear coordination
Can be visualized as perovskite ABO 3 structure with empty B-sites
Representative Oxides
ReO3 , UO 3 , WO 3
Used for gas sensing and electrochromic applications
Figure 1
...
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Module 1: Structure of Ceramics
Other Cubic Structures
1
...
3 CsCl Structure
MX type compounds, parent compound being CsCl
...
Radius ratio governs cubic co-ordination of both cations and anions
...
Motif: Anions (X): 0 0 0, Cations (M): ½ ½ ½
One formula unit per unit cell
...
34 (a) CsCl structure (b) Ball-stick
model
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9 Orthogonal Structures
Many superconductors follow the structures which are perovskite based i
...
, the structure
contains the perovskite structured units stacked along c-axis or [001]-direction in most cases
...
In some other compounds such as La-Sr-Cu-O, the structure is composed of alternating
perovskite and rocksalt structure units
...
Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly
...
9
...
1
...
Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu 3+ to
Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ions
and hence creates oxygen vacancies in the structure
...
e
...
Y3 Cu 3+3 O 9 →YBa 2 Cu 3+3 O 8 → YBa 2 Cu 2+2 Cu 3+O 7-x
Figure 1
...
(D
...
Smyth, PP
...
Research Update 1988, M
...
Yan, Ed
...
)
Here Cu coordination is of interest:
Cu2+ atoms have four-fold coordination along Cu-O chains
...
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36 Atomic coordination in
YBCO
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9
...
LaCuO 3
...
Substitution of La by Sr results in the compound La 2-x Srx CuO4 turning into a
superconductor with a Tc ~ 35K
...
37 (a) Origin of La2-x Sr x CuO4 structure, shown in
(b) as two perovskite and cells
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10 Structures based on HCP packing of ions
Similar to FCC packing of anions, many ceramic structures are also based on another type of
closed packing of anions i
...
hexagonal closed packed (HCP)
...
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Module 1: Structure of Ceramics
Structures based on HCP Packing of Ions
1
...
1 Wurtzite (MX) structured compounds
Compounds with M2+ X2- stoichiometry
Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO, SiC
...
Anions form an HCP lattice with ½ of the tetrahedral sites occupied by cations
...
Figure 1
...
e
...
Lattice type: Primitive, HCP
Motif: M: 0 0 0 and ; X: and
The filling of structure can be seen below
...
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Figure 1
...
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Module 1: Structure of Ceramics
Structures based on HCP Packing of Ions
1
...
2 Corundum (Al2 O3 ) Structured Compounds
M2 X3 type of compounds
- Alumina or Sapphire (Al2 O 3 ) is the parent compound
...
Coordination numbers: M: 6, X: 4
...
This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell and
start filling the interstices
...
40 Layered filling of Corundum
One unit-cell consists of six layers of oxygen ions
...
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41 View of {1010} plane of
Corundum
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10
...
Other compounds which follow this structure are CdTiO 3 , CoTiO 3 , CrRhO 3 , FeRhO3 , FeVO3 , LiNbO3 ,
MgGeO 3 , MgTiO3
...
Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer i
...
half of
the total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sites
and continue this order of substitution along the c-axis of the unit-cell
...
Coordination numbers: both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations
i
...
2 Fe and 2 Ti
...
42 Layered filling of Ilmenite
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A side view of the structure on {10-10} plane, as shown below, shows the columns of cations along the caxis with 2 /3 rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a
vacant site
...
43 Ilmenite structure on {10-10}
plane
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...
10
...
The parent compound LiNbO3 is ferroelectric in nature and hence, is technologically
important
...
Such materials are used in electro-optic devices
...
44 Atomic arrangement of a layer in
LiNbO3 structure
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...
45 Structure on {10-10} plane in LiNbO3
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...
10
...
It is formed by quasi-HCP packing of anions
...
The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice
...
Materials shows large and anisotropic refractive index and high birefringence
...
Figure 1
...
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Figure 1
...
48 Polyhedral model of Rutile
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Interstices can be completely or partially filled depending on the size of cations and
stoichiometry
...
Most ceramic compounds follow three types of common structures based on packing of
anions i
...
Structures based on FCC packing of anions
Structures based on HCP packing of anions
Primitive cubic or other structures
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These defects play an
important role in determining the properties of ceramic materials and in this context, the role of point
defects is extremely important
...
e
...
Subsequently, we will adopt a simple thermodynamic basis for calculating their
concentration in equilibrium and then will extend the Gibbs-Duhem relation for chemical systems to
the defects in ceramics considering them to be equivalent to the dilute solutions, an approximation
which is fairly valid
...
The Module contains:
Point Defects
Kroger-Vink Notation in a Metal Oxide, MO
Defect Reactions
Defect Structures in Stoichiometric Oxides
Defect Structures in Non-Stoichiometric Oxides
Oxygen Deficient Oxides
Dissolution of Foreign Cations in an Oxide
Concentration of Intrinsic Defects
Intrinsic and Extrinsic Defects
Units for Defect Concentration
Defect Equilibria
Defect Equilibria in Stoichiometric Oxides
Defect Equilibria in Non-Stoichiometric Oxides
Defect Structures involving Oxygern Vacancies and Interstitials
Defect Equilibrium Diagram
A Simple Procedure for Constructing at Brower's Diagram
Extent of Non-Stoichiometry
Comparative Behaviour of TiO 2 and MgO vis-à-vis Oxygen Pressure
Electronic Disorder
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...
K
...
Physical Ceramics: Principles for Ceramic Science and Engineering, Y
...
Chiang, D
...
Birnie, and W
...
Kingery, Wiley-VCH
Introduction to the Thermodynamics of Materials, David R
...
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1 Point Defects
Point defects are caused due to deviations from the perfect atomic arrangement or
stoichiometry
...
Usually, point defects in metals are electrically neutral whereas in ionic oxides, these are
electrically charged
...
Defects are present in most oxides and are easily understood
...
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2 Kroger–Vink notation in a metal oxide, (MO)
Kroger-vink notations are typically used to depict the atomic defects with charges
...
Regular Sites
M m: normal or regular occupied metal or
cation site
Oo : normal or regular occupied oxygen or
anion site
Point Defects (a • (dot) means a positive charge and a ' (prime) means a
negative charge)
Oxygen (anion) vacancy
VO
Metal (cation) vacancy
VM
Oxygen (anion) interstitial
Oi
Metal (cation) interstitial
Mi
Vacant interstitial site
Vi
Foreign cation
Mf
Foreign cation on regular
Mfm
metal site
Foreign
cation
interstitial site
on
Mfi
A normal cation or anion in
an oxide with zero effective MM x or O O x
charge
Charged oxygen vacancy:
VO • or VO ••
Charged metal vacancy
VM ' or VM ''
Charged metal or oxygen
Mi •• and O i ''
interstitial
Neutral cation
vacancies
and anion
electrons and holes
VM x or VO x
e' or h •
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3 Defect Reactions
Rules for writing defect reactions
Ratio of regular cation and anion sites is always constant
...
Electrical neutrality is to be always preserved
...
We will assume complete ionization of defects
...
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Module 2: Defect Chemistry and Defect Equilibria
Defect Structures in Stoichiometric Oxides
2
...
Various such combinations are possible such as
Cation and anion vacancies (VM and VO )
Vacancies and interstitial ion of same kind i
...
VO and O i or VM and Mi
Misplaced atoms interchanged (M O and O M ) - interchanged
Vacancies and misplaced atoms for the same kind of atom (VM + MO )
Interstitial and misplaced atoms i
...
,O i and MO
Interstitial atoms i
...
Mi and O i
Among all of these, the first two are most important as these are regularly seen in many important
oxides
...
2
...
1Schottky Disorder
This defect normally forms at the outer or inner surfaces or dislocations
...
Figure 2
...
Examples are rocksalt structured compounds such as NaCl, MgO, Corundum, Rutite etc
...
4
...
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Figure 2
...
It forms where cations are appreciably smaller then anions
...
The corresponding
defect reaction in that case would be
0
V0 ••+ O i ''
Examples of compounds showing this defect are AgBr type compounds such as AgBr, AgI
etc
...
4
...
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Module 2: Defect Chemistry and Defect Equilibria
Defect Structures in Non-Stoichiometric Oxides
2
...
Oxygen deficient (or excess metal)
ii
...
In such oxides, electrical neutrality is preserved via the formation of point defects and
electronic changes
...
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...
5
...
Formation occurs only at the surface
...
5
...
1 If oxygen vacancies are the dominating defects
Depicted by MO2-x (x is the extent of non-stoichiometry) and overall reaction as
MO2 MO2-x + x /2 O 2 ↑
Due to loss of oxygen, possible defect reactions would be
Electronic compensation leading to creation oxygen vacancies and of electrons
O0
VO ••+ ½ O 2 + 2e'
Ionic compensation leads to formation of oxygen vacancies and reduction of metal ions
on their sites
...
5
...
2 If metal interstitials are the dominating defects then,
Depicted as (M 1+y O 2 is the extent of non-stoichiometry)
Possible defect reactions are
Ionic compensation leading to the formation of metal interstitials and reduction of metal
ions on their sites
MM
Mi •••• + 4 M' M
OR
Electronic compensation leading to the formation of metal interstitials and free electrons
M
Mi •••• + 4e'
Creation of quasi-free electrons (extra charge is represented as M’)
Conduction occurs due to transport of electrons
Typically n-type conductors
...
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5
...
The following cases are possible:
2
...
2
...
Creation of holes
Conduction due to holes i
...
a p- type conductor
Examples of oxides showing this characteristics are MnO, NiO, CoO, FeO etc
...
5
...
2 If metal deficiency is dominating defect then
Oxides depicted as MO2+x
Oxygen interstitials can form due to following reaction
P-type conductor
Example can be an oxide like UO 2
...
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Module 2: Defect Chemistry and Defect Equilibria
Dissolution of Foreign Cations in an Oxide
2
...
6
...
The following scenarios are likely:
i
...
Alternatively for a metal deficient oxide MO, creates metal vacancies as
(ii)
iii
...
iv
...
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Module 2: Defect Chemistry
Concentration of Intrinsic Defects
2
...
e
...
1)
where ΔSC is the change in configurational entropy and is positive
...
r
...
n i
...
the concentration at which free energy is minimum
...
2)
where W is the number of ways in which defects can be arranged
...
e
...
3)
Assume that total number of lattice sites = N
Number of ways to arrange the vacancies, W v is
(2
...
5)
Total number of possible configurations
(2
...
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(2
...
8)
and total free energy change is
(2
...
10)
Figure 2
...
Hence, at equilibrium, we can safely write that
Now at equilibrium,
(2
...
This results in
(2
...
Hence Equation (2
...
13)
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14)
Similarly, for Schottky defects, you can work out that
(2
...
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Module 2: Defect Chemistry and Defect Equilibria
Intrinsic and Extrinsic Defects
2
...
8
...
2
...
2 Extrinsic behavior
Extrinsic defects are defects caused by impurities consisting of aliovalent cations
...
Only at very high temperatures, intrinsic behavior again dominates, and the
cross-over temperature depends upon the defect formation energy
...
8
...
So, at any given temperature NaCl will have
much larger defect concentration than MgO
...
Interestingly, while the highest achievable purity level in MgO is 1 ppm, in NaCl, it is 50
ppm
...
Thus the concentration of extrinsic defects is much greater than
intrinsic defect concentration in MgO
...
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9 Units for Defect Concentration
Defect concentration fraction, n/N , is nothing but the ratio of number of defects, n, relative to
number of occupied lattice sites N i
...
defect concentration fraction
...
Commonly used units for
concentration is #/cm 3 or cm -3
Typical defect concentration in ceramics ~ 1 ppm
...
Conversion of mole fraction to number per unit volume can be the following:
No
...
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Module 2: Defect Chemistry and Defect Equilibria
Defect Equilibria
2
...
10
...
This allows us to establish, for example, an equilibrium
diagram between defect concentration and pO 2 , helping us to identify various regions which may be
useful under practical conditions
...
So, if a chemical system consists of n 1 + n 2 + ---- +ni moles of constituents 1, 2, 3, ………
...
16)
Then, according to the Gibbs Duhem equation, at equilibrium
(2
...
18)
where ΔG o
...
e
...
At equilibrium, ΔG o = 0, , hence
(2
...
In addition, free energy can be expressed as
(2
...
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which leads to K = K0 exp (-ΔH 0 /RT), where K0 = ΔS0 and R is the gas constant
...
21)
This is an important outcome as it shows that we can treat the defects in a solid as solutes in a
solvent
...
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Module 2: Defect Chemistry and Defect Equilibria
Defect Equilibria in Stoichiometric Oxides
2
...
2
...
1 Schottky Defects
Defect reaction in an oxide MO is written as
Equilibrium constant for this reaction is
KS = [
] [V M '']
Here square brackets i
...
[ ] are used for concentration
...
22)
where ΔG S is the molar free energy of defect formation and is ΔH S - TΔSS , where ΔH S is the
enthalpy for defect formation and ΔSS is the entropy change which is mainly vibrational in nature
and can be assumed to be constant
...
23)
If Schottky defects dominate, then
[
]
(2
...
2
...
2 Frenkel defects
For an oxide MO
which leads to
(2
...
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At reasonably low defect concentrations when
=
[M i ••] and [V M ''] << MM and MM ˜1
Thus
[M i ••][VM '']=KF
(2
...
e
...
27)
I n a similar manner what we did above for Schottky defects, one can now write
[M i ••]= [V M '']
(2
...
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12 Defect Equilibria in Non-Stoichiometric Oxides
2
...
1 Oxygen Deficient Oxides
In the following sections, we take example of MO type oxygen deficient oxides with cases when
either oxygen vacancies may dominate or metal excess may dominate in the form of metal
interstitials or when both kinds of defects are simultaneously present
...
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12
...
1 Case I: When oxygen vacancies are dominant defects
Here, the defect reaction can be expressed as
The reaction constant will be
K=[
]
(2
...
From the above reaction, to maintain the electrical neutrality,
n e = 2 [V O ••]
...
30)
or
(2
...
32)
and
[
]=
(2
...
This makes sense because concentration of vacancies
will go down as we supply more oxygen to the material
...
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Figure 2
...
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Module 2: Defect Chemistry and Defect Equilibria
Defect Equilibria in Non-Stoichiometric Oxides
2
...
1
...
34)
Both [M M ] and [OO ] can be assumed to be ~1 if [M i ••] << [M M ] and [OO ]
...
35)
Thus
OR
ne
(2
...
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...
12
...
3 Case III: Simultaneous presence of oxygen vacancies and metal
interstitials
Such a scenario is often found in ceramic oxides like TiO2 , and Nb2 O 5
...
The
corresponding defect reaction is
OR
Assuming
, the defect equilibrium can be written as
[VO••] n e2 pO2 1/2 =K1
(2
...
38)
]
According to the electrical neutrality condition
ne
= 2[ V O••] + 2[ Mi ••]
(2
...
40)
And
(2
...
e
...
42)
As you can see, under such conditions, [M i ••] decreases more rapidly with increasing pO 2
...
••
••
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43)
and
[
]=
(2
...
Figure 2
...
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Module 2: Defect Chemistry and Defect Equilibria
Defect Equilibria in Non-Stoichiometric Oxides
2
...
2 Metal Deficient Oxides
Now we turn towards the case of MO type oxides with deficient of metal which can be reflected
either by metal vacancies or oxygen interstitials or presence of both
...
For MO oxide, assuming complete ionization of vacancies, we can write
whose equilibrium constant will be
(2
...
46)
According to the electrical neutrality condition
(2
...
One can do similar exercise for the cases when oxygen interstitial is the main defect and also when
there is mixed presence of metal vacancies and oxygen interstitials
...
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12
...
48)
If n e = n h ,
(2
...
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13 Defect Structures involving Oxygen Vacancies and
Interstitials
Depending upon the partial pressure of oxygen, an oxide may be oxygen deficient (or metal excess)
or metal deficient (or oxygen excess)
...
e
...
High pO 2 i
...
oxygen interstitials dominate
...
e
...
Assuming that both oxygen vacancies and oxygen interstitials are doubly charged (fully ionized), the
defect reactions can be written as follows:
At low pO 2
The defect reaction can be written as
+[
] + 2e'
The corresponding reaction constant, assuming [M M ] and [OO ] =1, would be
[
]
(2
...
51)
At intermediate pO 2
Stoichiometric defects are likely to prevail i
...
either via intrinsic ionization or Anti-Frenkel
defects
...
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and corresponding reaction constant is
(2
...
[
]
(2
...
54)
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13
...
These three
regions are regions of
Low pO 2 ,
Intermediate pO 2 , and
High pO 2
These regions depict oxygen deficit (or metal excess), stoichiometric composition and oxygen excess
(or metal deficiency) respectively
...
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13
...
1 Low pO 2 i
...
oxygen deficit
At large oxygen deficit, we can assume that
[
]
(2
...
50)
(2
...
53)
(2
...
52)
(2
...
56)-(2
...
55), we get the following condition
(2
...
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Module 2: Defect Chemistry and Defect Equilibria
Defect Structures involving Oxygern Vacancies and Interstitials
2
...
1
...
e
...
60)
In such a situation, from (2
...
61)
and from (2
...
62)
Now, from (2
...
63)
Now combining (2
...
63) and using (2
...
e
...
64)
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...
13
...
3 Stoichiometric Condition, i
...
, Intermediate pO 2
Case - I: Intrinsic ionization dominates i
...
The defect reaction is
The corresponding reaction constant is
OR
[
] and
(2
...
50)
(2
...
51)
(2
...
e
...
68)
Now, since
and [
] are independent of pO 2 , using (2
...
51) respectively, n e and
n h are given as
(2
...
70)
The above equations provide the limiting conditions of oxygen partial pressure separating three
regimes of oxygen pressures with variations of defect concentration vs pO 2 obtained
...
Such diagrams are
extremely important in defect chemistry to understand the dominating defects which govern the
physical processes
...
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14 Defect Equilibrium Diagram
2
...
1 Frenkel defects dominating at stoichiometric composition
The following diagram is obtained when Frenkel defect dominates i
...
the internal disorder of the
material dominates in the intermediate pressure range
...
e
...
Then, draw the electron and hole concentrations, n and p, in the low
and high pressure regions respectively since their relationship to Vo and O i is straightforward
...
This process yields the diagram as shown in the figure below
...
6
Concentration of ionic defects vs pO2 with
Oxygen Frenkel defects dominating at stoichiometric
composition
2
...
2 Intrinsic ionization dominating at stoichiometric composition
Using the proceedure similar to that explained in the previous slide except that in the central region
now n=p as intrinsic ionization dominates at the stoichiometric composition, we obtained the following
figure
...
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...
7
Concentration of ionic defects vs pO2 with
intrinsic ionization dominating at stoichiometric composition
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...
15 A Simple General Procedure for Constructing a Brouwer's
Diagram
1
...
This can be, to a large extent,
determined by crystal structure, solute concentration and electrical conductivity or diffusion
rates
...
e
...
e
...
2
...
g
...
Intrinsic defect formation mechanism
Oxidation or reduction (not both)
3
...
4
...
5
...
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16 Extent of Non-Stoichiometry
In highly stoichiomteric pure oxides such as MgO, Al 2 O 3 , ZrO 2 , the extent of oxidation or
reduction is very small
...
Changes in oxygen pressure have very little effect on the defect concentration
...
Oxides containing multivalent cations, such as transition elements, are much more prone to be
non-stoichiometric
...
e
...
Transition metal mono-oxide series Ni 1-x O, Co1-x O, Mn 1-x O and Fe 1-x O are the oxides in
which a fraction of the divalent cations is easily oxidized to the divalent state resulting in
cation deficiency, x
...
1% for
Mn 1-x O and ~0
...
FeO is seldom stoichiometric and it has a minimum nonstoichiometry of 0
...
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17 Example: Comparative Behavior of TiO 2 and MgO vis-à-vis
Oxygen Pressure
Magnesium oxide in intrinsic form, primarily contains Schottky defects
...
e
...
Further from electrical neutrality condition,
It should be noted that though actual concentrations can be very small but they still do vary with
temperature and oxygen pressure
...
e
...
21 atm
which is equivalent to a deficiency of x=0
...
04 ppm in Mg 1-x O
...
The
reduction reaction is
From the electrical neutrality condition
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...
cm-1
...
8 Tm , i
...
1690 K
In air
[
]=
which is equivalent to a deficiency (x) of 93 ppm
...
27% in TiO2-x and makes TiO2-x an n-type semiconductor
due to the resulting high electron concentration
...
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Module 2: Defect Chemistry and Defect Equilibria
Electronic Disorder
2
...
This occurs in materials having a forbidden energy gap between conduction and the valence band
and are categorized as semiconductors and insulators (see footnote**)
...
Disorder implies elevation of electrons into higher energy levels creating vacant states in lover
energy bands which are called as holes
...
5 eV e
...
Si has bandgap of ~1
...
5-3 eV
...
Figure 2
...
1 eV
NaCl
7
...
7 eV
MgO
7
...
4 eV
NiO
4
...
43 eV
FeO
2
...
25 eV
BaTiO3
2
...
8 eV
TiO2
3
...
44 eV
UO 2
5
...
2 eV
SiO2
8
...
1 eV
MgAl2 O 4
7
...
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** Basically, materials having a well defined band gap show conduction band, band of higher energy
and valence band, bands of lower energies with maximum of valence band and minimum of conduction
band separated by the forbidden energy gap i
...
Eg
...
At 0 K, all the states in the valence band are filled while the states in the conduction
band are empty
...
It is just that for basic physics reasons, carriers cannot reside in the
forbidden energy gap
...
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...
18
...
71)
(2
...
73)
Here me * and mh * are the effective mass of electron and holes respectively, k is the Boltzmann’s
constant, h is the Planck’s constant and T is the temperature
...
Electron and hole concentrations in the conduction and valence bands are equal i
...
(2
...
75)
where
...
Since,
the atomic density of solids is about 10 23 cm -3 , the density of states is about four order of
magnitude lower than that in semiconductors
...
75) is small and is actually zero if me * = mh * , then
Fermi level is quite close to the center of the band gap
...
(2
...
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Fermi level usually remains in the middle of the band gap but can shift up or down when materials is
doped
...
The above expressions have striking similarity to the concentration of lattice defects where
(2
...
The excitation of electrons across the band gap can be depicted by a chemical defect reaction as
follows
The equilibrium constant is
(2
...
79)
Here Ki is not unit-less unlike the reaction constant in the defect reactions because n e and n h have
the units of cm -3
...
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Module 2: Defect Chemistry and Defect Equilibria
Examples of Intrinsic Electronic and Ionic Defect Concentrations
2
...
19
...
7 eV
...
65 eV which decreases at a rate of 1 meV per K as MgO is
heated
...
1×10 31 kg
...
85 eV – (1570*1*10 3 ) eV = 6
...
Hence, n e = n h = 4
...
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...
19
...
These donor and acceptor atoms
basically create donor and acceptor energy levels very close to conduction and valence band
respectively, such that the difference with the band edges is approximately equal to kT at room
temperature
...
Obviously, defects with positive charge act as donors while those with negative charge act
as acceptors
...
9 Figure showing the positions of
impurity energy levels in the band diagram of
MgO
For example, oxygen vacancy can be viewed as donor according to the following reaction
The ionization energies are
These energy levels are situated with respect to the conduction and valence band edges in the band
gap
...
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Similarly, Cl acts as a donor while N acts as an acceptor
...
e
...
Y atom can replace either Ba or Ti due to its intermediate size
...
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Module 2: Defect Chemistry and Defect Equilibria
Examples of Intrinsic Electronic and Ionic Defect Concentrations
2
...
3 Electronic vs Ionic Compensation of Solutes
Here we will discuss which of the electronic or ionic compensation of solute incorporation in oxides is
favoured and what are the conditions determining this
...
This is due to the fact that an aliovalent impurity in
an ionic compound can be charge compensated by ionic defects (ionically compensated ) or by
electrons or holes (electronically compensated ) or by a combination of the two
...
We will take the example of Nb 2 O 5 doping in TiO2
...
e
...
Similarly, as the temperature reduces, oxidation is again favored
...
In any case, the electrical neutrality condition requires that
(4)
Similar effects are observed in care of titanates such as BaTiO3
...
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Module 2: Defect Chemistry and Defect Equilibria
Examples of Intrinsic Electronic and Ionic Defect Concentrations
2
...
4 Point Defects and Crystal Density in ZrO2 when doped with CaO
CaO doping in ZrO 2 allows stabilization of high temperature tetragonal or cubic polymorphs of ZrO 2
at room temperature as a metastable phase
...
When CaO in ZrO 2 dissolves substitutionally, it has to be compensated
by creation of a positive charge
...
Figure 2
...
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...
While
defects such as Schottky defects maintain the stoichiometry of the materials, most oxide ceramics
are prone to the non-stoichiometry
...
While defect concentration in the intermediate oxygen partial pressure (around
atmospheric conditions) are independent of the partial pressure of oxygen, the defects in nonstoichiotemetric oxides either at high or low pressures are strongly dependent on the partial pressure
of oxygen
...
Finally,
we looked at the electronic disorder and evaluated the conditions to compare the ionic and electronic
defect concentrations
...
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Module 3: Defects, Diffusion and Conduction in Ceramics
Introduction
In this module, we will discuss about migration of the defects which happens via an atomistic process
called as diffusion
...
As we
shall also see, the conductivity in ceramics is a sum of ionic and electronic conductivity and the ratio
of two determines the applicability of ceramic materials for applications
...
Subsequently we will
look into the conduction in glasses and look at some examples of fast ion conductors, material of
importance for a variety of applications
...
Finally, we will look at a few important applications for
conducting ceramic materials
...
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Physical Ceramics: Principles for Ceramic Science and Engineering, Y
...
Chiang, D
...
Birnie, and W
...
Kingery, Wiley-VCH
Principles of Electronic Ceramics, by L
...
Hench and J
...
West, Wiley
Electroceramics: Materials, Properties, Applications, by A
...
Moulson and J
...
Herbert, Wiley
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1 Diffusion
Diffusion causes changes in the microstructures to take place in processes such as sintering,
creep deformation, grain growth etc
...
Ionic conductors are used in variety of applications such as
chemical and gas sensors, solid electrolytes and fuel cell
...
Atomic diffusion rates and electrical conductivity are largely governed by defect types and
their concentration where concentration is a function of temperature, partial pressure of
oxygen or pO 2, and composition
...
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Module 3: Defects, Diffusion and Diffusion in Ceramics
Diffusion Kinetics
3
...
Thermodynamically speaking, this amounts to gradient in the chemical
potential which drives the migration of species from regions of higher chemical potential to lower
chemical potential so that system reaches a chemical equilibrium
...
These laws are briefly explained below
...
1,2
3
...
1 Fick's First Law of Diffusion
It states that atomic flux, under steady-state conditions, is proportional to the concentration gradient
...
1)
where
J is the diffusion flux with units moles/cm 2-s, and basically means the amount of material
passing through a unit area per unit time;
D is the proportionality constant, called as diffusion coefficient or diffusivity in cm 2 /s;
x is the position in cm; and
c is the concentration in cm 3
...
H
...
indicates that diffusion takes place from regions of higher
concentration to lower concentration i
...
down the concentration gradient
...
3
...
2 Fick's Second Law of Diffusion :
Strictly speaking it is not a law, but rather a derivation of the first law itself
...
It can be derived
from Fick's first law easily
...
(3
...
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where t is the time in seconds
...
1 Diffusion in Solids, Paul Shewmon, Wiley
2 Diffusion in Solids, Martin Glicksman, Wiley-
Interscience
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...
2
...
1 Schematic of the planes of
atoms with arrows showing the crossmovement of species
As shown in Figure (3
...
Flux from position (1) to (2) is written as
(3
...
of atoms at position (1) and G is the jump frequency i
...
4)
where n 2 is the number of atoms at (2) and G is the jump frequency in s -1
...
Now, the net flux, J, can be calculated as
(3
...
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(3
...
Figure 3
...
7)
Hence, flux can now be expressed as
where D = ½ λ 2 t with unit cm 2 /s in 1-D and can easily show to become D = 1/6 λ 2 t in a 3-D
cubic co-ordination scenario
...
9)
where γ is governed by the possible number of jumps at an instant and λ is the jump distance and
is governed by the atomic configuration and crystal structure
...
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Module 3: Defects, Diffusion and Conduction in Ceramics
Diffusion Kinetics
3
...
4 Temperature Dependence of Diffusivity
Now, equation (3
...
10)
where ν is the vibration frequency in s -1 , ΔG * is the activation energy of migration in J and k is
Boltzmann Constant (J/K)
...
11)
where ΔH * is the enthalpy of migration and ΔS* is the associated entropy change
...
10) in equation (3
...
Equation (3
...
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...
3 Examples of Diffusion in Ceramics
3
...
1 Diffusion in lightly doped NaCl
NaCl is often used a conducting electrolyte and it a good case for proving an example
...
13)
In addition, NaCl will also have certain intrinsic sodium and chlorine vacancy concentration (VNa ' and
VCl •) due to Schottky dissociation, depending on the temperature
...
14)
where ΔG Na * is the migration free energy for sodium vacancies and V Na ' is the sodium vacancy
concentration
...
However, the diffusivity dependence on temperature shows two regimes as shown
in figure 3
...
Figure 3
...
3
...
1 Low Temperature Regime:
Extrinsic region is dominant where vacancy concentration is constant as it is determined by the solution
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...
e
...
The diffusivity is given by
...
15)
In this regime, diffusivity exhibits a temperature dependence of
...
3
...
2 High Temperature Regime:
In this regime, the vacancy concentration is governed by intrinsic defect creation mechanism with
Schottky defect formation and hence, diffusivity exhibits a steeper slope with higher activation energy
which include not only the energy for defect migration but also for defect creation i
...
16)
At the point of crossover of two regimes, vacancy concentration due to dopants equals the thermally
created vacancies
...
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Module 3: Defects, Diffusion and Conduction in Ceramics
Mobility and Diffusivity
3
...
Absolute mobility
(3
...
Chemical mobility
(3
...
20)
For atomic transport, Einstein first pointed out that the most general driving force is the virtual force
that acts on a diffusing atom or species and is due to negative gradient of the chemical potential or
partial molar free energy
...
Absolute mobility, Bi is given by
(3
...
e
...
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(3
...
21), we have
(3
...
25)
where R is the gas constant
...
26)
OR
(3
...
27) into equation (3
...
28)
If we compare the above equation with Fick's first law, which is equation (3
...
29)
The above equation is called Nernst Einstein Equation
...
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Module 3: Defects, Diffusion and Conduction in Ceramics
Analogue to the Electrical Properties
3
...
30)
where f is potential, E is the electric field, Zi is the atomic number, e is the electronic charge and Zi e
is the total charge on the particle
...
v i , the velocity can be written as
(3
...
33)
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...
6 Conduction in Ceramics vis-à-vis metallic conductors: General
Information
While it is possible to apply Nernst-Einstein's relation to express the motions of both ions and
electrons to state the similarity between electrons and ions, in reality the motion of electrons
is quite different
...
The
extent of this drift velocity is governed by various scattering phenomenon such as scattering
from the lattice
...
34)
In such a condition, drift velocity increases in such a manner as expressed by Newton's law of
motion:
(3
...
Under steady-state conditions i
...
when carrier motion reaches a steady state,
(3
...
37)
where m* is the effective mass of the carrier
...
For details, please consult any book on the basic solid state physics as
mentioned in the Bibliography
...
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Module 3: Defects, Diffusion and Conduction in Ceramics
Ionic Conduction: Basic Facts
3
...
Conduction in these solids is often governed by concentration of impurities, dopants and point
defects
...
In ionic solids, proximity of the carriers to the ions results in the formation of a polarized
region i
...
, a polaron which may be longer or smaller than size of the unit-cell
...
Effective mass can be taken as equal to that of a free electron i
...
me * = mfree electron *
...
In a number of of ionic systems such as LiNbO3 , COFe2 O 4 , FeO etc
...
In such systems, motion of carriers is thermally activated i
...
via hopping mechanism
...
Mixed presence of Ni ions in +2 and +3
states leads to hopping type conduction of electrons between two states
...
4
...
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...
4 Example of change in resistivity
(ρ) of NiO upon doping with Li2 O
The movement of carrier between these differently charged states can happen by hopping
which is a thermally activated phenomenon
...
In contrast, in the band
model, carrier concentration is determined by temperature
...
33)
and is expressed as
(3
...
Typically carrier mobilities in strongly interacting ionic systems are of the order of ~0
...
Table 3
...
Table 3
...
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InSb
10 5
1700
AgCl
50
InAs
23,000
200
KBr ( 100 K
)
100
InP
3400
650
CdTe
600
GaP
150
120
GaAs
8000
AlN
--
10
SnO 2
160
FeO
SrTiO3
6
MnO
Fe 2 O 3
0
...
2
NiO
Fe 3 O 4
GaSb
--
2500-4000
~0
...
1
…
10 -4
10 -8
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8 Ionic and Electronic Conductivity
Electrical conductivity (si ) is defined as charge flux per unit electric field with units (O -1 cm -1 ) or S/m
...
39)
where J i is the flux of species i
...
e
...
29)
...
40)
Note:
As we saw earlier that temperature dependence of diffusivity shows two distinct regions: a low
temperature extrinsic region and a high temperature intrinsic region
...
While in
intrinsic regions, conductivity is governed only by defect migration as defect concentration is
independent of temperature, in the extrinsic region, defect concentration is temperature dependent
as well leading higher slope of log s vs 1/T plot consisting of energy for defect creation as well as
defect migration
...
8
...
41)
Fraction of total conductivity carried by each charged species is called as transference number, ti and
is expressed as
(3
...
43)
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...
2 lists the transference numbers for some conducting
oxide ceramics
...
0
10 -4
Na2 O
...
0
ZrO 2 +18%CeO 2
1500
--
0
...
48
ZrO 2 +50%CeO 2
--
0
...
85
Na2 O
...
SiO2 Glass
1 (Na+ )
-
-
0
...
9
15%(FeO
...
SiO 2
...
e
...
Typically conductivities of ionic conductors are in the range of 10 -5 to 100
S/m, depending on the temperature and are about 3 - 5 orders of magnitude lower than the
metallic conductors but about 10 - 15 orders of magnitude higher than the ceramic insulators
...
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...
9 Characteristics of Ionic Conduction
Long range migration of ionic charge carriers, the most mobile species, through the lattice
under application of an electric field for example migration of Na+ ions in soda-silicate
glasses
Dependent on the presence of vacant sites in neighbourhood of mobile defects/ions
...
When external field is absent, the thermal energy, kT, is required for counter migration of ions
and vacancies overcoming the migration energy Ea, as shown in Figure 3
...
Figure 3
...
4)
...
Ionic conductivity is promoted by
Small ionic size
Small charge i
...
less Coulomb interaction between ions
Favourable lattice geometry
Cations are usually smaller than anions and hence, they diffuse faster
...
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...
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Module 3: Defects, Diffusion and Conduction in Ceramics
Theory of Ionic Conduction
3
...
Here, we consider one dimensional movement of ions in a lattice as shown schematically in figure
below
...
Figure 3
...
44)
Where a is the accommodation coefficient related to the irreversibility of the jump and ν is the
vibrational frequency of the ions
...
Now when the field E is applied and if the ionic charge is z, then the force experience by the ion
would be zeE
...
F
...
d
...
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(3
...
46)
The above expressions shows that at any given temperature, the probability of the ion moving to
right is higher than to the left i
...
...
e
...
This velocity is given as
(3
...
48)
Now, under normal circumstances and normal field for ceramic conduction, the field are quite small,
a few V per cm and thus the field induced energy change is small as compared to the thermal
energy i
...
1/2 zeE
...
Using this we can modify above equation into
(3
...
d >> kT
(3
...
51)
where n is the ionic density per cc and total charge is ze
...
52)
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...
Thus
(3
...
E, we can write an expression for conductivity as
(3
...
The following table shows activation energies for diffusion (kcal/mole) for a few oxides
...
3
-
-
66
...
7
-
12
...
2
16
...
6
11
...
1
72
...
6
15
...
9
11
...
2
22
...
8
14
...
3
5
...
4
20
...
5
Good correspondence between activation energies for diffusion and dc conductivity shows that the
model we just derived is also correct
...
These also have typical activation energies between ~3
...
5
kCal/mole
...
5 (O-cm)-1 and have very low
activation energies, typically about 0
...
Compounds like Y2 O 3 , HfO 2 , SiO2 , Al 2 O 3 are far more insulating and have conductivities in the
range of 10 -5 to 10 -14 (O-cm)-1 in the temperature range of 400 K to 1000 K
...
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4 Principles of Electronic Ceramics, L
...
Hench and J
...
West, Wiley,
Table 4
...
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Module 3: Defects, Diffusion and Conduction in Ceramics
Conduction in Glasses
3
...
Conductivity increases as the temperature increased as barrier for migration for mobile
modifier ions is easily overcome by thermal activation
...
As the concentration of blocking ions increases, the conductivity decreases
...
11
...
Electrical conductivity is dependent upon the concentration of alkali ions (R: Na+ , Li + , K+ )
...
Typical electrical resistivity for soda-silicate glasses varies between10 -2 to 50 O-cm and
depends strongly upon the alkali content
...
In alkali earth compounds, highest conductivity is obtained by incorporating CaO
followed by MgO, SrO and BaO, however the differences are very small
...
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Module 3: Defects, Diffusion and Conduction in Ceramics
Fast Ion Conductors
3
...
Intrinsic FICs
Halides and Chalcogenides of silver and copper, e
...
- AgI is a silver ion (Ag + )
conductor
Alkali metal conductors such as non-stoichiometric aluminates e
...
ß-Al 2 O 3 or
Na2 O
...
Extrinsic FICs
Oxides with fluorite structure such as ZrO 2 are doped with aliovalent oxides creating a
large number of oxygen vacancies and are called as fast ion conductors such as
Y2 O 3
...
In these materials the enthalpy of migration, ΔH m , can be as low as ~
0
...
2
...
4: Examples of Fast Ion
Conducting Ceramics and Conducting
Ions
Material
Conducting Ions
Alkali ions
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Module 3: Defects, Diffusion and Conduction in Ceramics
Examples of Ionic Conduction
3
...
13
...
The carrier concentration can be expressed as (hint: write the defect reaction, refer to the Defect
Chemistry module)
In these materials, hole mobility is much larger than ionic mobility such that conductivity is essentially
electronic
...
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13
...
For working this out, we need to start with certain
parameters
...
4*10 11 cm 3
concentration of electronic defects ˜ 3
...
Diffusion coefficient of magnesium vacancies is estimated as
At 1600o C, one can calculate the mobility from the Nernst-Einstein Equation i
...
However, in reality, MgO is prone to having impurities such as Al 2 O 3 with Al concentrations of the order of
400 mole ppm or 2*1019 cm -3
...
Hence, Ionic conductivity is
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For example, in normal atmosphere i
...
air (pO2 = 0
...
Hole concentration can be
determined using the following defect reaction:
Here, from the mass conservation, we can see that n h = 2 VMg '' and hence the equilibrium constant is
and hence
Using the above equation, n h can be determined at 1600o C air
So, the total conductivity is
Hence, we can now work out the ionic transference number as
and
The system shows mixed electronic and ionic conduction even when vacancy concentration is
10 5 times higher than electronic ( 2*1019 cm -3 vs 5*1013 cm -3 )
...
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13
...
Cubic ZrO 2 is considered as a very good and fast ion conductor
...
Example: Take the case of Ca0
...
86, O 1
...
Bandgap of ZrO 2 is ~5
...
At high temperatures, the diffusion coefficient is given as
From the defect reaction
You can show that 14% molar concentration of CaO in ZrO 2 gives rise to an oxygen vacancy concentration of ~ 3
...
Now, ionic conductivity can be written as
Mobility can be derived by using Nernst-Einstein equation as the following
Hence, ionic conductivity is
Electronic conductivity is calculated as
At 1823 K, electron concentration, n e , using band model, is estimated to be 1
...
s
Hence, the electronic conductivity is
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s ionic increases only until about 12-13% of CaO doping and then decreases due to increased defect interactions
...
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Module 3: Defects, Diffusion and Conduction in Ceramics
Examples of Ionic Conduction
3
...
4 Conductivity in SrTiO3
Perovskite oxides SrTiO3 and BaTiO3 have two cation sub-lattices onto which aliovalent ions can
substitute
...
Typically, Ba/Ti or Sr/Ti is approximately very close to 1
...
To promote n-type conductivity in these materials, solutes such La 3+ , Nb5+ , and Ta 5+ are added to
BaTiO3 , giving rise to shallow donor levels in the bandgap
...
On the other hand, undoped materials often contain impurities such as Fe 3+ or Al 3+ substituting Ti4+
acting as acceptors
...
The above equation is ionically compensated reaction
...
e
...
If you are interested, you can refer to the work of Choi and Tuller, Journal of American
Ceramic Society, 71, 201, 1988 for discussion on defects and conduction in Ba-doped SrTiO3
under various conditions
...
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14 Electrochemical Potential
Usually in ionic systems, both the effects of gradients in chemical and electrical potential on
the mobilities are present
...
Total driving force for mass transport is defined as Electrochemical potential
...
55)
where Zi is the effective charge and F is Faraday's constant (=96500 Coulomb/mol)
...
e
...
56)
Flux, J, can be expressed as
OR
(3
...
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15 Nernst Equation and Application of Ionic Conductors
We have seen that the electrochemical potential is a combination of chemical potential and electrical
potential
...
Ionic transport is essential for the imposed oxygen pressure gradient to establish a gradient in the
oxygen concentration
...
Consider two sides separated by an electrolyte
...
e
...
58)
Electrical potential across the electrolyte is related to the chemical potential difference by
(3
...
60)
Thus,
OR
(3
...
For partially ionic conduction, equation (3
...
62)
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15
...
e
...
21 atm and the other side is at
pO 2 of 10 -9 - 10 -21 atm
...
25-0
...
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16 Examples of Ionic Conductors in Engineering Applications
Conducting ceramics are used in a variety of applications such as
SiC and MoSi2 as heating elements and electrodes
ZnO and SiC as varistors for circuit protection
YSZ, ß -Alumina as electrolytes in fuel cells and batteries
Materials like YSZ in gas sensing applications
While heating elements and varistors utilize the electronic conduction properties of the
ceramics, fuel cells and sensors typically utilize ionic conductivity characteristics
...
J
...
M
...
In the following sections, we will briefly discuss the major applications of conducting ceramics
viz
...
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16
...
Used to protect a circuit against sudden upsurges in the voltage or transients (Figure 3
...
The current-voltage characteristics are governed by the electrical
nature of grain and intergranular layers
...
8)
...
7 Schematic diagram showing use
of varistor in circuit protection
Figure 3
...
These IGLs contain defect states whose
role is to trap free electrons from the neighbouring n-type semiconducting zinc oxide grains
...
These boundaries between each
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9)
...
The respective barrier heights are f L and f R while the zero biased
barrier height is f 0
...
The whole mass of material resembles an electrical circuit network with back to back pair of
diodes, with each pair of diode in parallel with many others
...
When voltages are high enough to cause break down of the
diode junction, large currents flow due to thermionic emission as well as tunneling, as seen in
the above graph
...
9
Schematic of the electronic structure of ZnO
grain and intergranular layer (note the bias on the grains)
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16
...
Schematic of such a device is shown in Figure 3
...
g
...
This device can usually work with an efficiency of ~60% at 1000 ( o C) with an output power of
100 kWh
...
Anode material is typically porous Ni and cathode is a porous p-type semiconducting oxide
such as La 1-x Srx MnO 3+y
...
Figure 3
...
10
...
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Overall reactions that occur are:
It must be noted that high ionic conduction through the electrolyte is required to minimize the
resistive losses
...
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Module 3: Defects, Diffusion and Conduction in Ceramics
Examples of Ionic Conductors in Engineering Applications
3
...
3 Oxygen Sensors in Automobile Exhaust
Modern automobiles demands good fuel efficiencies and pollution control
...
One alternative is to use the catalytic converters, which however do not regulate the fuel
consumption
...
5, hydrocarbons and CO are decomposed efficiently but not
NO x , while below 14
...
When there are deviations in the air /fuel ratio from the stoichiometric value of 14
...
This differential provides a way to
use the activity of oxygen, i
...
to measure the combustion efficiency
...
The
difference between the pO 2 of the exhaust and air can be translated into an electrical signal
which is fed as an electrical input to the fuel injection system to optimize the air/fuel ratio
...
Second type is where one can measure the variation in the conductivity of the ceramic
vs pO 2
...
This also affects the electrical conductivity of the ceramic, which is generally
proportional to the number of current carriers, dependent upon T and pO 2
...
11 shows the schematic of an exhaust gas sensor using Y2 O 3 stabilized ZrO 2
which gives rise to sufficient voltages under a gradient of oxygen partial pressures
encountered in automobiles which can be used to regulate the fuel/air supply
...
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Figure 3
...
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Migration of ionic defects such as vacancies or interstitials is basically a diffusive process and is
governed by the law of diffusion
...
The diffusivity related to electrical
conductivity is shown by Nernst-Einstein relationship
...
In ionic systems, to utilize
the ionic conduction of materials, it is essential that ionic conductivity is a dominant contribution
towards the total conductivity which typically happens at either higher temperatures or at reasonably
high doping levels unless the electron concentration is extremely low
...
The presence of composition gradients of charged defects also leads to electrical potential
gradients and this when combined with the chemical potential gradients, gives rise to an expression
for electro chemical-potential, called Nernst equation
...
Ceramic conductors are used in
a variety of applications, for example, ZnO as varistors, doped ZrO 2 as solid electrolytes in fuel cells
and as oxygen sensors in the automobile exhaust
...
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Module 4: Dielectric Ceramics: Basic Principles
Introduction
Dielectrics are insulating or non-conducting ceramic materials and are used in many applications such
as capacitors, memories, sensors and actuators
...
First we will look the simple properties of dielectric materials
such as dipole moment, polarization, susceptibility, polarizability and polarization mechanisms
...
Subsequently, we will do detailed analysis of
dielectric properties for each of the polarization mechanisms under the influence of alternating field,
important from the point of understanding the behaviour of these materials in real conditions
...
The Module contains
Basic Properties: Dielectrics in DC Electric Field
Summary
Mechanisms of Polarization
Microscopic Approach
Determination of Local Field
Summary
Analytical Treatment of Polarizability
Summary
Effect of Alternating Field on the Behavior of a Dielectric Material
Summary
Frequency Dependence of Dielectric Properties: Resonance
Dipolar Relaxation i
...
Debye Relaxation in Polar Solids
Summary
Circuit Representation of a Dielectric and Impedance Analysis
Impedance Spectroscopy
Dielectric Breakdown
Basic mechanisms of breakdown
Summary
Suggested Reading:
Principles of Electronic Ceramics, by L
...
Hench and J
...
West, Wiley
Dielectrics and Waves, by Arthur R
...
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J
...
M
...
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Module 4: Dielectric Ceramics: Basic Principles
Basic Properties: Dielectrics in DC Electric Field
4
...
However,
there is a limited movement of charges leading to the formation of charge dipoles and the material,
in this state, is considered as polarized
...
The applied field can also align the dipoles that were already present in the material i
...
material
containing dipoles without application of the field
...
e
...
The net effect is called Polarization of the material
...
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Module 4: Dielectric Ceramics: Basic Principles
Basic Properties: Dielectrics in DC Electric Field
4
...
1 Electric Dipole
An electric dipole comprises of two equal and opposite point charges that are separated by a
distance d
...
1)
Figure 4
...
It has units of C
...
Total dipole moment per unit volume is defined as Polarization, P i
...
(4
...
m-2 i
...
charge per unit area
...
m
where N is the number of dipole per unit volume
...
In the subsequent sections, we will replace displacement 'd' by 'd' to avoid duplication by other
symbols
...
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Module 4: Dielectric Ceramics: Basic Principles
Basic Properties: Dielectrics in DC Electric Field
4
...
2 Polarization and Surface Charge
Figure 4
...
Assume that all the dipole moments are aligned in the same direction
...
While at the surface, there is a finite charge as shown by the small circle
...
The number of charges, n c , on the surface area A is equal to the number of dipoles contained within
a surface volume (V=A
...
Assuming homogeneous distribution of the dipoles, polarization can be written as
where the subscript ‘s’ implies the surface
...
3)
where ∑ s implies sum over the surface volume
...
4)
So the surface charge density s is
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5)
s is a scalar quantity if P
...
If polarization is not normal to the surface then
(4
...
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1
...
3 Parallel plate vacuum
capacitor
Figure 4
...
7)
where capacitance of the vacuum is given as C vac = (e 0 A)(d) where e o is the permittivity of free
space and is equal to 8
...
If one inserts a dielectric between plates, then capacitance gets modified as
(4
...
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...
9)
(4
...
Basically, e r signified some sort of interaction between the material and electromagnetic field
...
5 Q-V plot with effect of e r
Now the question is: Why does inserting a dielectric increases the capacitance of a capacitor?
We will see this in the following sections
...
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Module 4: Dielectric Ceramics: Basic Principles
Basic Properties: Dielectrics in DC Electric Field
4
...
4 Polarization Charges
In a parallel plate capacitor without any dielectric, the surface charge in vacuum s s is
(4
...
In the presence of a dielectric, the net charge density now becomes
(4
...
Hence,
(4
...
According to the electromagnetic theory, the surface charges on the plates can be defined as
dielectric displacement, D, such as
OR
OR
(4
...
E) and extra
charge resulting from polarization of the dielectric material, ρ
...
14)
OR
(4
...
16)
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...
Now the question arises: What is the reason for polarization? It can be said that it is basically due to
short range movement of masses i
...
electrons, or atoms or molecules under applied electric field
...
This also implies that the dielectric properties are also a function of the frequency
...
But for now, we will turn
our attention to the understanding of basics of mechanisms of polarization and qualitatively what it
means in terms of applied frequency
...
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Module 4: Dielectric Ceramics: Basic Principles
Summary
Summary
So far, we have learnt that the effect of an applied field on a dielectric material is to polarize it which
is quantified by parameters such as dielectric constant, e r ; polarization, P; and dielectric
susceptibility, χ
...
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2 Mechanisms of Polarization
Basically, there are four mechanisms of polarization:
Electronic or Atomic Polarization
This involves the separation of the centre of the electron cloud around an atom with respect
to the centre of its nucleus under the application of electric field (see (a) in figure below)
...
Here, external field leads to small displacement of
ions from their equilibrium positions and hence inducing a net dipole moment (see (b))
...
This usually happens at the grain boundaries or any other interface such as
electrode-material interface (see (e and f))
...
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Figure 4
...
While mathematical treatment of the first three mechanisms is rather straightforward, interface
polarization is not easy to quantify
...
This has a direct relation with the frequency of the applied field (see figure
below)
...
As a result, atomic polarization is the fastest and typically persists at
frequencies between ~10 13 -10 15 Hz
...
Interface or space charge polarization occurs at
frequencies below 10 Hz
...
Contribution from any
other mechanism will be on top of it
...
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...
7 Schematic figures between dielectric
constant vs frequency showing various
mechanisms
The following table shows the values of e r and n 2 for a variety of materials and the dominant
polarization processes in them:
Dominant
mechanisms
Material
er
n2
C
(Diamond)
~5
...
85
Ge
~16
16
...
9
2
...
77
Electronic,
Dipolar
Electronic
Ionic
and
So, you can see that while carbon and germanium being single elemental materials show
electronic polarization only and as a result their dielectric constants match well with the
values of n 2
...
We will discuss more
about these processes in the latter sections
...
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Module 4: Dielectric Ceramics: Basic Principles
Microscopic Approach
4
...
e
...
17)
This gives
(4
...
e
...
Further, polarizability of an ion or atom of type, i, in a dipole is defined as
(4
...
However, magnitude of local field can be modified quite significantly
the polarization of surrounding medium
...
20)
Note that equation (4
...
As a result, in general, susceptibility is
(4
...
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4 Determination of Local Field
The local field, Eloc , experienced by an atom or dipole or molecule usually differs from the applied
field, Eex owing to the polarization of the surrounding medium around a dipole or molecule
...
Although this local field which is nothing but the sum of applied field and some other fields and can,
in principle, be solved using Poison’s equation, coupling charge density and potential at each
location, the method is far from being simple
...
The understanding of this field can be obtained from the figure shown below which shows a sphere
of dielectric material, say containing about 10 atoms, removed
...
Figure 4
...
EL , Lorentz field at the center of the spherical cavity due to charges on the surface of the
cavity
( Note the difference between the two: one is talking of a sphere while it is composed of material and
another is talking of a material surrounding the spherical cavity from which material has been
removed
...
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d
Hence, Eloc is written as
(4
...
23)
Lorentz showed that for isotropic crystals,E center = 0
EL can be worked out for a cavity as
(4
...
25)
By substituting P = e 0 (e r - 1)E in (4
...
26)
Now combining equations (4
...
26), we get
(
+ 2)
(4
...
15) and (4
...
28)
The above equation is called Clausius-Mossotti relationship and is valid only for linear dielectrics
...
e
...
As a result, if one knows the dielectric constant by means to measurement, this form can be used to
calculate the polarizabilities of a variety of materials to quite an accurate estimate
...
e
...
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a ionic, a dipolar and a interfacial
...
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Module 4: Dielectric Ceramics: Basic Principles
Determination of Local Field
4
...
1 Analysis of the Lorentz Field (newly added)
Here, we are interested in calculating the field from the free ends of dipoles i
...
Lorentz field El, lined
along the cavity wall in the direction of applied field, as shown below
...
9 Schematic of field components for a
spherical cavity
Now, since each element dA contributes to the field, according to Coulomb’s law, the radial field
intensity is
Each dA’s angular position is between θ and θ+dθ and for each dA element, there is another dA
element on the other side of the sphere which produces same but opposite horizontal field
component
...
Now we can also rewrite
dA as
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Module 4: Dielectric Ceramics: Basic Principles
Summary
Summary
Here, we saw that polarization is caused by the movement of electrons or ions or molecules from
their equilibrium positions and is frequency dependent
...
Basically, there are four polarization mechanisms:
electronic, ionic, dipolar or orientation and interface and each of these mechanisms is characterized
by different polarizability
...
We also learnt that the local field inside a dielectric is not the same as the applied electric field
...
This leads to
derivation of Clausius-Mossotti equation which relates the microscopic dielectric properties to the
macroscopic dielectric properties
...
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...
5 Analytical Treatment of Polarizability
Here we discuss the simple analytical solutions for determining the polarizability and polarization for
each of the above polarization mechanisms
...
4
...
1 Electronic Polarization
To achieve this, let us first imagine an atom as a perfect sphere, having +Ze charge at the center of
the nucleus and an equivalent –Ze charge of electrons around it
...
Figure 4
...
E
(4
...
However, d has to be finite values because this force is balanced by Coulomb force of attraction
between opposite charges which can be calculated by simple electrostatistics and is given as
(4
...
e
...
31)
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...
32)
If dipole density was N, the polarization is given as
(4
...
15), we can get electronic susceptibility, χe , and electronic polarizability, a e , as
(4
...
This also shows the linear relationship between polarization and electric field for electronic
polarization mechanism
...
e
...
e
...
The equation (4
...
The values for atomic polarizability of some atoms are given below:
Halogens
Alkali
F
1
...
03
Cl
3
Na
0
...
5
K
0
...
7
4
...
1
...
e
...
What it says is that the electronic polarization effect is generally very weak and contributes very
little to the overall polarization
...
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Si which are covalently bonded and orbital do not have spherical symmetry and as a result the
dielectric constants can be higher (for example Si has about e r ~ 12)
...
Values for some materials where electronic polarization is the principle mechanism of polarization
are given below (source: "Solid State Physics” by N
...
Ashcroft and N
...
Mermin):
Carbon (diamond)
5
...
0
GaP
8
...
0
SiC
6
...
9
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...
5
...
Remember, in these
solids, no dipole rotation is allowed
...
Figure 4
...
Now we calculate the magnitude of ‘d’
...
35)
However, there is a force, F2 , in the other direction trying to restore the equilibrium between the
ions which is expressed as
(4
...
The spring constant can be expressed in terms of elastic modulus, Y, and can be expressed as
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...
37)
At equilibrium, F1 = F2 and by combining equations (4
...
37), we get an estimate of d i
...
(4
...
39)
where a i is ionic polarizability
...
40)
where N is the dipole density per unit volume
...
If this is
not the case, one needs to take the component of dipole moment in the direction of the field before
adding them together
...
For example, in
calcium fluoride, CaF 2 , a material used for making lenses for lithography machine, the dielectric
constant is approximately equal to n 2
...
1 mm
...
W
...
D
...
6
ZnS
5
...
8
CdSe
7
...
0
CdS
5
...
0
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Module 4: Dielectric Ceramics: Basic Principles
Analytical Treatment of Polarizability
4
...
3 Dipolar or Orientation Polarization
In case of dipolar polarization, we have materials where the dipoles are present independent of
each other, i
...
they don’t interact and they can be rotated freely by an applied field unlike in case
of ionic polarization
...
Due to the ability of molecules to move around randomly, liquids like water have very limited dipolar
polarization contribution despite having a permanent dipole moment for each molecule
...
Arrows depict the direction of dipole moment associated with each molecule
...
12 Dipoles in a polar
material
It is quite obvious by looking at the figure that in the absence of an applied field, all the moments
are randomly distributed (huge number for even a small amount of water, say 10 ml) and net dipole
moment would be zero
...
For instance, water does have a pretty large dielectric constant of ~80 which means that there is
obviously some orientation along the field
...
In reality, the tendency of dipoles to orient along into the field direction will be counteracted by
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...
Moreover, the dipole containing molecules move around at all times due to thermal energy and
the associated entropy i
...
disorder
...
The Later prevails in liquids more than in solids
...
As a result of intermolecular collisions and thermally induced random movement, the dipoles do not
completely align themselves along the field unless the field is extremely high
...
Figure 4
...
Thermodynamically speaking, we will minimize the free energy, G which is equal to H – TS where
H is the internal energy and S is the entropy
...
Figure 4
...
e
...
U is minimum when the dipole is completely aligned with the field i
...
θ = 0 and maximum if θ =
180o
...
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The energy U(θ) of a dipole with dipole moment, µ under an applied field, E, can be written as
(4
...
The total internal energy of
the whole material would be the sum of internal energies of all these cones
...
15 Schematic
representation of dipoles around the
electric field
Now we need to calculate the number of dipoles at an angle θ, N(θ) which can then be multiplied
with the energy for that θ and then be integrated between θ ranging from 0 to 180° to give us the
total internal energy
...
This is where Boltzman statistics (from classical thermodynamics) rescues us which allows us to
work out the minimum of the free energy using a distribution function
...
r
...
applied field
...
Now, at a given temperature, T, the number of dipoles with energy U can expressed as
(4
...
Equation (4
...
Now, we should calculate the component of the dipole moment parallel to the applied field using the
solid angle dO for a unit sphere in the segment between θ and θ + dθ
...
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Figure 4
...
d θ
...
e
...
43)
The average dipole moment, <µ E >, is calculated by adding up the contributions from all the solid
angles i
...
(4
...
45)
which together with (4
...
46)
To solve this equation, we make substitutions as ß = (µE/kB T) and x = cos θ in the above equation
which yields
(4
...
48)
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...
49)
The function cot h(x) is nothing but (e x + e -x ) / (e x - e -x )
...
Langevin function is plotted as shown below
...
17 Schematic plot of Langevin
function
As we can see, for large values of ß (i
...
large electric fields or very low temperatures, close to 0
K),
(4
...
e
...
51)
In general, ß is always much smaller than one i
...
ß << 1
...
52)
Where a d is dipolar polarizability
...
53)
These equations hold pretty well for small values of µ and E and/or large enough T which we can
presume as normal conditions
...
e
...
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the dipoles would be parallel to the applied field and then the average dipole moment would be
equal to the theoretical dipole moment
...
52) also shows the temperature dependence of dipolar polarizability as µ = a, unlike in
the case of electronic and ionic polarization
...
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Module 4: Dielectric Ceramics: Basic Principles
Analytical Treatment of Polarizability
4
...
3
...
g
...
Using the above equation, we can calculate ß = 0
...
However, at T = 30 K, ß = 2
...
But, at this temperature it is likely to have liquid into solid state and what we see now is only the
effect of ionic polarization and no longer at orientation polarization!
Similar analysis is applied to the study of magnetic dipoles (a misnomer) in response to a magnetic
field
...
There are, however, some solids like ferroelectrics where dipoles do
interact and can rotate to some extent giving rise to extraordinary effects as we will study later
...
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Module 4: Dielectric Ceramics: Basic Principles
Summary
Summary
Here, we calculated the electronic and ionic polarizability and the polarizations of the dielectric using
electronic charge dipoles and ionic lattice having charges coupled as springs
...
On the other hand,
ionic polarizability is inversely proportional to the elastic modulus which is related to the bond
strength
...
Finally we
calculated the dipolar polarization using random dipole model and Boltzman statistics
...
Next, we will investigate the behavior of dielectric under alternating fields
...
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Module 4: Dielectric Ceramics: Basic Principles
Effect of Alternating Field on the Behavior of a Dielectric Material
4
...
4
...
1 Behavior of an Ideal Dielectric
While most of the above discussion has been for d
...
or static fields, in most practical applications,
dielectrics are used under alternating fields
...
Let us apply a sinusoidal field
(4
...
55)
(4
...
This current
voltage relationship can also be understood from a phasor diagram as shown below
...
18 Phasor diagram for a perfect
dielectric
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...
6
...
57)
where G is the time period is given as
...
e
...
58)
What it means is that during the first cycle, the capacitor charges completely and during the other
cycle it completely discharges without any loss of charge
...
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...
6
...
There are two sources of this loss current:
Long range migration of charges due to ohmic conduction and is frequency independent i
...
d
...
in nature, and
Dissipation of energy due to dipole rotation or oscillation as there always is certain inertial to
their movement due their mass
...
e
...
Since, both of these current are in phase with the applied field, the loss current, Il , can be written as
(4
...
Hence, the total current, Il , is the sum of charging and loss current i
...
60)
Hence, the current in a real dielectric is a complex quantity that leads voltage by angle 90 o - d where
d is defined as the loss angle or dissipation factor
...
Naturally, d
will be zero if there was no loss current present
...
19 Real and Imaginary part of I
and V
When the field applied is static or d
...
i
...
ω = 0 then
where G dc = 1/R and R is the ohmic resistance
...
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An alternative way to express the real dielectrics passing both charging and loss current is to use
complex permittivity
...
61)
and
(4
...
63)
Using equation (4
...
64)
i
...
(4
...
66)
The first term in (4
...
Comparing equation (4
...
60) yields overall conductance, G, as
(4
...
68)
If Gdc << ωk'' C 0 then
(4
...
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...
8
2
BaTiO 3
500
150
Nylon
3
...
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Module 4: Dielectric Ceramics: Basic Principles
Effect of Alternating Field on the Behavior of a Dielectric Material
4
...
4 Power Dissipation in a Real Dielectric
Hence, a
...
conductivity can be expressed as
(4
...
71)
Now, the time average power loss can be expressed as
(4
...
73)
OR
(4
...
d, the dissipated power density in a dielectric would
be
One can now see that for static or d
...
fields, ω = 0, i
...
(4
...
76)
which is the standard equation for power dissipation in a material under dc fields
...
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Module 4: Dielectric Ceramics: Basic Principles
Summary
Summary
We have investigated the behavior of dielectric materials in alternating electric fields
...
On the
other hand, in real dielectrics, the current leads voltage by an angle 90° - d where d is the loss
angle
...
Next we will look into the frequency dependence of the dielectric properties
...
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Module 4: Dielectric Ceramics: Basic Principles
Frequency Dependence of Dielectric Properties: Resonance
4
...
This also means that if the frequency of the field is very high, say above 10 15 Hz, no
dipole system will be able to follow the field oscillations and in such a situation, no mechanism will
contribute and hence e r ~ 1
...
In case of electronic and ionic polarization, the charge dipoles can be considered behaving as
mechanical oscillators where charges are connected with linear springs whose restoring force
balances the force induced via the applied electric field
...
In case of orientation polarization, there is no direct mechanical restoring force
...
In other words, if a driving force is present, there is an equilibrium state with an (average) net dipole
moment
...
This process does not show any resonance phenomena and is characterized by its
relaxation time instead of a resonance frequency
...
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Module 4: Dielectric Ceramics: Basic Principles
Frequency Dependence of Dielectric Properties: Resonance
4
...
1 Description of Ionic and Electronic Phenomenon
If we assume charge dipoles behaving like a linear harmonic oscillator, then they also follow the its
equation of motion
...
77)
where mi * is the mass of the particle (i
...
charges or ions containing charges), i, x is the
displacement from the equilibrium, γ i is the friction coefficient on the species and ω 0,i is the natural
frequency of the particle
...
The field will actually be local field than just the applied field but for the sake of simplicity, we can
assume the system to behave like a gas consisting of N atoms per unit volume for which q i = -Zi e
...
c
...
If friction is zero, there will be no damping of oscillations
...
Case II :
When applied field is sinusoidally varying field i
...
(4
...
79)
Ignoring transient terms, the solution of equation (4
...
80)
The induced dipole moment per particle is
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...
81)
(4
...
If the friction term is neglected, then
frequency dependent polarizability is
(4
...
In case of ionic polarization, one takes the reduced mass which is given as {M + M- / (M + +
M-)} where M+ and M- are cation and anion masses respectively
...
The polarization can then be written as
(4
...
15) and (4
...
85)
and
(4
...
The subscript 8 signifies the susceptibility and dielectric constants at frequencies below the
resonance frequencies
...
Now, we can separate out the real and imaginary parts of the dielectric constants as following
(4
...
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(4
...
Figure 4
...
34) and (4
...
All now you need
to do is to balance the force inducing the moment and the restoring force to find out the spring
constant
...
This makes sense because for ionic polarization, the masses of ions are
several thousand times higher than electrons and 10 13 Hz is very close to the natural frequency of
lattice vibrations!
The shape of the plot should also help you understand the far right portion of the dielectric constant
versus frequency plot shown earlier when we discussed mechanisms of polarization
...
You must remember that the above discussion is valid for gases with non-interacting dipoles for we
assumed that local field was equal to the applied field
...
e
...
The changed resonance frequency is
...
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...
7
...
1 Electronic polarization vis-à-vis frequency
If ω<< ω 0,e the charges oscillate in phase with the applied field and contribute to dielectric constant
...
e
...
The charges are 90° out of phase with the applied field and do not contribute
to e r'
...
e
...
If ω 0,e < ω, then in this region field changes its direction too fast compared to the rate at which
electrons can move
...
e
...
4
...
1
...
The total dielectric constant is now e r,e+i' i
...
both due to ionic and electronic contributions
...
e
...
The charges are 90° out of phase with applied field and do not contribute to the
dielectric constant and we witness maximum loss i
...
high e r''
...
Hence, ions don’t respond and no ionic polarization results and e r' is e r,e' , i
...
the system
behaves as if there was no ionic polarization mechanism present
...
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Module 4: Dielectric Ceramics: Basic Principles
Frequency Dependence of Dielectric Properties: Resonance
4
...
2 Microscopic Examination of Polarizability
From the equation (4
...
89)
Here, we have replaced the subscript i with e for electrons
...
dx) and the force inducing the displacement
...
90)
which is nothing but equation (4
...
A similar analysis can be carried out for ionic polarization as well which is left to you as an exercise
...
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Module 4: Dielectric Ceramics: Basic Principles
Dipolar Relaxation i
...
Debye Relaxation in Polar Solids
4
...
e
...
In such cases, the structure of the solid plays an important role because movement of
charges may have to occur over many atomic distances and can be classified as diffusional
in nature
...
As a result, it may take considerable amount of time in the distribution of charges
...
21 Potential energy distribution of
ionic sites in a glass
For instance, in an ionic solid, application of a field leads to first almost instantaneous
development of ionic and electronic polarization followed by slow development of dipolar
polarization, Pd to a saturation polarization, Ps
...
22 Dependence of dipolar polarization on
time
We can approximately express the rate of change of polarization as
(4
...
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where 1/t is the proportionality constant
...
This equation can be derived using a simple bi-stable model in a similar fashion to
that used in Module 3 (section 3
...
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...
e
...
8
...
23 Schematic of a polar molecule going one
state to another and resulting energy well diagram
In a solid when field is applied to a polar material, the ions hop from ionic position to another as
shown above in figure 4
...
g
...
We consider a bistable dipole model
...
At any temperature above
0K, there is random oscillation of cation between these sites
...
92)
Upon application of field, the wells tilt in the direction of applied field, resulting in unequal probability
in two directions resulting a net flow of dipoles
(4
...
Under the application of an ac field, the change in the number of dipoles at site 1 = outflow to site 2
– inflow to site 1 i
...
(4
...
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(4
...
96)
Now, polarization P can be expressed as the product of the net ions moved and the dipole moment
i
...
(4
...
So, the above equation can be written as in the form of "Dipolar Polarization" by substituting P with
Pd (t) and µ 2 /kT by dipolar polarizability leading to
OR
(4
...
5 of the Chapter 5 of “Principles of Electronic
Ceramics” by L
...
Hench and J
...
West
...
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Module 4: Dielectric Ceramics: Basic Principles
Dipolar Relaxation i
...
Debye Relaxation in Polar Solids
4
...
2 Solution of relaxation equations
The initial and final conditions are
At t = 0, Pd = 0 and at some reasonably large t, P=Ps
Integrating within these limits yield
(4
...
Application of alternating fields to this polarization is not as simple as in the previous analysis
This is because the saturation polarization, Ps , is dependent on the instantaneous value of field and
thus will be time-dependent and the local field is a function of both position and time
...
100)
Here, we can define e rs ' as static dielectric constant or dielectric constant at very low frequencies
and e r8' is dielectric constant at high frequencies covering electronic and ionic polarization
...
Now, substitute E* and Ps into (4
...
101)
The integration of this equation yields us
(4
...
c
...
c
...
Since, electronic and ionic polarization are approximately frequency independent in this regime of
frequencies, we can write
(4
...
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Now ignoring the transient time dependent term of (4
...
96) into (4
...
104)
We also know that e r* = e r' - ie r'', hence, now we can separate the real and imaginary parts as
OR
(4
...
105b)
and the loss tangent is
(4
...
The graphical representation of these is shown below:
Figure 4
...
e
...
The equation (4
...
e
...
As the applied field frequency
approaches value ω = 1/t, e r' passes an inflection and then drops off to e r8' which is dependent
only on (Pi +P e )
...
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as depicted by the peak in e r'' magnitude of e r'' peak being (e rs '' - e r8')/2 depending upon the
number of oscillating charges and distance of motion
...
Now since, we have understood that polarization develops by temperature dependent diffusional
processes which also give rise to the d
...
conductivity, temperature dependence of the relaxation
time, t can be expressed as
(4
...
Combining Debye equations with (4
...
The plots below show this trend for Li 2 O
...
Figure 4
...
2SiO 2 glass due to
increasing temperature (Reproduced from Principle of
Electronic Ceramics, L
...
Hench & J
...
West, P206)
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...
26 Temperature dependence of the frequency
maxima vs time for Li2 O
...
L
...
K
...
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Module 4: Dielectric Ceramics: Basic Principles
Dipolar Relaxation i
...
Debye Relaxation in Polar Solids
4
...
3 Complete Picture of Frequency Dependence of the Dielectric
Constant
So, now we can plot the contributions to the dielectric constant from all the mechanisms
...
Figure 4
...
Although the real plots may look quite different (look for dielectric constant data for electronic
ceramics in journal publications), you can expect a correlation between the real and imaginary part
of the curve i
...
we can still clearly see the absorption peak
...
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...
For electronic and
ionic polarizations, we described charges as linear harmonic oscillators and solved the equation of
motion in an alternating field
...
These mechanisms contribute
only when the applied frequency is lower than this characteristic frequency
...
This maxima is also a
function of temperature in many dielectric ceramics
...
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...
9 Circuit Representation of a Dielectric and Impedance Analysis
Data analysis after dielectric characterization often requires modeling of dielectrics which is helped
by their representation as equivalent electrical circuits
...
Figure 4
...
108)
where
and
(4
...
110)
Now if we consider time constant for the segment R 2 C 2 as t2 = R 2 C 2 , then admittance can be
written as
(4
...
112)
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...
113)
which have the same form as Debye equations with C 2 = (e s - e 8 ) C 0 and C 1 = e 8
...
S
...
H
...
By this technique, they were able to identify and relate the observed relaxation effect with the
atomic and microstructural features of the materials
...
114)
where a is a parameter which describes the distribution of relaxation times in the material
...
e
...
e
...
104-4
...
As a result, if we plot e r'' (ω) vs e r' (ω)
complex plane, we get a semi-circle
...
e
...
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Module 4: Dielectric Ceramics: Basic Principles
Impedance Spectroscopy
4
...
Such technique of characterizing dielectrics is called Impedance
Spectroscopy
...
The maxima of the semi-circle occurs at
(4
...
105a) and (4
...
Figure 4
...
xThis relation can easily be obtained just by using the
Debye’s equation (4
...
Basically any point on the surface of the semi-circle of diameter (ers ' - e r8' ) in this complex plane
can be defined by these vectors
...
However, if there is the presence of a tail on the low frequency side, as shown by dashed line i
...
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...
In case a material has more than one contribution to the impedance, which is often the case with
polycrystalline ceramics where grain, grain boundaries and electrode-ceramic interface have distinct
contributions, then one can witness more than one semi-circle, often overlapping each other
...
One of the ways to model such a behavior
may be using three series-parallel RC elements circuit as shown below
...
30 Cole-Cole Plot and possible equivalent
circuit for a solid with multiple polarization
mechanisms
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11 Dielectric Breakdown
Every material is bound to fail or breakdown under certain conditions
...
Technically speaking, dielectric breakdown occurs when the electron density in the conduction band
becomes very high during the application of an electric field such that conductivity increases rapidly
resulting in a permanent damage to material
...
The most critical parameter is the field strength E in the dielectric
...
The (DC) current vs
...
31 Dielectric breakdown I-V
Plot
After reaching a critical field or breakdown field, Ec, a sudden increase in the current may, within a
few seconds or even quicker, completely destroy the dielectric resulting in something like a ‘burnt’
material
...
While electric field plays an important role, dielectric may also break in a gradual time dependent
manner and in such cases we would rather call it as ‘failure’
...
In such cases,
normally the breakdown field also decreases with time
...
You
can do this measurement rather easily by letting a small current pass through the samples and then
monitor the voltage needed as a function of time
...
A typical voltage-time curve may then look like this:
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...
32 Time dependent failure for a
dielectric
The values of breakdown fields for some materials are given below:
Material
Critical
Field
Strength
[kV/cm]
Oil
200
Glass, ceramics
200-400
Mica
200-700
Oiled paper
1800
Polymers
50-900
Al 2 O 3 film (100 nm)
16,000
Al 2 O 3 ceramic
200-300
BaTiO3 (bulk single crystal)
300
BaTiO3 (Polycrystalline ceramic)
120
SiO2 (in Integrated circuits)
> 10,000
Example:
For example, in thin film memory devices, SiO2 is used as a gate dielectric and has a thickness of a
few nanometers say 5 nm
...
This explains the importace of material form on the breakdown field
...
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Module 4: Dielectric Ceramics: Basic Principles
Basic mechanisms of breakdown
4
...
12
...
Actual breakdown field is larger than critical breakdown field, Ec , needed to cause a critical
breakdown electron temperature, Tc
...
It is independent of sample or its geometry or waveform type
...
4
...
2 Thermal Breakdown:
It occurs due to heat dissipation in the sample due to current flowing through defective parts
of the sample which in turn further increase the ionic defect concentration leading to
subsequent increase in the conductivity and then failure
...
It depends on the speed of application of field
...
Ambient temperature determines the electron temperature and not the electric field strength
...
The process can be quite slow, from minutes to ms, and is dependent on sample geometry
The shorter the pulse time is, the higher is the breakdown voltage
...
12
...
e
...
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...
Quite often it occurs in thin films
...
Other or Pseudo breakdown mechanisms are
Dielectric discharge
Electrochemical and/or mechanical breakdown
Dielectric discharge
In small pores which are always present in sintered dielectric ceramics, the field strength is
higher than the average field and as a result, microscopic arc discharge in the pores may be
initiated
...
As the pores
grow, the current in the arc increases leading to an increased sample temperature eventually
leading to failure
...
It is assisted by suitable atmospheric conditions such as humidity and pH
...
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Module 4: Dielectric Ceramics: Basic Principles
Summary
Summary
A dielectric material is characterized by parameters such as dielectric constant or dielectric
permittivity which is greater than 1
...
Polarization in dielectrics happens because of variety of reasons and is a
frequency dependent phenomenon as characterized by electronic, ionic and dipolar or orientation
polarizations
...
Frequency
dependence of dielectric constant is also characterized by the presence of resonance phenomenon
for electronic and ionic polarization mechanisms, occurring at frequencies typically above 10 12 Hz
and relaxation phenomenon for orientation or dipolar polarization in polar materials which is a
typically low frequency mechanism
...
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...
In addition, there are a few special classes of dielectric materials which show large dielectric
constants, non-zero polarization in the absence of electric field and nonlinearity in the dielectric
constant
...
e
...
Most of these materials happen to be oxides and as you can very well understand now, these
properties will be greatly affected by the defect chemistry and process variables
...
In this module, we will discuss origin of these properties with a crystallographic and thermodynamic
framework and associated mathematical representations along with a few examples of materials and
devices
...
L
...
K
...
E
...
M
...
J
...
M
...
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Module 5: Nonlinear Dielectrics
Classification based on Crystal Classes
5
...
e
...
The term centrosymmetric refers to a space group which contains an inversion center as one of its
symmetry elements i
...
for every point (x, y, z) in the unit cell, there is an indistinguishable point (x, -y, -z)
...
e
...
Out of these 20 non-centrosymmetric point groups, 10 belong to polar crystals i
...
crystals which
possess a unique polar axis, an axis showing different properties at the two ends
...
Spontaneous polarization depends upon the temperature
...
This is called pyroelectric effect
...
In some of these polar non-centrosymmetric crystals, the polarization along the polar axis can be
reversed by reversing the polarity of electric field
...
e
...
So by default, all ferroelectric materials are simultaneously pyroelectric and piezoelectric
...
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...
1 Classificatin of piezo-, pyro- and ferroelectrics
In the subsequent sections, we will discuss about the ferroelectric, pyroelectric and piezoelectric
materials explaining the fundamental physics, some key materials and applications
...
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Module 5: Nonlinear Dielectrics
Ferroelectric Ceramics
5
...
4H 2 O by Czech scientist Roger
Valasek in 1921
...
However, after a few decades, renewed technological interest led to much more
extensive studies and a better understanding
...
Among various categories of ferroelectric materials
following stand out either due to interest in the structure or in the properties:
Tri-glycine sulfate and isomorphous materials
Pottasium dihyrogen phosphates and isomorphous materials
Barium titanate and other perovskite structured compounds such as KNbO 3 , PbTiO3 etc
Complex oxides such as Aurrivillious compounds
Rochelle Salt and similar compounds
Ferroelectric Sulfates
Among the above, perovskite related compounds have been studied most, primarily due to their
exciting properties and reasonably high transition temperatures, making them attractive for various
applications
...
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...
3
...
e
...
This requires the material to be non-centrosymmetric whereas
dipole moment would be forced to be zero in a centrosymmteric material due to symmetry
considerations
...
The following is the figure showing the crystal structure of a perovskite structured material such as
BaTiO3
...
This displaced position of the central atom is the energetically stable position i
...
the free energy is minimum
...
2 Cubic and tetragonal perovskite structures with dipole
moment in latter represented by an arrow due to movement of
central B ion up or down along c-axis
When this kind of ferroelectric material is switched i
...
subjected to a bipolar electric field, it exhibits
a polarization vs electric field plot as shown below
...
e
...
These two values can be connected with the position of a central atom in the octahedra in the figure
above
...
The plot shows a hysteresis which is important for applications such as memory devices
...
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...
3 A typical ferroelectric hysteresis
loop
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...
3
...
Figure 5
...
There are two equilibrium positions in which a B ion can stay, but to change from one state to
another, energy must be provided to overcome an energy barrier ΔE
...
e
...
However, a coercive field, which is the field
required to bring the polarization back to zero, is needed when you switch to other direction of the
field
...
1)
Here, N is the number of dipoles per unit volume of the material, a is dipolar polarizability, and Tc is
defined as the Ferroelectric transition temperature or the Curie temperature
...
A paraelectric state is
essentially a centrosymmetric higher symmetry state where dipoles are randomly oriented in a
crystal giving rise to zero polarization
...
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Figure 5
...
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Module 5: Nonlinear Dielectrics
Ferroelectric Ceramics
5
...
3 Proof of Curie-Weiss Law
We can now say that in a ferroelectric material, reversible spontaneous alignment of electric dipoles
takes place by mutual interaction
...
Interestingly, this phenomenon happens below a critical
temperature Tc when crystal enters into nonsymetric state where thermal energy cannot randomize
the electric dipoles, presumably due to dipole-dipole interactions and local field
...
2)
Where local field E' = E + P/3e0 as given by Clausius-Mossotti relationship and e'r is the relative
dielectric permittivity and a is total polarizability
...
2) results in
i
...
or
(5
...
4)
Equation (5
...
This can be assumed to be a right condition for ferroelectrics as near the ferroelectric transition, they
exhibit very large susceptibilities and dielectric constant
...
Here define a = a d = C/kT where C is Curie’s
constant
...
5
...
Hence, we can further write
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...
5)
Below this Tc spontaneous polarization is prevalent and dipoles tend to align
...
6)
Thus, by now modifying equation (5
...
7)
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...
3
...
L
...
F
...
The approach is based around calculating the free energy of system and working around
other thermodynamic parameters to predict the nature of phase transition
...
These non-linear dielectrics exhibit various kinds of couplings between physical properties and can
be expressed mathematically
...
Ferroelectric Effect:
Electric charge in a polar material can be induced by application of an external electric field and can
be expressed as
(5
...
e
...
Note that the equation is
valid only for the linear region of the hysteresis curve
...
e
...
9)
where
is the piezoelectric coefficient and is third rank tensor with units C/N,
is the stress
applied
...
10)
Where d ijk t is in m/V
...
e
...
11)
where Pi is the vector of pyroelectric coefficient in cm -2 K-1
...
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(5
...
These
couplings between thermal, elastic or electric properties can be understood formally by adopting a
thermodynamic approach
...
From the laws of thermodynamics, the thermodynamic state of any crystal in a state of equilibrium
can be completely established by the value of number of variables, which in case of ferroelectrics
include temperature T, entropy S, electric field E, polarization P, stress s and strain e
...
For a ferroelectric system, the free energy G can be expressed in terms of ten variables as
(5
...
are the stress
We can get the value of the independent variables in thermal equilibrium at the free energy
minimum
...
Here, we select the origin of free energy for a free unpolarized and unstrained crystal
to be zero
...
14)
Note that only even powers are taken because energy is same for ±PS states
...
The equilibrium is found by establishing
i
...
(5
...
e
...
16)
If all of a, b, and c are positive, then P = 0 is the only root of the equation as shown below in the
figure
...
If we ignore higher power terms, then
(5
...
18)
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...
According to Landau-Devonshire theory, near the Curie point (T~T0 ) we assume
a = a 0 (T - T0 )
(5
...
13), only ‘a’ is dependent on temperature while other
constants are temperature independent
...
18) into (5
...
20)
In this expression, for all known ferroelectrics, both ao and c are positive while depending upon the
sign of b, the phase transition nature changes
...
6 (a) show the free energy vs polarization plot when T >> T0 i
...
when the crystal is in
paraelectric state
...
6 (b), representing the ferroelectric
state with non-zero polarization at zero field
...
6 Free energy vs polarization for (a)
paraelectric (above T 0 ) and (b) ferroelectric crystal
(below T 0 )
...
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Module 5: Nonlinear Dielectrics
Ferroelectric Ceramics
5
...
5 Case I: Second order Transition
When b is positive, the ferroelectric transition occurs at a temperature T = T0 and is called as
second order transition (do not get confused between T0 and Tc as the distiction will become clear
in the next section when we learn first order phase transition)
...
e
...
Two curves become closely related if ‘a’ changes continuously with temperature and changes sign
at Tc and can be shown together as in figure 5
...
Figure 5
...
Note how the sign of a changes with temperature and its effect
on the curve
...
20) and
retaining only two lowest order terms since all the coefficients i
...
a 0 , b and c are positive
...
21)
showing that the polarization decreased to zero at T = T0 as shown in figure 5
...
Dielectric susceptibility at T < T0 can be estimated as
(5
...
e
...
8
...
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...
8 Effect of temperature on polarization and
reciprocal susceptibility for a second order phase
transformation
...
15) for T < T0
...
23)
Now substituting
in (5
...
24)
Examples of ferroelectric materials showing a second order transition are materials like Rochelle salt
and KH 2 PO4
...
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Module 5: Nonlinear Dielectrics
Ferroelectric Ceramics
5
...
6 Case – II: First Order Transition
Another situation to consider is that when a < 0, b < 0 but c > 0
...
e
...
This gives rise to the following free energy vs polarization plot
...
9 Free energy vs polarization
schematic plot for a first order phase
transition
The most important feature of this phase transition is that polarization i
...
the order parameter
drops from P ≠ 0 to zero discontinuously at T = Tc and is called as first order phase transition
...
For example, solid-liquid phase transition is a first order phase transition while among
various ferroelectrics, barium titanate is a fine example of first order transition among
ferroelectrics
...
10 Polarization and reciprocal
susceptibility plot for a first order phase
transition
In order to compute the non-zero polarization (P0 ) and susceptibility at the transition, the value of
free energy (5
...
e
...
25)
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...
e
...
26)
The polarization and susceptibility at Tc are obtained by solving two equations and are given as
(5
...
28)
The fact that there are three minima at T=T c is reflected in whether the Tc is approached while
heating or cooling
...
This results in thermal
hysteresis when these materials are thermally cycled across Tc
...
Lines
and A
...
J
...
Jona and G
...
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Module 5: Nonlinear Dielectrics
Ferroelectric Ceramics
5
...
7 Ferroelectric Domains
In a ferroelectric crystal, it is likely that the alignment of dipoles in one of the polar directions
extends only over a region of the crystal and there can be different regions in the crystal with
aligned dipoles which are oriented in many different directions with respect to one another
...
You should not confuse ferroelectric domain walls with the grain boundaries
...
The types of domain walls that can occur in a ferroelectric crystal depend upon the crystal structure
and symmetry of both paraelectric and ferroelectric phases
...
On the other hand, a
tetragonal perovskite like PbTiO3 has Ps along the [001]-axis and here domain walls are either 180°
or 90° domain walls
...
11 Schematic representation of a 180° and 90°
domain walls in a tetragonal perovskite crystal such as BaTiO3
Formation of the domains may also be the result of mechanical constraints associated with the
stresses created by the ferroelectric phase transition e
...
from cubic paraelectric phase to tetragonal
paraelectric phase in PbTiO3
...
Combination
of both effects leads to a complex domain structure in the material with both 90° and 180° domain
walls
...
This electrostatic energy associated with the depolarizing field can be minimized by
splitting of the material into oppositely oriented domains or
compensation of the electrical charge via electrical conduction through the crystal
...
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Domains can also be seen by microscopy
...
Figure 5
...
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...
3
...
The
free energy change involved in the formation of a domain is given as
(5
...
The internal field opposes the applied field E and hence is called as depolarizing
fieldThis is dependent on the domain size (d) and is expressed as
(5
...
U w is the domain energy and can be expressed in terms of surface energy of the wall (γ), domain
width (d) and crystal volume (V) and is given as
(5
...
In (5
...
Therefore one only needs to consider U w and U d when ΔG is minimized
w
...
t
...
e
...
32)
Here, one can see that domain size is dependent on surface energy, crystal thickness and
polarization, showing a competition between the surface energy of the wall and polarization
(governing the depolarizing field)
...
Secondly, the larger the polarization is, the larger the depolarizing field will be and large
would be the driving force for the domains to form and hence smaller domains will be preferred
...
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Module 5: Nonlinear Dielectrics
Ferroelectric Ceramics
5
...
9 Ferroelectric Switching and Domains
Application of an electric field to a ferroelectric ceramic leads to the alignment of antiparallel or offaligned dipoles to reorient themselves along the field
...
This phenomenon of polarization reversal takes place by way of
nucleation and growth of favourably oriented domains into the unfavourably oriented domains and
associated domain wall motion
...
13
material
Characteristic hysteresis loop of a ferroelectric
If we assume that our hypothetical crystal has an equal number of positive and negative domains in
the virgin states, then the net polarization of the crystal will be zero
...
The process is something like this:
Initial polarization P increases linearly with the increasing electric field and the crystal behaves like
a dielectric because the applied field is not large enough to switch any of the domains oriented
opposite to its direction
...
Further increase in the field strength forces nucleation and growth of favourably oriented domains at
the expense of oppositely oriented domains and polarization starts increasing rapidly (BC) until all
the domains are aligned in the direction of the electric field i
...
reach a single domain state
(CD) when polarization saturates to a value called saturation polarization (PS )
...
Now, when the field is decreased, the polarization generally does not return
to zero but follows path DE and at zero field some of the domains still remain aligned in the positive
direction and the crystal exhibits a remanent polarization (PR)
...
Further increase of electric field in the opposite direction will cause complete reversal of orientation
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...
This relation between P and E is called a ferroelectric hysteresis loop which is an important
characteristic of a ferroelectric crystal
...
Domain switching in a previously polarized ferroelectric sample can also be viewed in the following
animation
...
14 Domain switching animation in a ferroelectric
material (Reproduced from DOITPOMS Library, University of
Cambridge, UK)
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...
3
...
The circuit is schematically drawn below
...
In this configuration, the voltage across C 0 is proportional to the polarization of the crystal
...
Figure 5
...
For device measurements, one needs to establish top and bottom contacts to the
ferroelectric materials which is typically achieved using thin platinum electrodes in thin films and
silver paste in bulk ceramics
...
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Module 5: Nonlinear Dielectrics
Ferroelectric Ceramics
5
...
11 Structural change and ferroelectricity in Barium Titanate (BaTiO 3 )
Barium Titanate (BaTiO 3 ) has a perovskite ABO 3 type structure
...
Figure 5
...
For Cubic form a=b=c while for
tetragonal a=b≠c
...
As a result, the six Ti-O dipole moments along ±x, ±y, ±z cancel each other
and the material in such a state is called paraelectric
...
Unilateral displacement of the
positively changed Ti4+ ions against surrounding O 2- ions occurs to give rise to net permanent dipole
moment
...
This transformation forces Ti ions go to lower energy off center positions, giving rise
to permanent dipoles (see the energy well diagram in section 5
...
6)
...
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17
phase
Distortion in BaTiO3 upon cooling from cubic
Because the distorted octahedra are coupled together in ferroelectric form, there is a very large
spontaneous polarization, ~25 µC/cm2 , giving rise to a large dielectric constant, ~160, and large
temperature dependence of dielectric constant
...
It transforms to
orthorhombic structure at ~5°C and then again to a rhombohedral structure at ~ -90°C and as result
of change in the symmetry, the polarization vector also changes from [001] for tetragonal to [110] in
orthorhombic and [111] in rhombohedral structure
...
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Module 5: Nonlinear Dielectrics
Ferroelectric Ceramics
5
...
12 Applications of Ferroelectrics
In addition to BaTiO3 , extensively studied ferroelectric materials have been PbTiO3 , Pb(Zr,Ti)O 3 ,
Bi 4 Ti3 O 12 and SrBi 2 Ta 2 O 9
...
Although many applications of ferroelectrics are for their piezoelectric properties (see section 5
...
Above Tc
where their dielectric constant increases linearly with temperature, they can be used for camera
flashes
...
3
...
1 Nonvolatile Memories
Since ferroelectric materials show a hysteresis loop and remnant polarization ±PR at zero field, these
two polarization states can be used as ‘0’ and ‘1’ states of binary data storage in memory devices
...
Other advantages are that ferroelectric switching is a very fast phenomenon
and hence memories can operate very fast
...
Figure 5
...
e
...
3
...
2 Camera Flashes
In this application, the battery charges the ferroelectric capacitor first
...
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...
19 Animation of a camera flash using a
ferroelectric (Reproduced from DOITPOMS
Library, University of Cambridge, UK)
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...
These materials also follow a Curie-Weiss behavior, i
...
a transition from ferroelectric
to paraelectric phase upon heating across a Curie temperature, Tc
...
Ferroelectric materials when switched under
alternating fields, show a typical hysteresis like behavior showing two stable polarization states +P R
and –PR which are useful for memory application where these can be used as stated ‘0’ and ‘1’ for
binary data storage
...
Typical examples of commonly
used ferroelectric materials are perovskite structured compounds e
...
PbTiO3 , BaTiO3 , Pb(Zr,Ti)O 3
...
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...
4
...
4 Actuators
In the precision engineering applications, precise linear or rotational movements are required for
achieving technological perfection
...
This ability makes these materials useful as
precise actuators for achieving very precise motions
...
One can achieve very high field in the multilayered materials using voltages lower than 150200 V, not very high voltages
...
Some of the examples of applications are
Piezoelectric motors consisting of piezoelectric elements which apply a directional force to an
axle, causing it to rotate
...
Scanning force microscopes use inverse piezoelectric effect to keep the sensing needle close
to the probe
Laser mirror alignment in the laser electronics helping maintain accurate optical conditions
inside the laser cavity to optimize the beam output
...
In inkjet printers where piezoelectrics are used to control ink flow from the print head to the
paper
...
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4
...
5 Frequency Standards
Here quartz is most commonly used material as its piezoelectric properties are useful as standard of
frequency
...
The quartz crystal has a very accurately defined natural frequency of vibration at which it prefers to
oscillate, and this is used to stabilize the frequency of a periodic voltage applied to the crystal
...
Both of these usually use a frequency multiplier to reach the megahertz and
gigahertz ranges
...
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Module 5: Nonlinear Dielectrics
Summary
Summary
Piezoelectric materials are noncentrosymmetric materials which show coupling of mechanical and
electrical properties
...
Interestingly, many ferroelectrics
have also found useful applications in such applications mainly because of their superior
piezoelectric properties
...
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Module 5: Nonlinear Dielectrics
Piezoelectric Ceramics
5
...
Direct piezoelectric effect
is the ability of some materials to create an electric potential in response to applied mechanical
stress
...
As a requirement, only materials with non-centrosymmetric crystal structure
can exhibit piezoelectric effect
...
An oscillating applied stress on a piezoelectric material can give rise to the field which can be
applied to an electrical load such as a bulb
...
You could not achieve the same while standing
...
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...
4
...
Here, the
polarization is directly proportional to the stress applied, as described by the equation
...
33)
where P is polarization, s is applied stress and d is piezoelectric coefficient (actually a third rank
tensor)
...
4
...
34)
where e is the strain induced, d is the piezoelectric coefficient and E is the applied electric field
...
20 Direct and Converse Piezoelectric
Effects
The direct piezoelectric effect is used as the basis for force, pressure, vibration and acceleration
sensors while converse effect is used as a basis for actuator and displacement devices
...
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Module 5: Nonlinear Dielectrics
Piezoelectric Ceramics
5
...
3 Poling of Piezoelectric Materials
As obvious from the previous sections (5
...
2), there are some piezoelectrics such as quartz
which are not spontaneously polarized but get polarized upon application of stress, while ferroelectric
which are anyway piezoelectric in nature are spontaneously polarized and show a change of
polarization upon application of stress
...
3
Barium Titanate
100-149
Lead Niobate
80-85
Lead zirconate titanate
250-365
So, you can observe from this table that the level of strain generated is not so massive but is still
important because of preciseness and reversibility of the effect
...
In the unpoled virgin state of the
material, the ferroelectric domains of single polarization direction are randomly distributed across the
material and in such a situation the net polarization would be zero
...
Poling i
...
application of a large electric field near Tc (just below Tc ) orients the domains along the
field and when the field is removed, the domain structure does not get back to the original condition
giving rise to a net polarization along a certain direction
...
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...
21 Poling of ferroelectrics and application of stress on poled
material
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...
4
...
Figure 5
...
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Module 5: Nonlinear Dielectrics
Piezoelectric Ceramics
5
...
5 Common PIezoelectric Materials
5
...
5
...
We discussed its structure and transitions
in the previous section (5
...
11)
...
This has been
used to control the piezoelectric properties around 0°C, and is important for underwater detection
and echo sounding
...
Ti substitution by
1-2 at % Co3+ leads to much reduced losses as high fields useful in ultrasonic applications
...
5
...
5
...
It has a perovskite structure with B sites randomly
occupied by either of isovalent Ti and Zr ions
...
Figure 5
...
The reason for
this is that this is closed to morphotropic phase boundary and here rhombohedral and tetragonal
structures co-exist
...
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few poling directions available making it a useful piezoelectric
...
This
in turn reduces the concentration of defect pairs which otherwise impede the domain wall motion
...
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...
4
...
These techniques are resonance or subresonance techniques
...
To a
first approximation, the electromechanical response of a piezoelectric material close to the
characteristics frequency can represented by the electrical equivalent circuit as shown in the figure
5
...
Simplest measurements are conducted by poling a piezoelectric long rod of length ~6 inch and
diameter ~¼ inch along its length
...
24 Schematic representation of (a) equivalent
electrical circuit of a piezoelectric sample close to its
characteristic frequency (b) Plot of electrical reactance of the
sample a function of frequency
The coupling coefficient, k 33 , is expressed in terms of series and parallel resonance frequencies (fs
and fp respectively) as
(5
...
36)
Measurements are limited to the specific frequencies determined by the fundamental vibration modes
of the sample
...
In such cases, the resonance in
the substrate driven by a thin film could be used to determine the piezoelectric coefficient of a thin
film
...
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...
This
includes measurement of direct effect i
...
charge developed on a piezoelectric material under
application of an external mechanical stress and measurement of converse effect i
...
measurement
of electric field induced displacements
...
Appropriate electrical circuits needs to drawn and modeled to clearly elucidate the
material properties
...
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Module 5: Nonlinear Dielectrics
Piezoelectric Ceramics
5
...
7 Applications of Piezoelectric Ceramics
Piezoelectric ceramics are used in a variety of applications utilizing either direct or converse
piezoelectric effect
...
4
...
1 Power Generation
Gas Lighter
Piezoelectric material can ignite the gases by generating a spark via an electric current
...
e
...
The piezoelectric are placed in a circuit with a spark gap
...
The force
brings together these two pieces which then gives rise to creation of charges
...
One must apply the force quickly otherwise the voltage generated disappears because the charges
leaks away through the piezoceramic, across its surfaces and via the apparatus
...
25 Schematic of operation of a gas lighter made using
piezoelectric material (you can also refer to the animation made by
DOITPOMS, University of Cambridge, UK)
...
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Power Transformer
A piezoelectric transformer works like an AC voltage multiplier
...
Piezo transformers can be quite compact high
voltage sources
...
This causes the bar to vibrate with vibration frequency chosen
to be the resonant frequency of the block, typically in the 100 kHz to 1 MHz range
...
One can
achieve the step-up ratios of more than 1000:1 using this technique
...
26 Schematic of a piezoelectric
transformer
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...
4
...
2 Piezoelectric Sensors
Here, typically pressure or force is used to create an electrical signal out of a piezoelectric material
...
Similar principle can also be used for pickup guitars and
microphones
...
To detect detonation in automotive engine by sampling the vibrations of the engine block
To detect the precise moment of fuel injection in an automotive engine
Detection of acoustic emissions in acoustic emission testing
...
Strain gauges
Medical applications using ultrasound waves
Kidney stone treatment
In this application, electricity of high frequency is applied to the sample which gives rise to a change
in the shape of the material
...
These powerful ultrasound waves can be used to shatter pieces of materials
inside the body such as kidney stone, which can then pass out through the urine
...
27 Ultrasonic waves creation and
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...
4
...
3 Ultrasound Imaging Using Transduction Effect
Another application is its use in the ultrasound imaging of the fetus where piezoelectric acts as a
transducer utilizing both direct and inverse effects
...
The reflected waves come back to
piezoelectric exploit the inverse effect which leads to creation of charges from the piezoelectric which
can then be modeled to generate an image of the fetus
...
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Module 5: Nonlinear Dielectrics
Pyroelectric Ceramics
5
...
If the latter is true,
then a pyroelectric material is also ferroelectric
...
Pyroelectricity, in itself, is the ability of materials to generate a voltage when they are heated or
cooled
...
If the temperature
is constant, then voltage gradually disappears due to leakage of charges through the material or air
or the apparatus
...
Pyroelectricity was first observed by the Greek philosopher Theophrastus in 314 BC who found that
Tourmaline attracted small pieces of straw and ash when it was heated
...
In 1747, Linnaeus first
related the phenomenon to electricity, although this was not proven until Franz Ulrich Theodor
Aepinus did so in 1756
...
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Module 5: Nonlinear Dielectrics
Pyroelectric Ceramics
5
...
1 Difference between Pyroelectric and Ferroelectric Material
Although both ferroelectric and pyroelectric materials must be non-centrosymmetric and polar, the
essential difference between them lies when an electric field is applied
...
However, when the material is
heated above Curie temperature, the atoms come back to their equilibrium positions (C)
...
29 Difference between a ferroelectric and pyroelectric
material
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5
...
e
...
e
...
37)
Here e is permittivity of the pyroelectric material and Ps is the spontaneous polarization
...
38)
Now, defining generalized pyroelectric coefficient, Δp g , as change in polarization with temperature,
we write
Δ
Δ
OR
(5
...
The last term in the above equation is the
temperature dependence of the permittivity of the material which we can measure
...
40)
The above equation neglects the change in dielectric constant with temperature and is valid only
when such assumption is true
...
The coefficient is usually negative
because polarization decreases with temperature
...
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...
5
...
The circuit
connecting a pyroelectric material (held inside an oven) with an amplifier and then measuring the
pyro-current
...
41)
Where Ip is the pyrocurrent and is given as
(5
...
Figure 5
...
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Module 5: Nonlinear Dielectrics
Pyroelectric Ceramics
5
...
4 Direct and Indirect Effect
Since all pyroelectric materials are naturally piezoelectric in nature, the thermal
expansion/contraction of the material while heating or cooling induces thermal stresses in the
material which in turn induce their own polarization and hence changing the overall polarization
...
Figure 5
...
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Module 5: Nonlinear Dielectrics
Pyroelectric Ceramics
5
...
5 Common Pyroelectric Materials
Most of the inorganic pyroelectrics (including ferroelectrics) are perovskite structured
...
*
The most common materials are tabulated below
...
m -2
...
75Pb(Mg 1/3 -Nb2/3 )O 3 -0
...
67 Sr0
...
5
...
1 Triglycine Sulphate (TGS)
High pyroelectric coefficient
Fragile and water-soluble, difficult to handle and cannot be used in devices where it would be
subjected to either a hard vacuum or high humidity as it tends to decompose
Can be modified to withstand temperatures above Curie point without depoling
Used in thermal imaging cameras
5
...
5
...
5
...
3 Perovskite Ferroelectric Ceramics
Generally robust and insensitive to moisture and vacuum
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* R
...
Whatmore, Pyroelectric devices and materials, Reports and Progress in Physics, Volume 49,
Page 1335 (1986)
...
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Module 5: Nonlinear Dielectrics
Pyroelectric Ceramics
5
...
6 Common Applications
5
...
6
...
To avoid the effects
due to thermal expansion, one needs to use a reference identical material to counter these
extraneous effects
...
Figure 5
...
5
...
2 Infrared or Thermal Imaging
Just like we use visible light to make a photograph, infrared (IR) radiation emitted by objects at
different temperatures is focused onto a sensitive plate to create thermal image of the object
...
The pyroelectric elements used in the devices are typically square plates with sides about a mm
long and thicknesses around 30 µm
...
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...
33 IR image of a dog (Ref: Wikipedia)
...
A special lens focuses the infrared light emitted by all of the objects in view
...
It only takes about one-thirtieth of a second for the
detector array to obtain to create a thermogram
...
This thermogram is translated into electrical signals which are sent to a signal-processing unit which
then translates the information from the elements into data for the display
...
The combination of all the signals from all of the
elements creates the image
...
34 Process of image
creation
5
...
6
...
This requires us to monitor the levels of pollution
...
Pyroelectrics being excellent detectors of IR radiation, can easily detect the level of IR radiation
which passes through a gas sample
...
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35 Gas detection using a pyroelectric
detector
(Courtesy: © DoITPoMS, University of Cambridge)
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These
materials are of immense use because of their ability to demonstrate surface voltage by subjecting to
changing temperature
...
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...
These materials possess extra-ordinary
properties such as strong magnetic coupling, low loss characteristics and high electrical resistivity
which is often related to their structure and composition
...
To build an understanding, in the following sections, we will first have a brief look at the basics of
magnetic properties, followed by various types of magnetism in materials and key characteristics and
differences
...
The Module contains:
Magnetic Moments
Macroscopic View of Magnetization
Classification of Magnetism
Diamagnetism
Paramagnetism
Ferromagnetism
Antiferromagnetic Materials
Ferrimagnetic Materials
A Comparison
Magnetic Losses and Frequency Dependence
Magnetic Ferrites
Summary
Suggested Reading:
Principles of Electronic Ceramics, by L
...
Hench and J
...
West, Wiley
Electroceramics: Materials, Properties, Applications, by A
...
Moulson and J
...
Herbert, Wiley
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1 Magnetic Moments
Magnetism in materials is crudely explained as mutual attraction between two pieces of a material,
say iron or iron ore
...
The strength of magnetism is quantitatively judged by a quantity called as
‘magnetic moment’
...
The major contributors of magnetic moment in a material are
Motion of electrons in an orbit of an atom
...
Spinning of electron around it own spin axis gives rise to a moment
...
The first two contributions are quite significant and contribute to most of the magnetic character of a
material while the third component, nuclear magnetic moment, is rather insignificant in the context of
most magnetic materials of practical interest and can be neglected
...
1
...
The field is related to the current by
(6
...
e
...
m -1
...
1 A current carrying
coil
Now based on classical physics, the current, I, is nothing but charge per unit time and is expressed
as
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...
2)
The time is nothing but the circumference divided by the velocity of electron, v, i
...
t = 2p r/v which
can be substituted in equation (6
...
3)
where r is the radius of the orbital
...
4)
where q is the charge of electron which equals to – e; v is velocity of electron; m is the mass of
electron and J is the angular momentum which is equal to mvr
...
4) shows that orbital moment can be expressed in the multiple of angular
momentum
...
1
...
For purely
quantum mechanical reasons, for an electron spinning around its own axis, the ratio of spin
magnetic moment to angular moment is twice as large as for orbital moment i
...
(6
...
1
...
(6
...
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(6
...
The latter is true for
many ferrites where orbital magnetic moment is thought as quenched and the dominating
contribution to magnetic moment arises from spin magnetization
...
Hence the angular moment for an
electron with a characteristic number, j is expressed as
(6
...
6×10 -34 J
...
The characteristics number j
can be orbital quantum number, l, for orbital moment and spin quantum number, s, for spin moment
...
9)
The value of one Bohr magneton is 9
...
m 2
...
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2 Macroscopic view of Magnetization
We will refer to the same geometry here where we have a current carrying coil
...
m-1
...
m-2
...
So now rewriting equation (6
...
10)
Now imagine if a magnetic material is inserted within the coil, then there is a current induced in the
magnetic material too, called Ameprian current, Ia which modifies the equation (6
...
11a)
Figure 6
...
11a), we can write
(6
...
e
...
H, we can write
OR
(6
...
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or
B = µo µr H
(6
...
Here, µ r = (1+ χm ) is the, in a similar manner to relative dielectric permittivity, e r
...
Naturally for vacuum, χm = 0
...
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3 Classification of Magnetism
Magnetic materials can be classified based on the values of magnetic susceptibility
...
Most diamagnetic
materials show very small negative susceptibility except superconductors in
superconducting effect when χm is equal to 1 which is very useful for applications such as
magnetic levitation
...
Susceptibility is positive but very small for paramagnetic materials
but can be very large for ferro- and ferri-magnetic materials
...
3 Classification of materials on the basis of
susceptibility
Susceptibility values of some of the common materials are provided below
...
E
...
Material
χ(SI) unitless
(cgs)
Unitless
Unitless
Bi
0
...
99998
Ag
0
...
99996
Ge
0
...
99999
Diamagnetic
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...
99999
Water
0
...
00008
Paramagnetic
Al
1
...
000026
Low
steel
carbon
Fe-3%Si (Grain
Oriented)
Ferromagnetic
Ni-Fe-Mo
superalloy
It should be noted, also as we will see, that except for diamagnetic materials, magnetic
susceptibilities are temperature dependent
...
This says that for diamagnetic materials, when an external field is applied, the magnetic
moment that is induced is in opposite direction to the field direction i
...
opposite
magnetization as shown below
...
It is just
that some materials like silver, gold, silicon are only diamagnetic i
...
they don’t have any
other effect present in them
...
e
...
This means that magnetic flux penetrates into the material as shown below
...
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...
4 Schematic diagram showing flux
penetration in magnetic materials with various
ranges of susceptibilities
...
For materials having only diamagnetic contribution, this turns out to be
negative
...
If taken too far, classical mechanics simply shows that all
the magnetic moments in any material cancel out
...
Those who are interested can go through Feynman Lectures on Physics which
provide a wonderful explanation of this conundrum (The Feynman lectures on physics,
Vol
...
P
...
B
...
L
...
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...
4 Diamagnetism
Diamagnetic materials are those in which the electron motions are such that they produce net zero
magnetic moment in the absence of any magnetic field
...
Examples of such materials are inert gases, hydrogen, many metals (e
...
Ag, Au, Cu etc
...
g
...
Figure 6
...
Imagine a circular orbit of radius, r, around an atom with its center coinciding that of the atom
...
Hence, according to Faraday’s law, as the magnetic field changes, it generates an electric field by
magnetic induction
...
14)
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...
6 Motion of an electron in an atom’s
orbit
This electric field produces a torque equivalent to -eE
...
e
...
r) which must be equal to the rate of
change of angular momentum, J, i
...
(6
...
Now integrating with respect to time with zero field, we get
(6
...
Now, since the motion of electron is taking place in the orbits, the change in magnetic moment
(Δµ m ) which is orbital in nature is given as
(6
...
18)
For atoms with spherical symmetry,
, hence,
(6
...
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So, here we have a magnetic moment which is negative in sign to the magnetic field strength
because it opposes the applied field
...
So, if there are N electrons per unit volume, then magnetization, M, is given as
(6
...
21)
This equation shows the opposite nature of the magnetic susceptibility of the diamagnetic
behaviour
...
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...
5 Paramagnetism
In paramagnetic materials, atoms have a permanent non-zero net magnetic moment due to the sum
of orbital and spin magnetic moments
...
Upon application of a field, the moments tend to align up in the direction of the field overcoming the
thermal barrier and giving a net positive magnetic moment in the direction of the applied field
...
Figure 6
...
Qualitatively, it is due to the electric field generated by the surrounding
ions in the solid
...
On the other hand, electron spins are weakly coupled and hence form a major
part of the magnetic moment
...
For example, Ni, with an atomic number of 28 has an electronic configuration of 1s2 , 2s2 , 2p 6 , 3s2 ,
3p 6 ,4s 2 and 3d 8
...
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...
These are shielded by the outer electrons from the crystal field and as result they show
both spin and orbital paramagnetism
...
22)
where N is the number of atoms per unit volume, µ m is the magnetic moment of an atom and C is
Curie constant
...
The derivation for Equation (6
...
Here we assume that each atom has a magnetic moment µ m whose magnitude is the same but
the direction can be random
...
B = -µ m B cosa where
a is the angle between the moment and field
...
8 Representation of magnetic
moment in a paramagnetic solid with respect
to the applied field
Boltzmann statistics in classical thermodynamics gives the probability of having any angle i
...
of
occupying any energy as exp(-µ m H
...
As a result, as you can also notice that it is more
plausible to have moment closer to 0° than 180° with respect to the applied field, simply because
that leads to smaller energy than those at 180°
...
8 which is proportional to exp(-µ m Hcosa/kT)
Considering a unit radius of the sphere (R=1), the area dA can be worked out as
(6
...
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or
dn =
(6
...
Assuming that γ = µ m H/kT, we can integrate equation (6
...
e
...
25)
Now, the total magnetic moment along the direction of magnetic field can be expressed
multiplication of number dipoles multiplied by the magnetic moment of each dipole along the field
direction (µm cosa) and is given as
(6
...
25) by (6
...
27)
Now, following a similar procedure as in Module 4 for orientation polarization, we can show that for
reasonably small magnetic field, moments are parallel to applied field and magnetization M can be
expressed as
(6
...
For most cases i
...
when
µ m B << kT, ,we can assume L(ß) = ß/3
...
28) as
(6
...
30)
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...
Figure 6
...
29 and 6
...
The same result can also be shown using quantum mechanism which
is not a part of this course
...
31)
which is similar to (6
...
Here, g is Landé-g factor
...
Kittel)
...
54
-
2
...
58
-
3
...
62
-
3
...
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Transition Elements
Mn3+
0
...
90
4
...
92
5
...
9
Fe 2+
6
...
90
5
...
63
3
...
8
Ni 2+
5
...
83
3
...
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Module 6: Magnetic Ceramics
Ferromagnetism
6
...
6
...
This phenomenon is observed below a critical temperature called as Curie
Temperature, above which the material behave like a paramagnetic material
...
10 Schematic representation of spins in a ferromagnetic
material
When a varying magnetic field is applied to a ferromagnetic material, the material exhibits a
ferroelectric-like hysteresis loop between magnetization and the magnetic field as shown below
...
Figure 6
...
However
some oxides such as chrmoimum oxide, CrO2 , are ferromagnetic oxides
...
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...
6
...
This is again similar to the case of ferroelectrics,
because these materials also contain the domains which are randomly oriented in a virgin material
...
The domain growth in ferromagnetic materials occurs by growth of favourably oriented domains at
the expense of unfavourably oriented domains unlike in ferroelectric materials where favourably
oriented domains nucleate and grow
...
To bring the magnetization
of the material back to zero, one needs to apply an extra field in the opposite direction which is
called coercive field of is - H c
...
A ferromagnetic material is a hard
magnet when it has large coercivity or soft magnet when coercivity is small
...
Hence equivalent points in a B-H curve would
be Bs (saturation induction) and Br (magnetic remanence)
...
Soft magnets with reasonably large Mr and
smaller H c are useful for transformer and motor cores where energy dissipation due to AC fields is
low while hard magnets are useful as permanent magnets and magnetic memories
...
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Figure 6
...
13 Formation of domains in a ferromagnetic
material
Again, the formation of domains and their size in these materials is basically due to balance between
various kinds of energies associated:
Exchange energy (we will see its origin later) which makes the magnetic moments align in one
direction without violating Pauli’s exclusion principle;
Magnetostatic energy in response to the flux lines at the surface of the material in the monodomain state which increases as alignment increases or in other words high surface magnetic
charges;
Magnetocrystalline or anisotropy energy which, due to coupling between the spins and crystal
lattice or spin-orbit coupling, is dependent on the crystal structure governing the direction of
magnetic moment orientation as there are some crystallographic directions along which the
sample is easy to magnetize than others such as for most cubic materials [111] is the easy
axis except for those containing cobalt which have [100] as easy axis
...
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Magnetoelastic energy due to changes in the lattice parameters of the material as a result of
spin-orbit coupling
...
Typically mono-domain state increases the magnetostatic energy significantly which is decreased by
the formation of domains
...
So, in ferromagnetic materials, not only do atoms possess the permanent magnetic moment, these
moments are also aligned along certain directions in an ordered fashion in regions called domains
and that is why a ferromagnetic material gives rise to a large non-zero magnetization even in the
absence of magnetic field
...
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6
...
A domain wall is often called as a Bloch wall, termed after F
...
A Bloch wall defines a region
between two neighbouring domains where the spin moments change gradually from one orientation
to another, as shown below in case of 180° or antiparallel domains
...
m -2
...
14 Domain wall structure in a ferromagnetic
material
As we wrote above, since favourably oriented domains grow at the expense of unfavourably oriented
domains, the Bloch wall also moves towards the left, for the configuration shown above
...
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Module 6: Magnetic Ceramics
Ferromagnetism
6
...
4 Molecular Field Theory
Now the question arises: What leads to this ordering and what is the effect of the ordering of
magnetic moment? This ordering or shall we say a neighbour effect gives rise to a modified internal
field experience by these magnetic moments (often denoted as magnetic dipoles) in a sort of a
similar manner as we witnessed in the derivation of Claussis Mossotti relation in dielectrics
...
This extra
field or called as Weiss molecular field, H w plays a crucial role in keeping the moments aligned in
one direction
...
Hence, this internal field can be expressed as
(6
...
23 and 6
...
33)
OR
(6
...
35)
where TC = γC and it has a unit of temperature and is called as Curie temperature
...
15 Schematic plot of
susceptibility and magnetization vs
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...
A positive value of TC indicates that molecular field is acting in the
same direction as of applied field and acts to align the magnetic moments parallel to each other, as
should be the case with a ferromagnetic material
...
For instance, in case of two electrons, electrons can arrange
themselves parallel or antiparallel
...
Hence, the electrostatic energy of an electron system is governed by the relative
orientation of the spins and the difference in energy defines the exchange energy
...
The exchange interaction is of short range, thus only nearest neighbor atoms produce the molecular
field
...
e
...
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...
7 Antiferromagnetic Materials
These are materials in which electron spins associated with magnetic atoms at particular
crystallographic sites are ordered yet oriented with respect to each other in such a manner that their
net magnetization is equal to zero
...
Examples include metallic manganese, chromium, various transition metal oxides such as
manganese oxide (MnO), forms of iron oxide (Fe 2 O 3 ), multiferroic perovskites like bismuth ferrite
(BiFeO 3 )
...
Below is the crystal structure
of MnO, drawn on (100) plane
...
Figure 6
...
Figure 6
...
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...
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...
8 Ferrimagnetic Materials
These are materials which again, like antiferromagnetic materials, show antiparallel alignment of
moments at particular atomic sites i
...
magnetic moment of one crystal sub-lattice is anti-parallel to
the other
...
The examples of such materials are various kinds of
cubic spinel ferrites such as NiFe 2 O 4 , CoFe2 O 4 , Fe 3 O 4 (or FeO
...
Other
examples are hexagonal ferrites likes BaFe 12 O 19 , garnets such as Y3 Fe 5 O 12 , represented by a
general formula R 3 Fe 5 O 12 where R, in addition to yttrium can be one of lanthanide atoms such as
lanthanum, cerium, samarium etc
...
18 Magnetic moment arrangements in magnetically ordered
materials
These materials also follow a temperature dependence of magnetization and susceptibility near
Curie transition (actually Néel transition) in a similar manner as shown by the ferromagnetic
materials
...
Figure 6
...
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Module 6: Magnetic Ceramics
A Comparison
6
...
e
...
The field dependent magnetization behavior of
these different types is shown below as well the dependence on temperature
...
Figure 6
...
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...
10 Magnetic Losses and Frequency Dependence
This is important mainly for ferromagnetic and ferrimagnetic materials when they are placed in an
alternating magnetic field
...
6
...
1 Magnetic Losses
We imagine a ferromagnetic or ferrimagnetic material with a hysteresis curve is subjected to an
alternating magnetic field, H = H 0 exp (iωt)
...
In an
analogy to the dielectric materials, two loss mechanisms are of importance:
Eddy current losses induced by the alternating magnetic field which are quite dependent on
the resistivity of the material; and
The domain wall motion requires as well as dissipates some amount of energy resulting in
what are called as intrinsic magnetic losses or hysteresis losses
...
While frequency, magnetic field
strength and maximum magnetic flux increase both types of losses, increasing the resistivity (ρ)
decreases eddy current losses in particular, losses are more when the material is conducting and
many ferromagnets tend to be quite conductive
...
36)
where f is the frequency of applied field, H c is the coercive field strength and Br is magnetic
remanence
...
37)
The total power loss is the sum of these two losses
...
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...
10
...
Now, the ease of domain wall movement can be described in terms of coercivity, i
...
the higher
the coercive field, the easier it is for domains walls to move during switching
...
In some materials, low frequencies are needed for easy switching while some
materials can witness easy switching at GHz or microwave region frequencies making them suitable
for microwave devices
...
e
...
6
...
2
...
These fields can be written as phasors,
such that
and
(6
...
Now, we can define the magnetic permeability as
(6
...
We can further simplify equation (6
...
40)
Hence, the loss tangent is defined as the ratio of the imaginary to the real part of the complex
permeability and is
(6
...
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...
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Module 6: Magnetic Ceramics
Magnetic Ferrites
6
...
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...
11
...
(See Module 1)
...
One unit-cell consists of eight formula units of AB 2 O 4 hence
containing a total of 32 octahedral interstices with one fourth occupancy and 64 tetrahedral
interstices with one eighth occupancy by the cations
...
From the
magnetism point of interest, the cations occupying tetrahedral sites have their spins oppositely
oriented with respect to the cations on octahedral sites (up and down depends upon your frame of
reference)
...
In compounds with inverse spinel strcture e
...
Fe 3 O 4 , NiFe 2 O 4 , half of B cations occupy
tetrahedral sites and all A and the remaining 50% B cations occupy octahedral sites
...
Hence the net magnetization of Fe 3 O 4 is 4µB per formula unit which is quite a large magnetic
moment
...
m -1 by simply calculating the net moment for the whole unit
cell and dividing that by the cell volume
...
This approach is also valid for mixed spinel compounds
...
As you can see that
since the moments of two sites are antiparallel, reducing the net magnetic moment of one site
would actually increase the net moment
...
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normal spinel, results in maximization of moment up to ~40 mol% ZnFe 2 O 4
...
e
...
e
...
Exceptions could
be Co-containing ferrites which are not only strongly magnetically anisotropic but also show large
coercive fields strengths
...
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11
...
The
model compound of this family is barium ferrite with formula BaFe 12 O 19
...
e
...
The structure is basically a mixture of cubic closed
packed and hexagonal closed packed layers formed by barium and oxygen ions
...
Out of 12 iron atoms of one formula unit, 9 occupy the octahedral sites, two occupy tetrahedral sites
and the remaining one is 5-fold coordinated
...
e
...
As we saw earlier,
each Fe 3+ ion has spin magnetic moment of 5µB simple math gives a net magnetic moment of 20µB
per formula unit leading to a magnetic moment of 40µB per unit cell
...
The material is typically categorized as a hard ferrite with
coercivity between 50-100 kA
...
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11
...
In the context of magnetic materials, garnets are represented
by a general formula Y3 Fe 5 O 12 , containing two magnetic ions, one typically being iron and another
being rare earth
...
The unit cell of Y3 Fe 5 O 12 is cubic and contains 8 formula units i
...
160 atoms, quite complex!
In garnet ferrites, orbital magnetic contribution of iron atoms is quenched due to shielding from
crystal field while lanthanide ions contribute to both orbital and spin magnetic moment, thus
contributing more to the total magnetic moment
...
e
...
We know that each Fe 3+ ion contributes 5µB which each
lanthanide atom, R, contributes a moment of magnitude µ Rµ B where µ R is the strength of moment of
R ion
...
As we see from the above schematic figure, net
magnetic moment would be dominated by rate earth ions when µ R is greater than 5/3
...
Typically the net magnetic moment drops as the temperature increases, especially for strongly
magnetic ions like Gd, Tb and Dy
...
2 Gd 1
...
Garnets can be quite useful materials in microwave applications because of their high electrical
resistivity and hence lower losses around microwave frequencies
...
The structural
parameters as well as magnetic properties can be tuned by tailoring the composition of the material
...
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Module 6: Magnetic Ceramics
Magnetic Ferrites
6
...
4 Properties of Ferrite Ceramics
As discussed above, ferrites can have a broad spectrum of properties depending upon the type of
ferrite and compositions
...
These values are quite dependent upon the microstructure and processing history of the material
...
m -)
Soft materials
Fe
150
80
2
...
93
690
60×10 -8
Mn-Zn
ferrites
500-10,000
5-100
0
...
50
90-280
0
...
10-0
...
4
0
...
3
450
Oriented
Barium
Hexaferrite
320
0
...
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Module 6: Magnetic Ceramics
Magnetic Ferrites
6
...
4
...
e
...
Typical soft ferrites used in transformer or electromagnetic cores contain nickel, zinc, and/or
manganese based ferrites
...
m, which leads to low eddy currents in the core, another
source of energy loss
...
The most common soft ferrites are Mn x Zn (1-x) Fe 2 O 4 , Ni x Zn (1-x) Fe 2 O 4
...
Mn-Zn ferrites, in comparison, have higher permeability and saturation induction
...
For instance, in Mn 1x Zn x Fe 2 O 4 and Ni 1-x Zn x Fe 2 O 4 , increasing the Zn
content leads to an increase in the magnetic
permeability just before the magnetic transition but at the same the magnetic transition temperature
also decreases
...
The increase in relative permeability is about an order of magnitude
in Mn 1-x Zn x Fe 2 O 4 for doping levels up to 50 at
...
Change in grain size also has a profound effect on the relative permeability with permeability
increasing with increasing grain size
...
Electrical resistivity of ferrites is again composition dependent
...
In ferrites, d-group elements are susceptible to valence fluctuations
...
This is also controlled very strongly by processing conditions such as firing
temperatures, atmosphere and rate of cooling after sintering
...
Any increase in the iron content i
...
a > 0 is compensated by the formation of Fe 2+ which creates
favourable conditions for electron hopping between Fe 3+ and Fe 2+ promoting n-type conduction
...
e
...
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...
23 Resistivity variation in Ni 1x Zn x Fe2+aO4-ßi as a function of Fe content
I n addition, the resistivity of nickel ferrites is also increased by addition of small amounts of Cobalt
...
Since Co
ions are sparsely located in the lattice, hopping of electrons between Co2+ and Co3+ states is
minimal
...
Both of these approaches make grain boundaries more resistive than the grains
...
10
...
Interested readers could go through Chapter 9
(Magnetic Ceramics) of Electroceramics; Materials, Properties and Applications by A
...
Moulson and J
...
Herbert
...
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Module 6: Magnetic Ceramics
Magnetic Ferrites
6
...
4
...
m-1 and are often
called permanent magnets
...
In addition to magnetization or remanence (Br) and coercive field, a permanent magnet is often
characterized by product B
...
e
...
Remanence in materials like hexaferrites which are strongly anisotropic can be affected by
processing
...
Similarly, coercivity is a strong function of grain size and it is
found for hexaferrites that it is maximum for grain sizes of about 1 µm
...
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Module 6: Magnetic Ceramics
Magnetic Ferrites
6
...
5 Applications of Magnetic Ceramics
In electronic inductors, transformers and electromagnets
Soft ferrites like Mn-Zn and Ni-Zn ferrites are used as core materials in these applications in
the frequencies ranging from a 100 kHz to 100 MHz
...
Most common radio
magnets, including those used in loudspeakers, are ferrite magnets
...
Ferrites are also used for power transformers which are used to transmit either over a single
frequency or within a range such as in ultrasonic generators
...
Equipment shielding
Here, due to their high impedance to high frequency currents, ferrite components of Ni-Zn and
Mn-Zn ferrites are able to prevent high frequency electrical noise due to electromagnetic
interference from exiting or entering the equipment
...
g
...
2-
...
The particles have single domains magnetized
along their major axes which are aligned in the plane of the tape
...
m -1
...
Here, the read/write head is not in direct
contact with the hard disk (in contrast to floppy disk) due to an air bearing (˜ 50 nm); air flow
is caused by the relative velocity between disk and head
...
in -2 and short access time
...
Ferrite powders are used in the coatings
of magnetic recording tapes
...
Absorbing materials
In stealth aircrafts, ferrite particles are used as a component of radar-absorbing materials or
coatings and in the absorption tile lining in the rooms used for electromagnetic compatibility
measurements
...
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...
Diamagnetism in most materials is characterized by small but negative magnetic
susceptibility while paramagnetic materials exhibit small and positive susceptibilities
...
While most ferromagnetic materials happen to be metals or alloys, many
spinel structured oxide ceramics, especially those containing iron and other magnetic elements, e
...
Fe 3 O 4 tend to be ferrimagentic
...
These ferrites
are useful for a variety of applications including electromagnets, shields, data storage devices and
microwave applications
...
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Module 7: High temperature Superconductors
Introduction
In addition to the dielectric and magnetic ceramic materials we discussed in earlier modules, there are
a few more ceramics which are rather different and exhibit unusual characteristics about which we
normally do not find any mention in the standard text books
...
This was soon followed by discovery of high
temperature superconductivity at 90 K2 in another oxide, Y-Ba-Cu-O, whose structure was shown in
Module 1
...
The Module contains:
Background
Meissner Effect
The Critical Field, H c
Theory of Superconductivity
Discovery of High Temperature Superconductivity
Mechanism of High Temperature Superconductivity
Applications
Summary
Suggested Reading:
Principles of Electronic Ceramics, by L
...
Hench and J
...
West, Wiley
1 Bednorz and K
...
Muller, Z
...
B, 64, 189 (1986)
2 M
...
Wu et al, Physical Review Letters, 58, 908 (1987)
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...
1 Background
Superconductors are materials which, when cooled below room temperature, exhibit a sudden drop
of electrical resistance of the material to exactly zero at a temperature called as critical temperature,
TC
...
Most first generation superconductors were elemental metals and metallic alloys
...
However, most of these are superconducting at temperatures below 30 K
...
For instance as
shown in figure 1, while in the non-superconducting (normal) regime, the electronic heat capacity is
T, at TC, it shows an abrupt jump and does not remain
proportional to the temperature i
...
C v
T-aT where a is a constant
...
Figure 7
...
They explained superconducting current as a superfluid of Cooper
pairs, pairs of electrons interacting through the exchange of photons
...
1 J
...
N
...
R
...
Rev
...
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Module 7: High temperature Superconductors
Meissner Effect
7
...
It was discovered by W
...
Ochsenfeld in Germany in 1933
...
The field penetrates up to a very
small depth decaying exponentially and is of the order of 100 nm or so, and is called as London
penetration depth, l
...
In a perfect conductor, an arbitrarily large current can be
induced, and the resulting magnetization exactly cancels the applied field giving rise to χ = -1
...
So when the material is in normal state at T >TC, the flux lines would
penetrate but when it reaches superconducting state at T
...
Figure 7
...
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Module 7: High temperature Superconductors
The Critical Field, H c
7
...
The field follows a parabolic form with temperature T and is given as
where H 0 is a constant while TC is the critical temperature i
...
the onset of the superconductivity
...
The difference between type-I and type-II superconductors is that while type-I superconductors
exhibit a distinct boundary between superconducting and normal state, type-II superconductors show
a region of mixed normal and superconducting states
...
Figure 7
...
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Module 7: High temperature Superconductors
Theory of superconductivity
7
...
Initial theories proposed a qualitative two conduction mechanisms i
...
a
normal fluid of electron and another superconducting fluid of electrons
...
An appropriate theory of superconductivity was proposed by J
...
H
...
R
...
Following this work, the theory has been called as BCS theory
...
While earlier it was thought that in the superconducting state, the electrons do not
interact with the lattice atoms destructively
...
The theory makes an important assumption that there exists an attractive force between the
electrons in typical type-I superconductors, which is due to Coulombic attraction between the
electron and the lattice
...
4
...
This is why it
is necessary that superconductivity is essentially a low temperature phenomenon
...
4 Formation of Cooper pair in a superconducting
material (Source: Oak Ridge National Laboratory online
reports)
7
...
1 Experimental Validation
The experiments were performed on the basis that if electrical conduction in mercury was purely
electronic, it should have no dependence on the nuclear masses
...
5)
...
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BCS theory based on electron-lattice interaction leading to the formation of Cooper pairs
...
At the end, one can observe that it is quite remarkable that an electrical
phenomenon such as the transition to zero electrical resistance should be associated with the purely
mechanical process of the lattice
...
5 Dependence of the critical temperature on the
atomic mass A (Reproduced from E
...
, Physical
Review, 78, pp 487 (1950))
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5 Discovery of High Temperature Superconductivity
In 1986, 75 years after the discovery of superconductivity, George Bednorz and Karl Müller at IBM,
Zurich demonstrated superconductivity in a perovskite structured lanthanum based cuprate oxide
which showed a TC of 35 K for which the inventors also won Physical Noble prize in 1987
...
This was realized by replacement of La by Y to give rise to YBa 2 Cu3 O 7-x (YBCO) which showed aTC
of ~92 K as first shown by Wu and his students at University of Alabama, Huntsville in 1987
...
e
...
15
...
6, when structure of YBCO changes from orthorhombic to
tetragonal
...
A list of important oxide superconductors is shown below with their structures and transition
temperatures
...
As you can see that
critical temperature is dependent very strongly on how the chemical substitutions are made into the
parent structure
...
T c (K)
Crystal
structure
YBa 2 Cu3 O 7
92
Orthorhombic
Bi 2 Sr2 CuO6
20
Tetragonal
Bi 2 Sr2 CaCu 2 O 8
85
Tetragonal
Bi 2 Sr2 Ca2 Cu3 O 6
110
Tetragonal
Tl2 Ba2 CuO6
84
Tetragonal
Tl2 Ba2 CaCu 2 O 8
108
Tetragonal
Tl2 Ba2 Ca2 Cu3 O 10
125
Tetragonal
TlBa2 Ca3 Cu4 O 11
122
Tetragonal
HgBa 2 CuO4
94
Tetragonal
HgBa 2 CaCu 2 O 6
128
Tetragonal
HgBa 2 Ca2 Cu3 O 8
134
Tetragonal
Compound
Y-based
Bi-based
Tl-based
Hg-based
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...
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Module 7: High temperature Superconductors
Mechanism of High Temperature Superconductivity
7
...
There has been
intensive research on developing a theory for high temperature superconductivity and consequently a
few mechanisms have been proposed, out of whch following two are well accepted
...
Spin fluctuation tests yield information on the symmetry of the pairing wave function which,
for cuprates, should be of the type d x 2-y 2
...
e
...
However, both of these models do not fully explain the high temperature superconductivity and there
is no clear consensus
...
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Module 7: High temperature Superconductors
Applications
7
...
Here, the huge magnetic field that can be
sustained by the superconductors is used to levitate and propel the trains
...
6 Magnetically levitated train in Japan
(Ref:
http://en
...
org/wiki/Maglev_%28transport%29)
Another application is magnetic resonance imaging (MRI)
...
These species then release this energy
at a certain frequency which can be detected and displayed in the form of an image by a computer
...
7
MRI image of a human knee
(Ref:
http://en
...
org/wiki/File:MR_Knee
...
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Zero resistance is potentially a fantastic property to have for conductors but since transition
temperatures are below RT, the commercial applications are not financially viable
...
However, a few firms in
US are trying to make taped wires where YBCO is grown on top of a nickel tape which provides it
the strength without compromising on the TC and current carrying capacity
...
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...
While most superconductors of early generation
were low TC materials, next generation superconductors, layered oxides based on perovskite
structure, were called as high TC as materials showing TC in the excess of liquid N 2 temperature
...
While the BCS theory explains the origin the superconductivity in the low TC superconductors
quite well, a well agreed theory for the high TC ceramic oxides is yet to evolve
...
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...
More recently there has
been another ordering mechanism proposed which is called as ferrotoroidic ordering
...
For example, it could occur in paraelectric ferromagnetic
materials or it can be mediated by other parameter such as strain
...
More recently, multiferroic materials have become of tremendous interests because of
potential device applications
...
However, there are challenges in finding a
material that would act as a perfect multiferroic
...
There are problems with respect to their fabricability, while their
transition temperatures are often impractical
...
In this module we discuss some of the
basic aspects of multiferroics and a few multiferroic materials
...
A
...
Phys
...
B, 104, 6694-6709 (2000)
M
...
Phys
...
Phys
...
Eerenstein, N
...
Mathur and J
...
Scott, Nature, 442, 759 (2006)
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...
1 Ferroic Material
Multiferroics are materials which possess more than one type of primary ferroic ordering in a
single phase
...
The critical temperature, in general, is also accompanied with a symmetry breaking
...
Another proposed ordering mechanism is ferrotordoicity which
exhibit arrangement of magnetic vortices in an ordered manner, called tordoization
...
1 explains the various possible scenarios
...
Magnetoelectric materials are those materials which
are simultaneously
electrically and magnetically polarizable, while Multiferroics are strictly those
materials which show
ferroelectric and ferromagnetic ordering
...
1 Classification of multiferroic and magnetoelectric
materials
While, strictly speaking multiferroism means only for those materials in which there is coupling of
more than one order parameter, now a days, researchers have also started including
antiferromagnetism as well as ferrimagnetism also with multiferroic materials
...
The examples are TbMnO 3 , TbMn 2 O 5 , HoMn2 O 5 , LuFe2 O 4 , BiFeO3 , BiMnO3 and
YMnO3
...
g
...
Given that the multiferroic materials show more than one ferroic ordering, the envisaged
applications are numerous
...
Recent reports also classify the multiferroics into Type I and Type II multiferroics
...
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multiferroics are those materials in which the source of ferroelectricity and magnetism is different
and the effects are fairly independent of each other, albeit with a small degree of coupling
...
However, the magnitude of polarization is
these materials remains very small, typically less than 10 -2 µC/cm2
...
These reviews have also been
source of much of the basic information in this module
...
A
...
Phys
...
B, 104, 6694-6709 (2000)
2 M
...
Phys
...
Phys
...
Eerenstein, N
...
Mathur and J
...
Scott, Nature, 442, 759
(2006)
4 D
...
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Module 8: Multiferroic and Magnetoelectric Ceramics
Historical Perspective
8
...
Rontgen found that a dielectric when moved in an electric field, became
magnetized and conversely it became polarized when moved in a magnetic field
...
The term magnetoelectric
was first used by Debye in 1926
...
Subsequently the research was carried on various materials and it is
now established that more than 80 compounds including Ti2 O 3 , GaFeO3 , boracites, phosphates
showed magnetoelectric effect
...
Subsequently many studies were made on various boracite
compounds
...
This was followed by studies on mixed perovskites, essentially solid solutions of two perovskite oxide
compounds
...
One of first such compounds to be discovered was a solid solution of Pb
and Pb (Mg 1/2 W 1/2 )O 3
...
Some other
candidates were lead based Fe or Co doped tungstates or tantalates which showed ferroelectricity
and antiferromagnetic ordering
...
Subsequently, the research focus was on other perovskite materials which are either manganites or
ferrites and have been more promising than previously research materials and will be discussed later
in this module
...
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Module 8: Multiferroic and Magnetoelectric Ceramics
Requirements of a Magnetoelectric and Multiferroic Material
8
...
For instance, for ferroelectricity, a material must be non-centrosymmetric to possess
spontaneous electrical polarization and there are only a limited number of point groups (out of 32)
which allow an unique polar direction
...
Out of these, 13 point groups allow occurrence of both the properties simultaneously
...
Electrically, while a ferroelectric material must be an insulator, it is not a constraint for a
ferromagnetic material
...
However, there are a few magnetic oxides, such as half metallic magnets and ferrimagnetic oxides
which show reasonable spontaneous magnetism while simultaneously being semiconducting or
insulating
...
In contrast, magnetic systems require d-orbitals to be partially occupied for
magnetic ordering to develop
...
Among type I multiferroics, multiple mechanisms of ferroelectricity have been proposed 5
...
In contrast, in
materials like BiFeO3 , ferroelectricity is believed to arise due to the ordering of lone pairs of Bi in one
direction such as [111]
...
e
...
This is seen in materials like TbMn 2 O 5
...
Another factor that could be analyzed is the size of small cation, especially in the perspective of
perovskites
...
Another contrast between ferroelectric and ferromagnetically ordered systems is that the way
structure is distorted
...
The latter is almost absent in most ferroelectrics as it
has been postulated that Jahn-Teller distorted structure may have less driving force for off-center
displacement of B-ions in the octahedra
...
e
...
On the other
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...
So, in summary, while there is no constraint on various material parameters which prevent materials
from being multiferroic i
...
simultaneously ferroelectric and ferromagnetic, a multiferroic does not
possess either time reversal or space inversion symmetry
...
Khomskii, Physics, 2, 20 (2009)
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...
4 Magnetoelectric Coupling
Landau theory describes the magnetoelectric effect in a single phase material through expansion of
the free energy expression as
(8
...
Here e and µ are the dielectric
permittivity and magnetic permeability respectively
...
1)
are the temperature dependent electrical polarization, Pi s , and magnetization, Mi s
...
The next two
terms consisting of ß ijk and γijk are third rank tensors and represent higher order coupling
coefficients
...
1) with respect to electric and magnetic fields respectively leads to
polarization and magnetization which are as follows:
and
(8
...
This coefficient basically quantifies the
dependence of polarization on magnetic field or of magnetization on the electric field
...
In addition to direct coupling, there may be instances of indirect coupling mediated by strain
...
However, more
recently, in cubic SrMnO 3 and EuTiO3 , strain mediated ME effect is observed in single phase
...
However, such measurements do not provide any mechanistic
insight into the coupling constant
...
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...
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Module 8: Multiferroic and Magnetoelectric Ceramics
Type I Multiferroics
8
...
As mentioned earlier, Type I multiferroics are the
materials which have different sources of ferroelectricity and magnetism with the two effects being
quite independent of each other
...
In this section, we will mainly have a look at most studied compounds:
Bismuth Ferrite (BiFeO 3 )
Bismuth Manganite (BiMnO 3 ) and
Hexagonal Manganites
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...
5
...
7
In its ferroelectric state, as shown in Figure 8
...
965 Å and
= 89
...
8 Above the Curie temperature, the structure changes to a high symmetry cubic
phase
...
2 Schematic diagram of the crystal structure of
BiFeO3
The material has been shown to be ferroelectric at room temperature in both single crystal and thin
film form high remanent polarization, more than 50 µC/cm2
...
11 On the other hand,
while magnetic character of pure phase in single crystal form is antiferromagnetic, there have been a
few controversies on magnetism in thin films
...
12 The material is also prone
to containing defects as well as difference valencies of Fe which can alter the material properties
...
However, properties of BiFeO3 can also be altered by making chemical
substitutions
...
These effects are attributed to the breaking of spin spiral upon doping
...
6 R
...
Gerson, and W
...
James, Solid State Commun
...
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7 P
...
Polomska, I
...
Szymanski, J
...
C 13, 1931 (1980)
8 G Catalan and J
...
Scott, Advanced Materials, 21, 2463 (2009)
...
Lebeugle, D
...
Forget, M
...
Bonville, J
...
Marucco, and S
...
Rev
...
10 J
...
B
...
Zheng, V
...
B
...
Liu, D
...
Vaithyanathan, D
...
Schlom, U
...
Waghmare, N
...
Spaldin, K
...
Rabe, M
...
Ramesh, Science 299, 1719 (2003)
11A
...
Simoes, A
...
M
...
S
...
S
...
Longo, and J
...
Varela, J
...
Phys
...
12H
...
Bibes, A
...
Bouzehouane, E
...
Khodan, J
...
Contour, S
...
Wyczisk, A
...
Lebeugle, D
...
Viret, Applied Physics Letters 87 (7),
072508 (2005)
...
A
...
A
...
Kopcewicz, M
...
V
...
Borisov, W
...
M
...
Lopes, Y
...
Pogorelov, J
...
Araujo, R
...
Rubinger, N
...
Sobolev, J
...
Vieira, and A
...
Kholkin, J
...
Magn
...
321, 1692 (2009)
...
Mukherjee, R
...
Garg, V
...
Bhargava, J
...
Phys
...
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Module 8: Multiferroic and Magnetoelectric Ceramics
Type I Multiferroics
8
...
2 Bismuth Manganite (BiMnO 3 )
Bismuth manganite is an interesting multiferroic material with a perovskite structure
...
The material shows ferromagnetic
ordering below 105 K attributed to the orbital ordering of B-site ions i
...
Mn 3+ ions and a
magnetization of 3
...
15 The material has a perovskite triclinic structure which
changes to monoclinic structure at ~450 K and then to a non-ferroelectric orthorhombic phase at
~770K
...
The bulk form of material has been shown to exhibit multiferroic
behavior near 80 K17 and negative magneto-capacitance effect in the vicinity of magnetic transition
temperature (T m ) with -0
...
16 The problem which arises
with this material is that it requires high pressures in bulk form17
...
18
15H
...
Atou, and Y
...
Kimura, S
...
Yamada, M
...
Takano, and Y
...
Moreira dos Santos, S
...
R
...
S
...
K
...
N
...
Rao, Solid
State Communications, 122, 49-52 (2002)
18W
...
D
...
F
...
D
...
Phys
...
87, 101906 (2005)
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...
5
...
These materials simultaneously
exhibit ferroelectricity and antiferromagentic ordering of magnetic Mn ions
...
20 In spite of having a chemical formula, ABO 3 ,
similar to the perovskites, hexagonal manganites have altogether different crystal and electronic
structure
...
R ions, on the other hand, have
7-fold coordination unlike the cubic coordination in perovskites
...
Figure 8
...
Figure 8
...
21
Crystal field level scheme of Mn 3+ ions in hexagonal RMnO 3 is also different from that of Mn 3+ ions
with octahedral coordination
...
As a
result, four d-electrons of Mn 3+ occupy two lowest lying doublets and unlike Mn 3+ ion in octahedral
coordination, there is no degeneracy present
...
22
Hexagonal RMnO 3 are found to possess considerably high ferroelectric transition temperature (> 500 K)
...
Table 1 lists the ferroelectric and
magnetic transition temperatures, spontaneous polarization (PS ) and effective paramagnetic moment
µ eff of some common RMnO 3 along with their structural parameters
...
In case of YMnO3 , it was observed that off-centering of Mn 3+ ion from the center of
22
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...
Apparently it turns out that R ions (Y, here) contributes most toward ferroelectricity by having large R-O
dipole moments
...
Similar to BO6 octahedra in perovskite oxides (ABO3 ), MnO 5
trigonal biprism in RMnO 3 , tilts and rotates in order to ensure closest packed structure
...
22 Since the mechanisms of ferroelectric and magnetic ordering in the above materials
are quite different in nature, giant effect of magnetoelectric coupling is understandably not present
...
1 Lattice parameters, Neel temperature (T N) and ferroelectric Curie
(T C ) temperature, effective paramagnetic moment (µeff ) and spontaneous
polarization (P s ) of some common hexagonal manganites
...
833
11
...
139
11
...
142
11
...
112
TmMnO 3
T C (K)
PS
µ eff (in
µB)
(µC
...
5
76
873
11
...
6
11
...
5
-
6
...
37
86
>573
8
...
1
YbMnO 3
6
...
36
87
993
6
...
5
LuMnO 3
6
...
37
96
>750
5
...
5
-
19S
...
Pirogov, M
...
, Nature 451, 805 (2008)
20H
...
Yakel, W
...
Koehler, E
...
Bertaut, et al
...
Crystallogr
...
Zaghrioui, V
...
A
...
, Physical Review B 78, 184305 (2008)
22D
...
Khomskii, Journal of Magnetism and Magnetic Materials 306, 1 (2006)
23J
...
Park, 1st APCTP Workshop on Multiferroics (2008)
24K
...
Malm, N
...
, Materials Chemistry and Physics 112, 1029 (2008)
25L
...
Wang, S
...
Feng, J
...
Zhu, et al
...
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Module 8: Multiferroic and Magnetoelectric Ceramics
Type II Multiferroics
8
...
Moreover, very strong coupling between ferroelectric and
magnetic order parameters has also been observed
...
reported 26 presence of
spontaneous polarization in the magnetized state of the TbMnO 3
...
It is in the commensurate state between 7 and 27 K, the
material show ferroelectricity
...
27 Subsequently variety of other materials have also been investigated such as Ni 3 V2 O 8 ,
MnWO 6 showing this effect
...
26T
...
Goto, H
...
Ishizaka, T
...
Tokura, Nature, 426, p55, (2003)
27N
...
Park, P
...
Sharma, J
...
Ahn, S
...
Cheong, Nature 429, 392-395
(2004)
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...
7 Two Phase Materials
Another method to for achieving high degree of magnetoelectric coupling is to mix ferroelectric (e
...
BaTiO3 ) and ferromagnetic (e
...
CoFe2 O 4 ) materials and utilize the strain between two phases to
introduce magneto-electric coupling
...
e
...
For a few systems, the data is shown in the table
below
...
2
Magnetoelectric coupling constant data for selected two-phase
magnetoelectric systems
Type of system
Materials
Composite28
BaTiO 3 and CoFe 2 O4
Laminated composite 29
Terfenol-D in polymer
matrix
and
PZT
in
polymer matrix
3,000
Laminate 30
Terfenol-D/PZT
4,800
Laminate 31
La 0
...
3 MnO3 and PZT
Laminate 32
NiFe 2 O4 and PZT
Epitaxial
Coupling
constant
(mV/cm-Oe)
50
60
1,400
thin
film BaTiO 3 and CoFe 2 O4
--
thin
film BiFeO3 and CoFe 2 O4
--
structures33
Epitaxial
structures34
In two phase structures, as evident from some of references, one can create large changes in the
magnetization owing to strain due to the ferroelectric phase transition of the ferroelectric material
during film growth or one can also attempt to alter the magnetic structure by applying a field the
piezoelectric material which thereby generates a strain in the magnetic material in the vicinity
...
M
...
G
...
R
...
H
...
Mater
...
-W
...
, Appl
...
Lett
...
30N
...
-W
...
Zhai, and Y
...
Phys
...
, 84, 3516–-3519 (2004)
...
Srinivasan, Phys
...
B 65, 134402 (2002)
...
K
...
, Appl
...
Lett
...
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...
Zheng et al
...
34F
...
, Nano Lett
...
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Module 8: Multiferroic and Magnetoelectric Ceramics
Summary
Summary
Multiferroic and magnetoelectric materials are a new class of materials which show interdependence
of magnetic and electric properties on each other
...
While a multiferroic material has to be a single phase material, magnetoelectric materials
can be single phase as well as a mixture of two phases showing interface mediated magnetoelectric
coupling
...
However, the applications are
realized yet due to lack of materials and difficulty in achieving the desired effects in the available
materials
...
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...
Both of these aspects depend heavily upon how these ceramics are processed
...
e
...
One of the major differences between thin films process and bulk processes is that typically
bulk processes are treated as equilibrium processes while thin films processes are generally nonequilibrium phenomenon thereby altering the kinetics of phase formation
...
In this section, we will focus on a brief review of most common synthesis methods for
fabricating the electroceramics, in both bulk as well as thin films form
...
David Kingery, H
...
Bowen, Donald R
...
J
...
M
...
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Module 9: Synthesis Methods
Bulk Preparation Methods
9
...
Figure one show a typical flow chart of the bulk preparation
processes and are shown below
...
Figure 9
...
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Module 9: Synthesis Methods
Bulk Preparation Methods
9
...
1 Selection of Raw Materials
One of the important factors in the fabrication of bulk ceramic is the choice of raw materials i
...
starting ingredients
...
It is
important to choose powders with the highest possible purity, as it is evident from defect chemistry
that impurities, even in small quantities, lead to significant changes in the electrical characteristics
...
It may vary from 0
...
One can
conduct the thermogravimetric analysis (TGA) to verify the weight change upon heating due to
moisture loss to accurately determine the moisture content
...
While smaller particle size promotes their reactivity,
tendency of agglomeration also increases as the particle size decreases
...
9
...
2 Powder Preparation
After carefully weighing the powers in desired proportions depending on the desired stoichiometry,
the powders are mixed together
...
Care should be taken to avoid any contamination at this stage while breaking the agglomerates as
effectively as possible
...
The mixed powders can also be prepared using chemical route if
the starting precursors are chemical precursors such as alkoxides, nitrates or citrates by dissolving
them in appropriate solvents such as methoxyethanol, glacial acetic acid or citric acid, followed by
mixing under controlled conditions
...
Again,
thermal analysis of the mixture gives an idea of the temperatures needed for drying and calcination
...
1
...
Calcination is a high temperature phenomenon in which powders are heated to
high temperature in a chosen ambience and then held for a long time to achieve the phase
formation
...
Calcination also reduces the diffusion requirements and shrinkage that
occur during sintering and densification stages
...
1
...
Sometimes binders are also required to allow the particle to remain bound to
each other
...
Wet processes for shaping include slip casting and
band casting where a slurry is cast into molds followed by drying
...
1
...
Typically this
stage is characterized by the use of high temperatures and accompanied by shrinkage during the
process
...
This
requires good control of heating and cooling rates during sintering process
...
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sintered compacts are provided finishing operations either by machining, glazing or metalizing
...
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Module 9: Synthesis Methods
Thin Film Preparation Methods
9
...
Consequently, many areas of interest have emerged such
as ferroelectric devices, optoelectronic devices, CMR devices etc
...
1)
...
Although in the past these materials have been used as bulk materials for many
applications, it is the thin film form of these oxides which makes them more attractive for various
applications
...
Superconducting oxides such as YBCO (yttrium barium copper oxide) for SQUIDs2 or
ferroelectric oxide such as PZT (lead zirconium titanate) for non-volatile memory applications3 show
excellent potential in device applications
...
Significant progress has been made
towards studying the growth mechanisms of the epitaxial thin films by various techniques such as
RHEED, LEED, and STM
...
But there are some issues which are yet to be tackled effectively
...
Other issues to be dealt with are
reproducibility of composition in complex oxide deposition, phase stability, high quality of epitaxy and
atomic level flatness of the films whilst maintaining the commercial viability of the device
...
1
Some important metal oxides and their applications
Property
Materials
Application
High temperature YBa2 Cu3 O7 ,
Power
transmission,
superconductivity Bi mSr 2 Can-1 Cun O2n+m+2 Communications,
Microwave Devices
Ferroelectricity,
Piezoelectricity
Pb(Zr x Ti1-x )O3 ,
Optics
Nb 2 O5 -SiO 2 -Na 2 O-
SrBi 2 Ta 2 O9
Ba2 O2-TiO 2
Magnetism
(La x Ca1-x )MnO3 ,
Ferrites, Garnets
Thermal
coatings
barrier ZrO2 , Al 2 O3
Memories,
Devices,
Actuators
Data
All
optical
devices
Storage
Sensors,
switching
Magnets (hard and soft),
Tunnel junctions
Heat resistant coatings e
...
for heat exchangers
Over the years various processes have been developed for the deposition of metal oxide thin films
...
PVD processes include laser ablation,
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...
There have been
many extensive reviews on the deposition of epitaxial oxide films
...
It still remains a major challenge for most of the thin film processing groups to
bring the deposition temperature down to acceptable limits
...
In this section, different processing techniques are discussed with primary emphasis on the sputtering
and laser ablation processes
...
G
...
A
...
Phys
...
H
...
C
...
J
...
Clarke, Nature, 352, 482 (1991)
3 J
...
Scott and C
...
Paz de Araujo, Science, 246, 1400 (1989); P
...
Larsen, R
...
A
...
M
...
Terashima, Y
...
Ijima, K
...
Hirata, K
...
Kamigaki,
and H
...
Rev
...
, 65, 2684 (1990)
5 A
...
Y
...
W
...
-P
...
Catana, E
...
Gerber, Appl
...
Lett
...
Weiss and M
...
Rev
...
Venkatesan, Thin Solid Films, 216, 52 (1996)
9 R
...
Somekh, Z
...
Barber, and J
...
Evetts, in ‘Concise Encyclopedia of
Superconducting and Magnetic Materials’, Pergamon Press, Oxford, p431 (1992)
10S
...
Krupanidhi, J
...
Sci
...
, A10, 1569 (1992)
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...
2
...
In this process the target atoms are ejected by the bombardment of energetic
species (usually inert gas ions) on it, as shown in Figure 9
...
The sputtered atoms travel through the
plasma and undergo many collisions with the plasma species (electrons, neutrals, gas ions, and
metals ions) and subsequently deposit on a strategically placed substrate
...
Sputtering process can employ dc or rf power
depending upon the electrical conductivity of the target
...
Usually a magnetic
field (use of magnetron target) is employed to enhance the efficiency of plasma utilisation and sputter
efficiency
...
Sputtering process using a magnetron is
called magnetron sputtering
...
11
Figure 9
...
Reactive sputtering in a mixture of oxygen and argon has been a very important process for the
deposition of oxides since the 1970s
...
12
Application of RF sputtering facilitated the use of an oxide as the starting material and target, but
given the variation in the sputter yields of different elements, the resulting films were not found to be
stoichiometric
...
Work has shown that the nature of growth species changes from atomic to molecular as the
sputtering gas pressure increases
...
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morphological changes e
...
formation of various features such as pits, ripples, cones and craters on
the film surfaces
...
These studies also predicted that resputtering can be
minimized by thermalizing (reducing the energy of energetic species originating from the target) the
energetic species in the plasma either by sputtering in high pressure environment or by off-axis
sputtering
...
Some of the considerations in selecting a particular target are target purity, deposition rates, and
stoichiometry related issues
...
The main issues which
are to be addressed are stoichiometry control, epitaxy, nature of the substrates (can involve use of
buffer layers), and oxygen activity during the deposition
...
1%
...
20 However, one of the major drawbacks is slow deposition rate
which makes this process quite expensive
...
11B
...
(1980)
12J
...
Erskine and A
...
Vac
...
Tech
...
M
...
Sci
...
, 4, 149 (1991)
14S
...
Rossanagel and J
...
Cuomo, in ‘Thin Film Processing and Characterisation of High
Temperature Superconductors’, eds
...
M
...
, Colton R
...
, and Feldman L
...
, American
Institute of Physics Conf
...
, 165, AIP, NY, 106 (1988)
15D
...
Kester and R
...
Mater
...
, 8, 1928 (1993)
16R
...
Somekh, in ‘Physics and Materials Science of High Temperature Superconductors’,
Dordrecht, The Netherlands, Kluwer Academic Publishers, 327 (1990)
17E
...
Tomlinson, Z
...
Barber, G
...
Morris, R
...
Somekh, and J
...
Evetts, IEEE Trans
...
Magn
...
Geerk, G
...
Meyer, Mater
...
Rept
...
Linker, X
...
Xi, O
...
Li, and J
...
, 69, 249 (1989)
20O
...
I
...
B
...
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Module 9: Synthesis Methods
Thin Film Preparation Methods
9
...
2 Pulsed Laser Ablation
In this process, a pulsed laser beam is focussed onto the surface of a target at an oblique angle so
that a substrate can be placed directly in front of the target surface (see Fig
...
3)
...
A mixture of Ar and O 2 or pure O 2 gas is used for the
deposition of oxides with a pressure range of 20-50 Pa in most cases
...
Most commonly used lasers are UV excimer lasers such as ArF
(193 nm), KrF (248 nm), XeF (351nm) and as it can be seen from these lasers, the wavelength of the
laser is controlled by lasing gas composition
...
This technique has been successfully employed for deposition of high Tc superconductors,
ferroelectric thin films, and other oxide films
...
21 Among some recent studies, laser ablation
has been successfully used to produce epitaxial cuprate films by layer by layer growth mode
...
Several reviews has been written on the growth of high Tc superconducting films9,22
and
ferroelectric thin films
...
3 Schematic diagram of a pulsed laser deposition
setup
The most important aspect of laser ablation is the presence of a low number of process variables as
compared to other physical and chemical vapour processes
...
The important parameters to be controlled are laser power density, substrate target distance,
and ambient oxygen pressure
...
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Some of the advantages of this process are
excellent control over stoichiometry using single-phase multi-elemental target,
efficient target use,
it can be easily coupled with in-situ growth monitoring methods such as RHEED,
it can work at high working pressures,
possibility of depositing multilayers,
possibility of depositing films at higher deposition rates, and
absence of bombardment effects on the growing film unlike in sputtering
...
19G
...
X
...
Meyer, O
...
Geerk, Solid State Commun
...
Auciello, A
...
Kingon, and S
...
Krupanidhi, MRS Bulletin, June, 25 (1996)
21D
...
Chrisey and G
...
Hubler, Pulsed Laser Deposition of Thin Films, 'Wiley-Interscience
Publication, New York (1994)
22T
...
Jackson and S
...
Palmer, J
...
D: Appl
...
, 27, 1581 (1994)
23O
...
Ramesh, MRS Bulletin, June, 31 (1996)
24D
...
Misra and S
...
Palmer, Physica C, 176, 43 (1991)
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...
2
...
Vacuum evaporation is a low energy process as
compared to sputtering because energy of species reaching the substrate is very low, of the order of
0
...
For deposition of oxides, reactive evaporation is a very important method
...
The critical factors involved in the evaporation processes
are vacuum level and the background oxygen pressure (pO2 ) which are instrumental in controlling
the impurity level and stoichiometry of the film
...
25 Use of active oxygen
has been shown to improve the film quality considerably
...
26 Epitaxial
DyBa2 Cu3 O 6+x films were grown by MBE and it was shown that by controlling the local nucleation
rate and lateral growth rate, it was possible to achieve layer by layer growth
...
Although these methods are complex in nature and expensive, they
offer advantage of in-situ growth monitoring by RHEED/LEED
...
Ritter, J
...
Sci
...
, 12, 225 (1966)
26H
...
Matsukara, IEEE Trans
...
, 27, 1009 (1991)
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...
2
...
First, a homogeneous solution is prepared by mixing the stoichiometric quantities of
precursors of desired substances (typically nitrates, acetates, citrates and metal alkoxides) in
solvents (typically glacial acetic acid, citric acid, 2-methoxethanol)
...
It is also possible that initially each precursor needs to be
dissolved in separate solvents which are then mixed together later
...
Use of stabilizing agents such
as ethanolamine may be necessary to avoid precipitation of the ingredients into a complex
...
The second stage involves spin coating of solution on appropriate substrates
...
For the first coating, a few drops of the solutions are
placed on top of the substrate kept on vacuum enabled chuck of a spin coater such that the
substrate in completely covered with the solution
...
4) so that excess solution is wiped away and a thin layer of solution remains on top of the
substrates
...
These temperatures can be determined by carrying out thermal
analysis of the solution
...
After achieving the desired thickness,
the final film is annealing in a selected environment (typically air, nitrogen or oxygen) for a fixed
period of time to achieve film crystallization
...
Figure 9
...
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...
2
...
The reaction vapours
and gas are flowed over a substrate, usually heated between 400-1000°C, produce the film species
which subsequently deposit on the substrate as shown by Figure 9
...
The residual gases and volatile
products are pumped away from the system
...
However
its further innovations like metallo-organic chemical vapour deposition (MOCVD) and plasma
enhanced MOCVD (PE-MOCVD) processes offer advantages of depositing films at relatively lower
temperatures as compared to those employed in PVD processes
...
CVD
processes also have the advantage of depositing films over large area substrates
...
Some other
problems with these processes are control of composition, accurate knowledge of phase diagrams,
toxicity of precursors
...
There are several reviews existing on the CVD
deposition of the high TC superconductors 27
...
29
...
5 Schematic diagram of metallo-organic chemical
vapour deposition (MOCVD) deposition process; from ref
...
27M
...
Molsa and L
...
Sci
...
, 6, 627 (1993)
28R
...
A
...
D
...
Lu, R
...
Feigelson, R
...
Route, and J
...
Young, J
...
Growth, 128, 781 (1993)
29M
...
J
...
Dormans, MRS Bulletin, June, 37 (1996)
30S
...
Dey and P
...
Alluri, MRS Bulletin, June, 44 (1996)
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...
2
...
It involves the precipitation of a crystalline phase as the film from a
supersaturated melt onto a substrate which acts as both the template for epitaxy and a physical
support for the film
...
However,
for film growth as the liquid cools down, some of the constituents precipitate out of the solution and
grow epitaxially on a suitable substrate
...
LPE offers several advantages over other epitaxial deposition processes such as low-cost
operation, good control of composition and thickness, and faster deposition rates
...
LPE could be an attractive process for
making coated conductors based on YBCO films and is being extensively studied
...
Yamada, Supercond
...
Tech
...
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Module 9: Synthesis Methods
Thin film deposition: Issues
9
...
To reduce the deposition temperature for depositing epitaxial films is the
foremost challenge for most of the processing techniques, barring some chemical processes
...
Choice of device compatible and cheap substrates and
deposition over larger area substrates are the main bottlenecks in the commercial success
...
In this section, it has been attempted to
briefly elucidate these problems and the ongoing efforts to handle them
...
3
...
7-0
...
2-0
...
These temperatures are very high when
considering the processing compatibility with semiconductor circuit processing
...
Usually if the deposition temperature is lowered, then epitaxy deteriorates
...
It has been shown for ion beam sputtered PZT films that
bombardment of the growing film by low energy ions in ion beam sputtering and use of active oxygen
species helps to reduce the crystallization temperature of PZT to some extent
...
9
...
2 Phase Stability and Stoichiometry Control
Because of the thin film growth processes in general being non-equilibrium in nature, it becomes very
important to know about the phase evolution under a given set of process parameters and how a
desired phase can be obtained if they are altered accordingly
...
Most of the perovskite oxides are not only
complex oxides, but they also have some specific oxygen level for desired properties e
...
superconducting oxides like YBCO
...
The work of Feenstra et al
...
Chemical complexity of these oxides also poses problems regarding the correct stoichiometry of the
films
...
This problem has been tackled by using multi-elemental
targets
...
Laser ablation has emerged as one of the most promising processes to address this problem with
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...
33
9
...
3 Substrate Effects
The choice of substrate is a very important factor in the growing epitaxial oxide films
...
SrTiO3 (100)
and MgO (100) have proven to be useful substrates for growing highly epitaxial films of YBCO
...
They
are also expensive
...
but the search is still on for the best which is inexpensive, should not react with the film at the
film deposition temperature and should be able to give high quality epitaxial films with excellent
properties
...
This problem has forced researchers to look for other alternative
electrodes such as LSCO, YBCO etc
...
These problems have been, to some extent, alleviated by
the use of buffer layers
...
34 Various materials have been used for this purpose such as
cerium oxide (CeO 2 ) for YBCO35, and strontium ruthenium oxide (SrRuO 3 ) for PZT 36 and resulted
in remarkable improvement in the device performance
...
3
...
15-20°
...
The better the epitaxy of the film, better is the
surface roughness
...
37 Stoichiometry control has also an
important role to play in controlling the film structure
...
38 The effect of atomic oxygen on the growth and surface roughness of YBCO films
grown on SrTiO3 (100) by MBE process, has also been studied
...
So there are challenges to produce epitaxial oxide films with smoother surfaces and
perfect orientation which require careful control of processing conditions
...
Feenstra, T
...
Linemer, J
...
Budai and M
...
Galloway, J
...
Phys
...
Dam, J
...
Rector, J
...
Kars, and R
...
Morris, ‘Buffer Layers for High Temperature Superconductors’, in High Temperature
Superconductors, eds
...
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Module 9: Synthesis Methods
Summary
Summary
In the summary, depending on the application, electroceramics can be prepared by either in the bulk
form or in thin film form
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Thin films' processes vary vastly depending upon the
way the thin films are made
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The advantage of
thin film processes is that one can achieve the desired phase formation and structural perfection at
much lower temperatures without much use of the material
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file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture35(40)/35_12
Title: ELECTROCERAMICS
Description: This coursework consist of 41 lectures , Suitable for material science , mechanical,electrical students , chemistry. 1: Structure of Ceramic Materials 5 2: Defect Chemistry and Equilibria 7 3: Diffusion and Conduction in Ceramics 7 4: Linear Dielectric Ceramics 8 5: Nonlinear Dielectric Ceramics 6 6: Magnetism and Magnetic Ceramics 5 7: Superconducting Ceramics 1 8: Multiferroic and Magnetoelectric Ceramics 1 9: Synthesis Methods 1 Total number of equivalent lectures 41 total lectures. Total no of pages : 376.
Description: This coursework consist of 41 lectures , Suitable for material science , mechanical,electrical students , chemistry. 1: Structure of Ceramic Materials 5 2: Defect Chemistry and Equilibria 7 3: Diffusion and Conduction in Ceramics 7 4: Linear Dielectric Ceramics 8 5: Nonlinear Dielectric Ceramics 6 6: Magnetism and Magnetic Ceramics 5 7: Superconducting Ceramics 1 8: Multiferroic and Magnetoelectric Ceramics 1 9: Synthesis Methods 1 Total number of equivalent lectures 41 total lectures. Total no of pages : 376.