Lecture handout including QS - Department of Materials Science ...

Natural Sciences Tripos Part IA
MATERIALS SCIENCE
Course B: Materials for Devices
Name............................. College..........................
Dr Zoe Barber
Michaelmas Term 2013-14
14
IA

Contents
Introduction
1
Magnetic Materials
36
Synopsis
2
Diamagnetism, Paramagnetism
36
Text Books & Websites
3
Ferro-, Antiferro-, Ferrimagnetism
37
Ferromagnetic Materials
38
Liquid Crystals
4
Curie temperature
38
Nematic Liquid Crystal structure
4
Magnetocrystalline anisotropy
38
Polarized Light
6
Domain formation
38
Birefringence & Polarized Light Microscopy
8
Shape Anisotropy
39
The Michel-Levy Chart
10
Magnetostriction
39
Extinction Positions &
Domain wall, Bloch wall
39
Permitted Vibration Directions
11
Ferromagnetic Hysteresis
40
Determination of Sign of Birefringence
11
Aside on magnetic parameters
42
Birefringence in Liquid Crystals
11
Hysteresis loops: hard versus soft
42
Further types of Liquid Crystals:
Ferrimagnetism & Spinel Structure
43
Smectic & Chiral Nematic
12
Inverse Spinel Structure
43
Liquid Crystal Displays
14
Solid Ionic Conducting Materials
44
An Introduction to Polymer Structures
16
(a) Diffusion Current
45
Flexible chains: conformation
17
(b) Drift Current
46
How big is a polymer molecule
19
Nernst-Einstein Equation
46
Examples of common polymers
20
Solid State Ionic Conductors
48
Polymer Microstructure & Properties
21
Yttrium Stabilized Zirconia, YSZ
48
Crystallinity
22
δ-Bi 2 O 3
48
Applications
49
Dielectric Properties of Materials
23
1. Oxygen Concentration cell
49
Polarisation Mechanisms
23
Lambda Sensor
50
Polarisation & Capacitance
23
2. Oxygen Pump
51
Dipole formation & Symmetry
25
3. Fuel Cell
51
Piezoelectrics
27
The hydrogen economy
53
Pyroelectrics
28
Ferroelectrics
29
Glossary
54
Polarisation & Structure
29
Phase Transitions in BaTiO 3
30
Switchable Dipoles
31
Dipole ordering & Domains
32
Reversible Polarisation, hysteresis
33
Applications, memory device
34
PZT
35

BH1 Course B: Materials for Devices BH1
Course B: Materials for Devices
The functional properties of materials, such as magnetism, polarization and conduction, are intimately
related to their crystal structure. Furthermore, many of these properties are also anisotropic –
meaning that they are dependent upon the specific direction within a crystal structure. It is imperative
that we understand these structure–property links in order to understand fully materials properties,
and be able to control and optimize these properties. Only through such an understanding can we
hope to exploit the full range of possible device and materials applications, keep coming up with
smaller and smaller music systems, faster computers, and overcome some of the environmental issues
currently troubling the world.
The course begins with the remarkable properties of Liquid Crystals: somewhere between liquids
(isotropic, no long range order) and crystals (anisotropic and ordered). The anisotropy of liquid
crystals leads to important optical effects, which are exploited in display technology.

BH2 Course B: Materials for Devices BH2
Synopsis of Course B
1. Liquid Crystals & Polarized Light: rigid polymer molecules, nematic structures, the order
parameter. Plane polarized light and birefringence. Permitted vibration directions, optical path
difference & phase difference, Polarized Light Microscopy.
2. Birefringence in Liquid Crystals: the Michel-Levy chart, extinction positions, compensators.
Schlieren texture and disclinations. Smectic and chiral / cholesteric liquid crystals. Liquid Crystal
Displays.
3. An Introduction to Polymer Structure: repeated monomer units, long chain molecules,
methods of representation. Conformation, flexibility, molecular size. Side groups, tacticity,
microstructure & properties. Crystallinity in polymers.
4. Dielectrics: polarisation mechanisms, permittivity & dielectric constant, capacitance. Symmetry &
properties.
5. Polarisation; Piezo-, and Pyro-electrics: Piezoelectric motor & generator effect, pyroelectric
applications.
6. Ferroelectricity: polarisation & structure, switching, phase transitions in BaTiO 3
. Dipole
ordering & domains, domain walls, poling.
7. Ferroelectric Hysteresis & Applications: domain reversal & hysteresis loops. FE memory
devices, materials requirements. PZT phase diagram.
8. The Origin of Magnetism: electron orbitals & spin; dia-, para-, ferro-, antiferro- & ferrimagnetism.
Magnetocrystalline anisotropy, domain formation, shape anisotropy, magnetostriction.
9. Ferromagnets: domains and domain walls, ferromagnetic hysteresis. Tailoring magnetic
properties for applications. Ferrimagnetism: the spinel & inverse spinel structures & magnetite.
10. Solid Ionic Conducting Materials: conduction in solids, vacancy mediated ion hopping,
activation energy, ion flux. Diffusion current (concentration gradient) & drift current (electric field).
11. Solid State Ionic Conductors: the Nernst-Einstein equation. Arrhenius plots and ion
conductivity. Yttrium stabilized zirconia, formation of oxygen vacancies. Oxygen concentration cell.
12. Applications of Ionic Conductors: Lambda sensor, oxygen pump, fuel cells. Materials
requirements, the hydrogen economy.

BH3 Course B: Materials for Devices BH3
Recommended Text Books & Websites
Parts of the following may be helpful:
Basic Solid State Chemistry A R West (Wiley, 2 nd edition) Pq62, and Pq62a Tripos Ref.
For polymers and liquid crystals –
Liquid Crystalline Polymers A Donald, A Windle & S Hanna (C.U.P., 2 nd edition) AN6b.140,
AN6c.140, and AN6c.140a Tripos Ref.
Physical Properties of Polymers James Mark et al (C.U.P., 3 rd edition) AN6c.135
Liquid Crystals Peter J Collings (I.O.P. Publishing, 2 nd edition) Ng134a, and Ng134b Tripos Ref.
Dielectrics, polarisation, conduction, magnetism, ceramics –
Electroceramics A J Moulson & J M Herbert (Wiley, 2 nd edition) AN2a.111
Introduction to Ceramics W D Kingery, H K Bowen & D R Uhlmann (Wiley, 2 nd ed.) AN2a.13,
and AN2a.13a Tripos Ref.
More general interest –
Materials Science and Engineering W D Callister (John Wiley, 2003, 6 th ed.) AB168a, and AB168
Tripos Ref.
Introduction to Materials Science J P Mercier, G Zambelli & W Kurz (Elsevier) AB218b, and
AB218a Tripos Ref.
Also, many of the DoITPoMS Teaching & Learning Packages:
Introduction to Anisotropy
Atomic Scale Structure of Materials
(see Single Crystals: Optical Properties)
Crystallinity in Polymers
Crystallography
Dielectric Materials
Diffusion
Ferroelectric Materials
Ferromagnetic Materials
Fuel Cells
Lattice Planes and Miller Indices
Liquid Crystals
Optical Microscopy (look at Polarised Light)
Introduction to Photoelasticity
Piezoelectric Materials
Pyroelectric Materials
Polymer Basics
http://plc.cwru.edu Resources related to Polymers and Liquid Crystals.
Crystal Maker Software: see link on website for this course.

BH4 Course B: Materials for Devices BH4
Liquid Crystals
• Crystalline materials have long-range order. This means that they are anisotropic: their properties
(e.g. refractive index, thermal expansion coefficient, electrical conductivity) may differ, depending
upon the direction of measurement.
• In contrast, liquids, which have no long-range order, are isotropic: they are invariant with respect
to direction. The absence of long-range order means that all directions are equivalent.
• Somewhere in between there are Liquid Crystals, which are anisotropic liquids. Their anisotropy,
or directionality, comes from molecular shape.
Liquid crystals often consist of rod-shaped molecules, with a rigid long axis:
e.g. Methoxybenzylidene Butylaniline, or MBBA
The central region of the
molecule (between the
rings) is rigid, whilst the
ends are flexible.
These rod-shaped
molecules are free to
move relative to each
other and flow past
each other, so that
there is no long-range
positional order.
However, because of their shape, they tend to line up, with their long axes lying roughly parallel, i.e.
they have some orientational order. This alignment stems from packing considerations, and also
electrostatic interactions.
Nematic Liquid Crystal (LC) structure
No positional order.
Long range orientational order, defined by a
Director, D.
D
This leads to useful anisotropic optical
properties, which are the basis for electronic
displays.

BH5 Course B: Materials for Devices BH5
The degree of orientational order in a LC is temperature dependent:
• at high temperature thermal agitation overcomes the alignment interaction, leading to randomisation
of the molecular orientation, i.e. the liquid crystal becomes a ‘normal’ isotropic liquid.
• at low temperature the system may crystallise into a ‘normal’ crystal structure.
crystal liquid crystal liquid
Director, D
highly ordered,
no intrinsic order,
no translational freedom
increasing temperature,
isotropic
decreasing order
An order parameter, Q, is used to describe the degree of orientational order.
Q = 3 cos2 θ −1
2
Director, D
θ
where
.... represents an average over all molecules.
With all molecules aligned:
cos 2 θ = 1 ⇒
Q = 1
For an isotropic liquid, with randomly oriented molecules: cos 2 θ = 1 €
3 ⇒ Q = 0
€ €
Consider a hemisphere:
The probability of the molecule lying at an angle, € θ, is proportional € to
the circumference, 2πr = 2πd sinθ r
(Higher angles are more likely.)
A small increment in angle, dθ , corresponds to a d
surface area of 2πr × d × dθ = 2πd sinθ d dθ
To calculate cos 2 θ , we integrate over the surface of
the hemisphere (from θ = 0 to π / 2 ), and divide by the
total surface area (for the hemisphere, this is 2πd 2 ):
∫
0
π 2
cos 2 θ 2πd sinθ d dθ
= − 1 π
⎡⎡ ⎤⎤ 2
2πd 2 3 cos3 θ
⎣⎣
⎢⎢
⎦⎦
⎥⎥ = 1 3
0
€

BH6 Course B: Materials for Devices BH6
Polarized Light
Light is a transverse electromagnetic wave with an electric field, E, oscillating in a direction orthogonal
to the propagation direction:
The direction of polarization refers to the
vibration axis of the E field. The plane of
polarization is the plane containing the
vibration axis and the direction of propagation.
The wave sketched here is called a plane wave,
since the vibration axis is always in the same
plane.
polarization direction
(In unpolarized light the field vibrates in all
directions perpendicular to the direction of
propagation.)
Light passing through a sample:
Electromagnetic waves passing through a polymer sample couple to electron density in the polymer
molecules. Light incident upon a polymer molecule will couple strongly (have a larger interaction) in
one vibration direction and weakly in the perpendicular direction.
In an isotropic polymer sample (e.g. at high temperature,
in the liquid state) the molecules are randomly aligned,
and hence there will be no net effect upon the polarization
of the transmitted light.
However, if the polymer molecules are preferentially
aligned (i.e. the polymer is anisotropic, as in a nematic
LC), there will be one direction in the sample along which
the vibration vector couples strongly (called the slow axis,
since the light will be slowed down most significantly)
and another, perpendicular, direction in which it couples
The isotropic state:
no orientational order
weakly (called the fast axis, since the light is slowed down less). These two directions, which are both
perpendicular to the direction of propagation, are called permitted vibration directions (PVDs).
Since the speed of light in the material, v, is different in these 2 directions, the refractive index, n, is
different along the 2 perpendicular vibration directions in the material:
n = c v

BH7 Course B: Materials for Devices BH7
Depending upon the structure of a polymer molecule, the slow direction may be along the length of
the molecule, or perpendicular to the length. e.g. (for light propagating perpendicular to the plane of
the paper):
→ n 1
↓ n 2
polyethylene polystyrene
€
n 1
> n 2 fast
n 1
< n 2
slow
slow
fast
The birefringence is defined as the difference between the refractive indices:
Δn = n 1
− n 2
Whether or not a polymer sample exhibits non-zero birefringence will depend on the degree of chain
orientation:
random, isotropic = non-birefringent
aligned (or crystalline), anisotropic = birefringent. €
Polarizers or Polaroid materials are used as polarizing filters to produce linear polarized light.
These are polymer films (e.g. polyvinyl alcohol, PVOH, doped with iodine), in which the long-chain
polymer molecules are uniaxially aligned. The molecules absorb light vibrating parallel to their long
axes, and transmit that vibrating perpendicular:
[ Note: the lines
drawn to illustrate a
polarizer generally
indicate the resultant
direction of polarization
of the light,
rather than the
alignment of the
absorbing structure.]
Direction of alignment
of polymer molecules ↓
Crossed polars:
(all light will be blocked)
polarizer
analyzer

BH8 Course B: Materials for Devices BH8
Birefringence and Polarized Light Microscopy
Light passing into an optically anisotropic, or birefringent, material is resolved into two components,
along the two permitted vibration directions (PVDs): fast (lower n), with its vibration direction parallel
to the fast direction in the material, and slow (higher n), with its vibration direction parallel to the
slow direction. For example, plane polarized light entering a sample:
A Polarizer defines
the direction of
polarization of the
incident light
2 perpendicular
PVDs in sample
The fast component (lower n) travels faster than the slow component (higher n)! Hence, the two
components take different times to reach the end of the sample. Or, put another way, there is an
optical path difference (o.p.d.) between them:
Direction of
polarization of
incident light (at 45°
to vertical axis)
o.p.d. = Δn.t
€
As illustrated above, this difference in speed between the fast and slow components leads to a phase
δ
difference, δ, between the light in these two vibration directions:
2π = Δn.t
λ

BH9 Course B: Materials for Devices BH9
Let’s think about viewing a sample which is placed between crossed polars. When the slow and fast
components exit the sample and recombine, the vibration direction of the resultant wave depends
upon the relative phase difference between them.
• if the optical path difference is λ / 2 , the phase difference will be π, and the direction of polarization
will be exactly perpendicular to the original:
Po
polarized incident
light (45° to vertical)
Allowed polarization
direction of Analyzer, placed
o.p.d. is exactly half
at 90° to Polarizer, after light a wavelength
has passed through sample.
Viewing through crossed polars, light will pass through the Analyzer
• if the optical path difference is λ, e.g. if the birefringence is greater (see (a) below); or the
wavelength is shorter (b); or the sample is thicker (c), the phase difference will be 2π (which is
equivalent to no phase difference at all), and the direction of polarization will be the same as the
original. For crossed polars, this means that no light passes through the analyzer.
(a) (b)
(c)
o.p.d. is exactly one wavelength
(or an integral
number of wavelengths).

BH10 Course B: Materials for Devices BH10
Remember - the phase difference is dependent upon the birefringence of the sample, the wavelength
of the light, and the sample thickness:
δ
2π = Δn.t
λ
Look at a sample between crossed polars (polarizer ⊥ r analyzer):
• if section is isotropic (non-birefringent), no light will be transmitted through the analyzer.
• if sample is birefringent, and the optical path difference is an integral number of wavelengths,
Nλ, then there will be zero phase difference between the 2 components, the plane of polarization
will be unchanged, and hence no light of wavelength λ will be transmitted.
⇒ if a birefringent sample is illuminated with white (i.e. polychromatic) light, one wavelength of
that light will be lost through the condition described above. Therefore the light observed will
consist of the full optical spectrum minus that specific wavelength (the complementary colour).
The colour will depend upon the birefringence and the thickness of the sample:
The Michel-Levy chart
Very low retardation:
(very thin section, or very
low birefringence).
→
→
Move through higher
‘orders’ of colours as
o.p.d. = λ, 2λ, 3λ etc.
High retardation: several different wavelengths
are ‘lost’ (with different integer values, e.g.
o.p.d. = 2λ 2 = 3λ 1 ), & colours become washed out
•
•
•
•
•
•
•
•
•
•
•
Retardation = optical path difference = Δn × t

BH11 Course B: Materials for Devices BH11
Extinction positions and Permitted Vibration Directions
If the incident light has its plane of polarization parallel to one of the permitted vibration directions in
a birefringent sample, then there will only be light transmitted through the sample in this one vibration
direction. Hence there are not 2 components of the light travelling through the sample, and there
can be no phase difference. The emerging light will have the same plane of polarization as it had
originally and, between crossed polars, no light will be transmitted.
The sample will appear black, and this is called
an extinction position.
By rotating a sample between crossed
polars in order to find this condition,
the permitted vibration
directions of the sample
can be determined.
Samples are best observed
with their PVDs at 45° to
the polarizer / analyzer.
Determination of the sign of the birefringence (i.e. which direction is fast, and which is slow)
This is done by adding a compensator: an optically anisotropic crystal with a known birefringence
(e.g. quartz), to the light path. With both the sample and compensator aligned such that their PVDs
are at 45° to the polarizer / analyzer, there are 2 possible situations:
ADDITION: fast direction of compensator is parallel to fast direction of unknown sample
⇒ total optical path difference increases, & observed colour will be higher in the Michel Levy chart.
SUBTRACTION: fast direction of compensator is parallel to slow direction in unknown sample,
⇒ total optical path difference decreases, & observed colour will be lower in the Michel Levy chart.
Birefringence in Liquid Crystals
Light passing through a sample of nematic Liquid Crystal is resolved into two components:
slow
The permitted vibration directions are parallel and
perpendicular to the director, D.
↑
D
fast
e.g. light vibrating parallel to the director
(principally along the long axes of the molecules)
may have a larger refractive index than that
vibrating in the perpendicular direction. This means
that it would travel slower.
Note: if the direction of propagation is parallel to the Director (i.e. along the optic axis of the
sample), then the section is isotropic (and hence non-birefringent).

BH12 Course B: Materials for Devices BH12
A sample of nematic LC will not necessarily have a single, uniform director across its whole volume,
but instead may be broken down into smaller, differently oriented regions, or domains.
Where differently oriented domains meet there is
a ‘glitch’, called a disclination. Observation of a
LC sample between crossed polars typically
shows a schlieren texture: bright regions of
nematic LC with uniform director, separated by
dark boundary regions where the director is
aligned with the polarizer or analyzer (or is
parallel to the light path), and hence the sample is
in extinction.
Further types of Liquid Crystals
There are several different classes of LC, based upon their overall tendency for molecular alignment.
Smectic: molecules organise into layers
⇒ orientational order, some positional order
Chiral nematic: helical twist
Smectic A
D parallel to
layer normal
Smectic C
↑
D
D
D
The molecules in a Chiral Nematic (also called cholesteric) have their long axes (and hence director)
in a plane (in the above, this is shown as the horizontal); the director rotates along the axis perpendicular
to this plane, tracing out a helix. For polarized light propagating along this axis, the plane of
polarization rotates with D. The pitch is the distance along the axis taken for a complete 360° rotation
of the director. It can vary over a very wide range, e.g. ~100 nm - 100 µm, depending upon the LC
molecules involved, the degree of polymerization, and the concentration in solution.

BH13 Course B: Materials for Devices BH13
This twisted structure occurs because the molecules are chiral (lack inversion symmetry) [see AH68],
and show an asymmetric packing preference.
e.g. Poly(γ-benzyl-L-glutamate) (PBLG):
with R:
Monomer:
Polymer:
The polymer molecule is an α-helix, i.e. is asymmetric (there’s a difference between a right-hand and
a left-hand helix). This leads to the observed twist in alignment, i.e. twist of the director.
dark →
(light propagation
along
symmetry axis
of molecules
⇒ nonbirefringent)
Observation of a chiral nematic LC between crossed
polars, with light travelling perpendicular to the axis of
the helix, shows a stripy pattern, often appearing like a
fingerprint:
dark →
strongly
birefringent
⇒ bright

BH14 Course B: Materials for Devices BH14
In another example of a chiral molecule, the steepness of the twist decreases (i.e. the pitch increases)
as m increases, and the chiral centre (the asymmetric, or ‘handed’ part of the molecule) moves away
from the mesogenic core (the ‘stiff’ part of the molecule):
mesogenic core
chiral centre
i.e. a long molecule will be ‘less’ asymmetric, have little angular twist with its neighbours, &
therefore have a long pitch.
- a shorter molecule → strongly asymmetric → steeper twist (larger angle between neighbours), &
Liquid Crystal Displays
hence a short pitch.
The director of a nematic LC can be encouraged to lie along a particular direction by creating grooves
on a surface in contact with it (this is generally done with an aligned polymer coating; but in one of
the samples in BP1 parallel scratches were made with fine emery paper). The LC molecules will tend
to lie along the length of the grooves. If the LC is sandwiched between 2 glass plates, with grooves at
90° to each other, the molecules (and hence the director) will twist across the sandwich, resulting in a
precisely defined twisted nematic structure:
Viewed from the top
In order to achieve this a small amount of chiral nematic dopant is generally added to the LC.
Polarized light entering the bottom of the
sandwich has its plane of polarization twisted
through 90° as it reaches the top. If the
polarizer and analyzer are parallel to the
grooves at the bottom and the top, respectively,
light will be transmitted through the LC cell:
the ON state.

BH15 Course B: Materials for Devices BH15
Application of an electric field across the LC cell
affects the orientation of the molecules. There is charge
separation (the opposite ends of the molecule develop
opposite charges: an induced dipole moment) and, as a
result, the molecules tend to line up along the field
direction, i.e. perpendicular to the original plane of
polarization. This is called the Freedericksz transition.
Although the molecules near the grooved surfaces
maintain their orientation parallel to the grooves, those
molecules towards the centre of the cell become
oriented parallel to the applied field. Hence, the
twisting of the plane of polarization is disrupted, and light cannot be transmitted through the cell:
the OFF state.
Light enters through the front of the display and is reflected from the back mirror. In front of the
mirror are a polarizing filter, a ‘back’ (transparent) common electrode, and the twisted nematic LC
cell. Light coming out of the front of the cell will have had its plane of polarization twisted through
90° from the polarizer, and this can pass through the analyzer. Shaped electrodes between the cell
and the analyzer can be independently powered: if powered, then the LC in the specific region
between the two electrodes will untwist and block the light.
In a colour display there is fluorescent backlighting, and colour filters create red, green and blue subpixels.

BH16 Course B: Materials for Devices BH16
An Introduction to Polymer Structures
Polymers consist of large molecules, or macromolecules, which are built up from repeating smaller
structural units, or monomers. The number of repeated units in a polymer molecule may be of order
100 – 10,000. This can lead to very long chain molecules, e.g. up to several microns, although the
chains are generally twisted and coiled up, rather than stretched out lengthwise. In addition to linear
chains, polymer molecules may be branched or cross-linked.
Polyethylene, or polythene (C 2
H 4
) n
, is the simplest example:
Monomer unit:
Polyethylene chain:
However, this representation can be misleading, since molecules are 3-dimensional structures. The
bonds from a carbon atom are oriented in an approximately tetrahedral arrangement:
2.5 Å
Space-filling models give, perhaps, an even more realistic representation, although it may be difficult
to visualise the bond angles:
Monomer unit:
Section of chain:

BH17 Course B: Materials for Devices BH17
Flexible chains: conformation
Chains are not rigid, but can twist and rotate.
Rotation about the C-C bonds leads to
changes in conformation.
In polyethylene the lowest energy
conformation (called trans) has neighbouring
C-C bonds “staggered”:
… the highest energy conformation
has the C-C bonds aligned:
The bonds from adjacent C atoms are misaligned,
with the subsequent C atoms in the
chain as far apart as they can be.
There are 2, equivalent, intermediate energy conformations (called gauche), with the bonds
misaligned, but the subsequent C atoms closer than in trans:
A plot of the energy of all
possible conformations
looks like this:
high energy states
between minima
low energy conformations
at minima

BH18 Course B: Materials for Devices BH18
A polyethylene chain with all bonds in the (lowest energy) trans conformation will look like this:
But, in reality, it will typically have a range of different conformations, which lead to twisting and
coiling:
2.5 Å
5 Å
20 nm
The energy increase associated with non-trans conformations and, in particular, the energy barrier
required to change conformation, means that increased thermal energy (i.e. increased temperature)
leads to a higher probability of conformational changes (i.e. chain rotation) taking place. At higher
temperatures polymer molecules become more flexible, and can twist, coil and uncoil around each
other. At lower temperatures they are unable to flex. This temperature scale depends upon the
polymer molecules concerned (depending upon their structure, some are ‘stiffer’ than others).

BH19 Course B: Materials for Devices BH19
How big is a polymer molecule
A polymer chain can be modelled as a series of shorter, rigid segments (each of length l ) that can
rotate freely where they join.
“Random walk”:
r 1
• If the chain consists of n segments, and it
were stretched out straight, its length, L, would
be nl.
• end-to-end vector,
• A useful measure of the size of the polymer molecule is given by the
2
root mean square of R n
: R n
€ €
r n-1
r 2
r 3 r 4
R n -1
R n
r n
€
also,
R n
=
R n−1
=
n
∑ and
• The average value € of the end to end vector,
R n
, is zero: R n
= 0
€
i=1
r i
n−1
∑r i
i=1
R n
= R
n−1
+ r
n
1 2
where γ is the angle between the vector
R 2 n
= ( R n−1
+ r n ) ⋅
Rn−1 + r n
€ 2
= R n−1
+ € 2R n−1
× l cosγ
( ) = R 2
n−1
( ) + l 2
The average value of cosγ must be zero, and therefore,
( ) + r n
+ 2 R n−1
⋅ r n
and the last segment,
R n
2
2
= R 2
n−1
€ = R 2
n−2
r n
+ l 2
€
+ l 2 + l 2 etc.
R n−1
And, by induction:
mean square end-to-end distance,
R n
2
= nl 2
root mean square,
R n
2
1 2
= n 1 2
l
We might assume that each segment of the chain, € l, corresponds to a single C-C bond. But such
segments can never be completely freely jointed (bond angles are limited, and adjacent segments can
simply get in the way of each other). € Hence real polymer chains are always stiffer than shown by this
model using a single C-C bond as a segment length, and their stiffness is dependent upon the nature
of the monomer units involved.
The model can be used more generally if we define l, the segment length, as the length scale below
which the chain is effectively straight and rigid – this is called the Kuhn length. The simple
polyethylene structure has a Kuhn length of about 3.5 times the C-C bond length, whilst polystyrene,
which has more bulky side-groups, has a Kuhn length of 5 × C-C (it’s stiffer).

BH20 Course B: Materials for Devices BH20
A few examples of common polymers with different side-groups:
Monomer
Polyvinylchloride (PVC)
Polypropylene
Polystyrene
The side-groups (Cl, or CH 3 , or C 6 H 5 in the above examples) may be arranged in different ways
along the chain, which defines the molecule’s configuration, or tacticity. Considering the chain in the
all trans conformation …
... in the isotactic arrangement, the side-group is always on the same side:
…. the syndiotactic arrangement has the side-group on alternate sides:
… and the side-groups distributed randomly corresponds to an atactic arrangement:

BH21 Course B: Materials for Devices BH21
Polyvinylidene fluoride (PVDF)
e.g. all trans conformation:
The resultant charge imbalance, across the molecule, leads to technologically important electronic
properties (to be discussed later in the course).
Polymer Microstructure and Properties
Linear chain polymers typically form a network of entangled chains. At relatively high temperatures,
or in a dilute solution, conformational changes mean that the chains can flex and coil, and slide
around each other. As the temperature is lowered there is less thermal activation and, as the polymer
contracts, less space for the chains to move around in, so that these processes are restricted and the
polymer becomes more and more viscous. Large side-groups, side branches and cross links between
chains can reduce mobility further.
Small molecules can be added to a polymer to space out the chains and increase mobility. These
small molecule additives are called plasticizers.
A rubber is a cross-linked network: the chain segments between cross-links are free to twist and flex
by conformational changes (rubbery-ness!), but they cannot slide past each other to give permanent
shape changes.

BH22 Course B: Materials for Devices BH22
Crystallinity
Many polymers can exist in crystalline, or partially crystalline form. The chains pack into a regular
structure by folding over and lining up with each other. Typically a polymer chain will meander
between different ordered, crystalline layers, or lamellae, which are separated by disordered, noncrystalline
(or amorphous - see AH124) regions. The relative thickness of these different regions
(crystalline vs. amorphous) defines the percentage crystallinity of such a semi-crystalline polymer.
crystalline lamellae
amorphous interlayers
Crystallisation is easier for polymers with very regular chains (e.g. isotactic and syndiotactic
molecules), and is limited by irregularity (e.g. atactic molecules) and/or bulky side-groups that hinder
packing.
Highly crystalline polymers are stiff (e.g. 95 – 99% crystalline polyethylene, which is also brittle),
since the molecules are restricted (‘locked into’ their ordered arrangement).
The density difference between crystalline and amorphous regions leads to light scattering, and hence
opacity.

BH23 Course B: Materials for Devices BH23
Dielectric Properties of Materials
A dielectric material supports electrostatic charge. It must therefore be non-conducting, but able to
become electrically polarised.
Electric field, E ↑
(V m -1 )
Polarisation occurs by the creation of electric dipoles, i.e. separation of opposite charges, at the unit
cell level.
Polarisation Mechanisms
+ + + + + +
- - - - -
[Electric field, E, is defined as the direction of
motion of a positive charge]
Electronic: distortion of the electron
cloud with respect to the positive nucleus
of an atom (occurs, to some extent, for all
atoms; dominates in, e.g. noble gases,
diamond).
Ionic: elastic distortion of ionic bonds, i.e.
relative displacement of positive and
negative ions (dominates in ionic crystals,
e.g. NaCl)
Orientational, or Molecular: rotation of
pre-existing permanent dipole moments
(dominates in dipolar liquids, e.g. H 2
O)
random dipoles … … … tend to line up
Different mechanisms may occur simultaneously.
Polarisation & Capacitance
For charges of opposite sign, + q, separated by a distance r, each charge centre has a
dipole moment, µ (C m) (which is a vector)
€
µ = qr
Dipole moment per unit volume, or polarisation,
where
n = number of dipoles per unit volume
€
P = nµ (C m -2 ) -q +q
We can also define polarisation, P, as the charge per unit surface area,
Q
A
r
€

BH24 Course B: Materials for Devices BH24
If an electric field, E, is applied to a sample of a dielectric material, there is a resultant total charge
density, D:
D = εE = κε 0
E (units of C m -2 )
ε = permittivity of the material (‘polarisability’) ε 0 = permittivity of free space (8.85 x 10 -12 F m -1 )
The Dielectric Constant of the material, κ , is defined as: κ = ε (dimensionless)
ε0
€
D (which is sometimes called a displacement field), is made up of 2 components:
D = ε 0
E + P P = polarisation of the material
therefore, polarisation of the material,
P = D − ε 0
E = κε 0
E − ε 0
E
Capacitance
(units of Coulomb / volt: C V -1 , or Farads)
€
A capacitor, which is made of a dielectric, stores energy in the form of an electrostatic field, or charge
€
P = ε 0
E( κ −1)
Capacitance, C, depends on the dielectric material & the geometry: C = Q V
(a) For an empty parallel plate capacitor (i.e. with a vacuum between the plates):
_ _ _ _ _ _ _
+ + + + + +
+
€
A = surface area of plates
L = distance between plates
applied voltage, V = E × L
E = field
⇒ charge density (free space) = ε 0
E and charge on plates, Q = ε 0
E × A
C = Q V
= ε 0 EA
EL
= ε 0
A
L
(b) If a dielectric is placed between the plates, the electric field polarises the dielectric and leads to a
charge density on its surface, P. There is now a higher total charge density on the plates (D):
-
+
_ _ _ _ _ _ _ _ _
+ _ + + + + +
- - - - - -
+ + + + + + + + +
+ +
-
+
D = ε 0
E + P & hence
capacitance,
ʹ′ C = Q V
C ʹ′ is increased:
= ( ε 0
E + P) × A
E × L
ʹ′ C = ε 0
κ A L
= ε A L

BH25 Course B: Materials for Devices BH25
Examples of dielectric materials:
Low dielectric constant, κ:
vacuum; dry air; many pure, dry gases
↓ many ceramics; paper; mica; polyethylene; glass; distilled water
High κ: some metal oxides, e.g. BaTiO 3
P
For a linear dielectric: (may get dielectric breakdown
at very high field, e.g. due to
E
ionisation & current flow)
Dipole formation & Symmetry
If a crystal structure has an inversion centre, or centre of symmetry, then it is centrosymmetric (see
AH62). Visualising centrosymmetric structures:
z
x
y
Every component must exist, reflected through the centre of symmetry.
A unique direction in a crystal structure is a lattice vector which is not repeated by the symmetry
present (see AH67). In centrosymmetric crystals there can be no unique direction (and hence no
possibility of a resultant electric dipole moment without the presence of an electric field).
Non-centrosymmetric structures do not necessarily include a unique direction, e.g.:
ZnS - sphalerite, or zinc blende
Cubic F, with 1 / 2 tetrahedral interstices filled
(see Data Book & AH31)
This is a non-centrosymmetric structure; but with
no unique direction.
1 / 2 1 / 3 4 / 4
1 / 1 2 / 2
3 / 1 4 / 4
1 / 2
o S
• Zn
Note, however, that the directions are asymmetric: [111] is not equivalent to ⎡⎡
⎣⎣111⎤⎤
⎦⎦ ; and
⎡⎡111 ⎣⎣
⎤⎤
⎦⎦ is not equivalent to ⎡⎡
⎣⎣111
⎤⎤
⎦⎦ ; etc.

BH26 Course B: Materials for Devices BH26
A material which can demonstrate a dielectric polarisation (e.g. due to charge separation) in the
absence of a field is a polar material. It must contain a unique direction.
A centrosymmetric crystal is non-polar, but a non-centrosymmetric crystal is not necessarily polar:
(a) Non-polar; non-centrosymmetric: (b) Polar; non-centrosymmetric
-Q
-Q
-Q
+4Q
-Q
→
-Q
+4Q
-Q
… displacement of central ion …
-Q
+ve charge sits at centre of balance +ve charge is offset ⇒ charge imbalance
of –ve charge, ⇒ there is charge ⇒ dipole moment & spontaneous
balance, & zero dipole moment
polarisation
If a stress is applied to sample (a), inducing a shape change (e.g. if it is squashed or extended in one
direction), the positive and negative charge centres will move relative to each other, i.e. there will be a
change in polarisation. This phenomenon is called piezoelectricity. Conversely, application of an
electric field will lead to a change in the positions of the ions, and hence to a shape change.
If the temperature of sample (b) changes, there will be a change in the relative positions of positive
and negative charges (e.g. due to thermal expansion / contraction), and this will lead to a change of
electrical polarisation. This is called pyroelectricity. All polar crystals are, to some extent,
pyroelectric: there is some relative movement of charges as temperature changes, which cannot be
symmetric along a unique direction; therefore there is a net change in polarisation.
In some materials the electrical polarisation of a sample such as (b) may be switched by an external
electric field. This is ferroelectricity: on application of an electric field the crystal can become
permanently polarised, and this polarisation can be reversed by applying the field in the opposite
direction.
Ferroelectric materials are a subset of pyroelectrics which, in turn, are a subset of piezoelectrics:
non-centrosymmetric
& polar
Piezojust
needs to be
non-centrosymmetric
Ferro-
Pyro-
non-centrosymmetric,
polar & switchable

BH27 Course B: Materials for Devices BH27
Piezoelectrics
Change in polarisation on application of a stress:
Polarisation due to stress, P = Q A = dσ
d is the piezoelectric coefficient σ is the stress (tension or compression)
Resultant voltage across piezoelectric, V = Q C
€
Examples of Piezoelectric materials:
(units of C N –1 ) (units of N m –2 )
= QL
κε 0
A
= dσ L
κε 0
(see BH24)
quartz (SiO 2 – see AH61); & many other crystals containing tetrahedral groups, e.g. ZnO, ZnS – zinc
blende (application of stress distorts the tetrahedra). All pyroelectrics & ferroelectrics are piezoelectric.
Piezoelectric Transducers:
Generator effect
- application of stress changes P and leads
to a voltage change.
Generator effect
e.g. igniters (pressure on a thin sheet induces a
high voltage (~kV), which is enough to cause
electrical breakdown across a small (~1 mm)
air gap); energy harvesting, & flashing lights
on trainers; guitar pick-up.
applied stress
Motor effect
- application of an electric field leads
to a shape change.
applied electric
field
Motor effect
e.g. watches (a quartz crystal is made to vibrate at a precise, fixed frequency); medical devices (e.g. an
array of quartz resonators focus sound waves onto a tumour to destroy it, or reflect off an object and
therefore image it); displacement transducers (the ability to make very small, very precise mechanical
movements leads to actuators with very high spatial resolution).

BH28 Course B: Materials for Devices BH28
Pyroelectrics
Change in polarisation with temperature. Polar impurity molecules in the atmosphere can adsorb onto
the surface and mask the true surface polarisation ⇒ measure changes in P:
ΔP = ΔQ A = pΔT
p is the pyroelectric coefficient (C m -2 K -1 )
Voltage change developed across pyroelectric, ΔV = ΔQ
€ C
€
Examples of Pyroelectrics:
€
€
= pΔTL
κε 0
= ΔQL
κε 0
A
wurtzite (hexagonal form of ZnO, ZnS); boro-silicates; all ferroelectric materials are also pyroelectric.
ZnS - wurtzite structure
Hexagonal P, with 1 / 2 tetrahedral interstices filled (see Data Book & AH21)
As drawn, only upward pointing tetrahedra (triangular based pyramids) are filled.
[001] is a unique direction.
Infra-red detector (burglar alarm)
The detector is a sheet of pyroelectric. As T changes (burglars radiate ~ 60W), polarization changes,
and hence the voltage across the sheet changes (typically a few mV): ΔV ∝ ΔQ (= ΔP × A)
infra-red radiation
e.g. LiTaO 3
reflecting electrode
absorbing electrode
Also, thermal imaging cameras.
compensating element
active element

BH29 Course B: Materials for Devices BH29
Ferroelectrics: Stable, spontaneous polarisation, which can be reversed by an external electric
field. The unit cell is spontaneously polarised (acquires a dipole moment) below a specific
temperature, called the Curie temperature, T c , due to a change from high to lower crystallographic
symmetry on cooling (the lower temperature, lower symmetry form is Ferro-Electric, FE.)
e.g. salts Rochelle salt: NaKC 4 H 4 O 6 .4H 2 O
Potassium Dihydrogen Phosphate (KDP): KH 2 PO 4 ; KNO 3
polymers polyvinylidene fluoride (CH 2 CF 2 ) n (PVDF) – see BH21
oxides
BaTiO 3 (see AH32), Pb(Zr x Ti 1-x )O 3 (PZT), SrBi 2 Ta 2 O 9 (SBT)
Polarisation & Structure
In FE crystals a displacive phase transition occurs on cooling through T c . The crystal becomes noncentrosymmetric
resulting in charge asymmetry and dipole formation.
e.g. starting with the cubic, ABO 3 , perovskite structure: e.g. BaTiO 3 (see Data Book & AH32)
1 / 2
1 / 2 1 / 2 1 / 2
1 / 2
A
(B
O 2-
€
ideally:
if r A
+ r O
>
r A
+ r O
=
2( r B
+ r O )
2( r B
+ r O ) i.e. the A cation is relatively large with respect to the B cation, the unit
cell size is larger, and there is more room for the smaller B ion to move in the BO 6 octahedron.
Drawing € the unit cell with the A cation at the origin:
high T ; cubic
y
z
x
Below T C ;
c-axis
lengthens
Hence, ferroelectricity, which may occur as a result of B ion displacement, is often found for large A
cations (e.g. K + or Pb 2+ ) and small B cations (e.g. Ti 4+ or Ta 5+ ). Below T c , the B cation (+ve)
moves from the centre of (–ve) charge in (what is defined to be) the (plus / or minus) z-direction; a
dipole moment is formed; and the c-axis extends (the structure becomes tetragonal).

BH30 Course B: Materials for Devices BH30
Phase Transitions in BaTiO 3
On cooling, the onset of ferroelectricity is triggered by a phase change from a high symmetry (cubic)
structure to a lower symmetry (tetragonal) structure at the Curie temperature, T c
. On further cooling,
the structure transforms to even lower symmetry (see AH58):
rhombohedral orthorhombic tetragonal
← COOLER ←
< -90°C distortion
-90°C < T < 5°C,
5°C < T

BH31 Course B: Materials for Devices BH31
For BaTiO 3 :
Polarisation, tetragonal, cubic,
P (µC cm -2 ) FE non-FE
Dielectric
Constant,
κ
κ a
κ c
[Note: in BP2 you’ll find that
KNO 3
behaves differently!
- polarisation decreases as
temperature decreases.]
In FEs the dielectric constant,
κ, is dependent upon the field,
E, so that polarisation is not
linear with field (cf. with linear
dielectric, BH25).
In BaTiO 3
κ is temperature
dependent, with a peak at T c .
This is where the crystal
structure changes abruptly.
Dielectric constant is also a strong
function of composition; the presence
of impurities or dopants can lead to
significant variations.
Ferroelectrics: Switchable Dipoles
κ
doped BaTiO 3
pure BaTiO 3
T C
Ferroelectricity stems from the fact that some ions in the crystal sit in double-well potentials:
Both are stable and,
in the absence of an
electric field, are
equally likely.

BH32 Course B: Materials for Devices BH32
Dipole ordering & Domains
All such ions in a local region (a domain) will tend to sit on the same side of the double well (adjacent
dipoles want to line up with each other); spontaneous polarisation (ordering of the dipoles) below
T c is a co-operative phenomenon. [Remember: Net electric dipole per unit volume is polarisation, P.]
But, stray fields (due to surface charge) have an energy cost, which can be reduced by the formation
of differently oriented domains:
P
→
reduced stray field
A sample may contain many domains of varying orientation, and hence net P is zero. For example,
when a non-centrosymmetric phase forms (e.g. tetragonal BaTiO 3
in a parent cubic structure), the
transformation begins in different regions of the sample, and will be randomly oriented.
But, there is an energy associated with the interface between
differently oriented domains: domain wall energy
There is therefore an energy balance, or compromise:
stray field energy (high for large domains) vs. domain wall energy (high for many small domains)
The angle between domains depends on the symmetry of the crystal and the preferred dipole
direction. For the cubic → tetragonal transition in BaTiO 3
, polarisation can occur along any of the 3
previously cubic directions, and in a +ve or –ve sense, giving 6 possible domain orientations:
90° domain boundary
90° walls are strained
because the c-axis is
aligned with the a-axis
(a ≠ c)
180° domain boundary
If a ferroelectric sample is held in a sufficient electric field it will become poled, i.e. dipoles will
preferentially line up in the allowed crystallographic direction most closely parallel to the field.
e.g. polycrystalline sample:
(random grain → after
orientations) poling

BH33 Course B: Materials for Devices BH33
Reversible polarisation: P can be switched by applying a large enough field, E. Cycling the
field leads to hysteresis (the P – E curve does not re-trace itself when the field is reversed):
1 Unpolarised, randomly oriented sample.
As for a dielectric, P
is proportional to
field near the origin.
↓
Reversible domain
wall motion
2 Favourably oriented domains grow at the
expense of their less favourably oriented
neighbours ⇒ sharp rise in P.
P = 0
Polarisation
(Cm -2 )
4
→ Irreversible domain wall motion
3
2
5
1
3 saturation polarization, P sat ,
reached (one, aligned domain)
Electric Field ↑
(MVm -1 )
Switching occurs by Domain Walls sweeping
through the crystal. They may be PINNED by
6
crystal defects (leading to higher coercive
field, and higher loop area).
Area of loop is proportional to the energy
required to switch the polarisation.
4 remove field → polarisation remains principally aligned ⇒ remanent polarisation, P r , remains at
zero field, E = 0 (sample has been poled).
5 reverse field →oppositely oriented domains nucleate & grow; need the coercive field, E c , to
return to zero polarization, P = 0.
6 further field reversal → continued growth of domains, & saturation in the opposite direction.

BH34 Course B: Materials for Devices BH34
Applications of Ferroelectrics
Ferroelectrics are used in many commercial devices simply for their pyro- or piezo- electrical
properties, or just as dielectrics: perovskite ferroelectrics have high κ values ⇒ BaTiO 3
capacitors,
e.g. for camera flashes (> 50% of the ceramic capacitor market).
FE was discovered in 1921, and it’s always been obvious that the +P and –P states could be used to
encode 0 and 1 for computing, but only relatively recently (~1995) has this become a reality.
⇒ memory devices: FE-RAM - non-volatile, low voltage, small size & cost, fast, radiation hard.
Materials requirements: high κ ; high P sat & P r ; (relatively) small E C ; high T C
e.g. LiNbO 3
, PbTiO 3
, Pb(Zr x
Ti 1-x
)O 3
(PZT), SrBi 2
Ta 2
O 9
(SBT), fabricated as thin films
(~ a few hundred nm) in order to minimise switching voltage.
Wide-scale applications (cell phones, smart cards, video games) depend upon materials science:
optimisation of composition, microstructure, interface quality with electrodes, control of nano-scale
defects that pin domain walls.
FE switching in a thin film memory device element:
Thin Film, e.g. ~10 – 300 nm
Start with a single-domain polarisation state [0]
Reverse field ↓
Nucleation of the opposite polarisation
initiated at the surfaces
Forward growth of needle-like domains
parallel to the applied field
Sideways growth of domains
Reversed polarisation state [1]
(switching takes ~ 50 ns)

BH35 Course B: Materials for Devices BH35
PZT- Pb(Zr x Ti 1-x )O 3
One of the most widely used piezoelectrics is PZT: Zr and Ti ions occupy the perovskite B-site at
random. Properties can be changed by varying x. Depending on the composition and temperature,
different phases of PZT are thermodynamically stable:
← Morphotropic Phase
Boundary
→
(x = 1) (x = 0)
e.g. ~50 mol.% PbTiO 3
/ 50 mol.% PbZrO 3
, at room temperature: the system is close to the
tetragonal / rhombohedral phase boundary. Hence there are 6 possible tetragonal distortions,
and 8 possible rhomobohedral distortions, along which polarisation states can form (see
BH30). This means that there are a total of 14 possible orientations along which the polarisation can
be directed, and hence this composition can be easy poled & switched.

BH36 Course B: Materials for Devices BH36
Magnetic Materials
Magnetic moments are produced by spinning electrons orbiting the atomic nucleus. Some atoms /
ions behave as magnetic dipoles.
Electron spin +
orbital angular
momentum
e
-
+
The magnetisation of a material is a function of how strong the separate moments are, and how well
aligned they are. The relative orientation may be influenced by neighbouring moments and by an
external field. They may be randomly oriented, or perfectly aligned, or somewhere in between.
For atoms with complete shells, all electron states are full (paired up, with opposite spins), so both
orbital and spin angular momentum average out to zero. Hence magnetism depends upon electrons in
incomplete shells (valence electrons).
Magnetisation, M = magnetic moment per unit volume: i.e. A m 2 per m 3
Magnetic Moment:
current × area
which has the same units as an applied Magnetic Field, H i.e. Am -1
M = χH
Susceptibility, χ (dimensionless)
current,
A
area,
m 2
Diamagnetism
€
Orbital shells are filled, no unpaired electrons - with no external field there can be no net moment. In
an applied magnetic field electron orbits react to oppose the external flux ⇒ moments develop to
oppose the field ⇒ χ is negative, e.g. ~ −10 -5 (e.g. graphite, noble gases)
Paramagnetism
Some unpaired electrons in partially filled shells, so
dipoles exist, but they’re isolated and non-interacting
⇒ they are randomly oriented.
With no external field the overall moment is zero.
In an applied field there is partial alignment of
moments in the direction of the field ⇒ χ is positive,
but small, e.g. ~10 -5 (e.g. many metals, Na, O 2
)

BH37 Course B: Materials for Devices BH37
Ferromagnetism
Many unpaired electrons, i.e. partially filled shells ⇒ strong
interaction between moments. Moments tend to align with
each other (parallel moments have lowest energy). This is a
quantum mechanical effect: the magnitude of the exchange
interaction depends upon the overlap of electron wave
functions, & therefore depends upon the crystal structure as
well as electron spin (not simply dipole–dipole interaction).
Moments align with an applied field, and orientation can
become permanent, i.e. remain aligned without an external
field. Parallel alignment of moments leads to a large net
magnetisation, even in zero field. e.g. χ (Fe) = 5.10 3
Cooperative Alignment (e.g. Fe, Ni, Co)
Antiferromagnetism
Strong interaction between moments, but anti-parallel
moments have the lowest energy.
The two sets of moments are exactly equal & opposite
⇒ net magnetic moment = 0
The only AF element at room temperature is Cr.
More complex forms of magnetic ordering can exist in ionic
compounds, and these are controlled by the crystal structure.
e.g. may have two magnetic sub-lattices in which the moments
align anti-parallel with each other:
Ferrimagnetism
As for antiferromagnetism, but the moments on the two
sub-lattices are unequal ⇒ a net magnetisation.
e.g. magnetite: Fe 3
O 4
, which has two different types of
Fe lattice site.

BH38 Course B: Materials for Devices BH38
Ferromagnetic Materials
Magnetisation
Curie Temperature, T C
Thermal agitation competes with exchange
interaction at high temperature.
Ferromagnetic
Temperature
Paramagnetic
χ becomes very high around T C
:
the magnetic moment changes
rapidly at the transition.
Magnetocrystalline anisotropy M
Interaction of the magnetic moment with the crystal
lattice means that the preferred direction of the magnetic
moment is dependent upon the crystalline direction.
Energy is lowest when the magnetisation points along a
specific crystallographic direction.
⇒ easy and hard crystallographic axes:
Fe (bcc) Ni (fcc) Co (hex.)
easy
↓
hard
easy
hard
Saturation magnetisation
H
Domain formation A ferromagnetic material is not necessarily magnetised …..
Stray fields have an energy cost, which can be reduced by the formation of magnetic domains. Each
domain is spontaneously magnetised, but they’re in different orientations.
→ → decreasing stray field → →
→ increasing domain wall energy →

BH39 Course B: Materials for Devices BH39
Shape Anisotropy
Magnetisation is also a function of the sample shape,
e.g. lower energy for magnetisation to lie along the length
of a long, thin sample (think of the stray field lines). This
also applies to grains within a sample.
Magnetostriction
Upon magnetisation a crystal experiences a strain, and will change its dimensions. Conversely,
application of a stress can lead to a change in magnetisation. Dimensional changes in differently
aligned neighbouring domains lead to elastic strain energy.
Magnetostriction coefficient, Λ: fractional change in length on changing magnetisation from zero to
saturation. This may be +ve, or –ve. e.g. Ni: Λ = –5.10 -5 along & –2.7.10 -5 along
Domain wall, or Bloch wall
- transition region between two differently oriented domains
e.g. for 180º wall: Wide Wall (gradual twist of moments) or Narrow Wall (abrupt change)
Wall width is fixed by a balance between:
Exchange Interaction Energy: This is
the energy term that encourages moments
to align with each other - misaligned
moments have high energy.
The exchange energy component is
minimised by keeping misaligned spins
apart (i.e. wide walls).
Magnetocrystalline Energy or Anisotropy Energy:
This is the energy term that encourages moments to
align with the easy axis of the crystal - moments misaligned
with the easy axis have high energy.
The magnetocrystalline energy component is minimised
by aligning spins along the preferred
crystalline direction (which leads to abrupt walls).
• Materials with high exchange energy & low magnetic anisotropy energy: ⇒ exchange energy
term will dominate, & domain walls will tend to be wide (in order to minimise this energy term).
• Materials with high magnetic anisotropy energy & low exchange energy: ⇒ anisotropy term will
dominate, so energy is minimised by reducing this anisotropy energy term, i.e. having abrupt walls.
Typical magnetic domain
wall widths are ~5 – 100 nm
Comparison with Ferroelectrics (see BH32):
Anisotropy in FE crystals is very large, so it is energetically unfavourable to have a gradual change in
polarisation direction at a domain wall ⇒ FE domain walls tend to be very narrow (e.g. ~1 – 10 nm).

BH40 Course B: Materials for Devices BH40
Small particles:
A domain structure won’t form in a small particle, or grain, if domain wall energy > stray field
energy. The domain structure of a polycrystalline material depends upon grain size: grains may be
too small to support a domain boundary ⇒ form uniformly magnetised, single domain grains.
Ferromagnetic Hysteresis
In the presence of an applied field, domains with magnetisation antiparallel to the field DO NOT
spontaneously flip. It is energetically cheaper for the magnetisation to switch by moving domain
walls. Plotting Magnetisation, M as a function of Applied Field, H:
Extrapolation of the curve from M sat back
to the axis defines the Spontaneous
M
a
b
c
d
dipole rotation
growth of favourably
Magnetisation (the net magnetisation oriented domains
within a uniformly magnetised microscopic
volume, in zero field) irreversible wall motion
reversible wall motion
H
H H H H
a b c d
*Domains in unmagnetised
sample cancel
out ⇒ zero net
magnetisation, M
*Moments are aligned
along easy axes
*Wall motion and M
are reversible
*Favourably oriented
domains grow by wall
motion
*M increases sharply
*Wall motion is irreversible
(due to pinning
by imperfections)
*Whole sample is
aligned as a single
magnetic domain
(along easy axis, not
along external field
direction)
*Moment is pulled
away from easy axis,
into line with external
field
*Saturation magnetisation
M sat produced
by rotation of moments
to lie along H

BH41 Course B: Materials for Devices BH41
Cycling the field back and forth produces hysteresis:
Residual magnetisation remaining at zero
applied field (H = 0) is the remanent
Reduce applied field ⇒ moment relaxes back to
easy crystallographic axis, but then the domain
remains stable as field is reduced further
magnetisation, M r
magnetisation
M sat
Continued field reversal ⇒ domains
grow in the opposite direction
M = 0 (multi-domain umagnetised
state) at the coercive field, H C .
applied field
Further field reversal ⇒
saturation in the opposite
direction
M sat
• area within the loop ∝ energy dissipated during one
cycle, i.e. 2 × energy needed to switch M
• imperfections (grain boundaries, impurities, vacancies, etc.) pin domain walls, ⇒ increase H C

BH42 Course B: Materials for Devices BH42
Aside on magnetic parameters:
If a magnetic field, H, is applied to a material, the resultant magnetic flux density, B, is
B = µH = µ 0 ( H + M ) with units of Tesla, T
µ = permeability of material; µ 0 = permeability of free space (4π×10 -7 H m -1 ); M = χH (BH36)
( ) = µ 0
H( 1+ χ) and
µ = µ 0 ( 1+ χ)
B = µ 0
H + χH
€
Although B (also called the inductance) is the parameter that we measure, it is useful to consider M,
the magnetisation, since it can be calculated from first principles:
€ M (A m -1 ) = magnetic moment per atom (A m€
2 ) × no. of atoms per unit volume (m -3 )
Hysteresis loops
Different shapes for different applications:
M
M
M
H
H
Large M r and H c for
permanent magnets.
High, stable magnetisation.
Well defined switching field,
(i.e. sharp change in magnetisation)
for memory devices.
High M sat and low H c
for easy switching in
transformer cores
Best soft magnetic materials: want very small loop area. Therefore get rid of imperfections
(crystalline defects, grain boundaries, impurities, vacancies, strains, etc.) that would pin domain walls.
Use long heat treatments (annealing) to form (ideally) ‘perfect’ crystal, through which domain walls
sweep without hindrance. Or, use very homogeneous, amorphous materials (e.g. H C
≈ 3 A m -1 ).
Best hard magnetic materials: want to pin domain walls. Therefore incorporate imperfections,
defects & stresses by quenching (cooling very rapidly from a high processing temperature, which
doesn’t allow re-organisation). Or, by compressing fine powders (sintering) to leave many
independent particles (e.g. H C
≈ 1.10 6 A m -1 )..

BH43 Course B: Materials for Devices BH43
Ferrimagnetism & the Spinel Structure
Spinel: nearly cubic close packed oxygen ions, with two types of cation sites.
2 × 2 × 2 fcc cells:
e.g. MgAl 2 O 4
A-site
Mg 2+
Mg 2+ in tetrahedral interstices
Al 3+ in octahedral interstices
B-site
Al 3
Normal Spinel
1 / 8
of the tetrahedral
A-sites are occupied by
divalent ions
1 / 2
of the octahedral
B-sites are occupied
by trivalent ions
The Inverse Spinel structure e.g. Fe 3 O 4 (magnetite):
Fe can be 2+ or 3+ and Fe 3 O 4 is actually Fe 2+ O.Fe 3+ 2 O 3
Fe 2+ ions are on B-sites / Fe 3+ ions are divided equally between A (tetrahedral) & B (octahedral)
sites,
i.e. (Fe 3+ ) A (Fe 2+ Fe 3+ ) B O 2 -
4
Spin moments of all Fe 3+ ions on B-sites are aligned parallel to each other, but opposite to the spin
moments of the Fe 3+ ions on the A-sites. Hence Fe 3+ spin moments exactly cancel.
Spin moments of all Fe 2+ ions are aligned parallel to each other, giving the total moment, responsible
for the net magnetisation.
⇒ magnetite is Ferrimagnetic

BH44 Course B: Materials for Devices BH44
Solid Ionic Conducting Materials
Ceramics, which are compounds between metallic and non-metallic elements (e.g. oxides, nitrides,
carbides), are generally electrically insulating (all electrons are bound to atoms or bonds):
e.g. Al 2
O 3
: resistivity, ρ = ~10 10 Ωm or conductivity, σ =10 -10 Ω −1 m −1 or Sm -1
c.f. Cu: ρ = 1.6×10 -8 Ωm σ = 6×10 7 Ω −1 m −1
However, some ceramics can conduct via ionic conduction.
e.g. the ionic conductivity of (ZrO 2
+ 8 mol.% Y 2
O 3
) is ~ 0.1 Ω −1 m −1 @ 500° C
An example of a ceramic structure:
interstitial
+
vacancy
anion
(Frenkel defect)
+
cation
vacancies
(Shottky
defect) → charge neutrality
Ions / atoms aren’t completely stationary on their lattice sites; the higher the temperature, the larger
the amplitude with which they vibrate. They can migrate through the lattice by swapping position with
other ions / atoms, and this migration is greatly enhanced by the existence of vacant sites.
Ions migrate by hopping into vacant lattice sites:
Ionic mobility depends upon:
- whether an adjacent site is empty, and
- the energy barrier between lattice sites
An ionic current flow can result from:
(a) a concentration gradient (of ions, or vacancies): diffusion current
(random diffusion evens out the gradient) and / or
(b) an electric field, E: drift current

BH45 Course B: Materials for Devices BH45
Flux of atoms or ions (atomic / ionic flux), J:
number of atoms or ions crossing unit area,
in unit time, i.e. m -2 s -1
(a) Diffusion current (due to a concentration gradient)
Fick’s 1 st law of diffusion: flux, J = −D ∂n
∂x
D = diffusivity, or diffusion coefficient (m 2 s -1 ) n = concentration of diffusing ions (m -3 )
Diffusion current density: € j = −qD ∂n (A m -2 )
∂x
q = z e = charge on diffusing ion
(z = charge number or valence; e = electronic charge = 1.6×10 -19 C)
€
This applies only to steady-state diffusion in a uniform concentration gradient (in one dimension):
Atoms / ions move down the concentration
gradient (oppositely charged vacancies move
in the opposite direction) by random diffusion.
n
- - - -
energy
lattice sites
+ +
+ +
x
distance
Illustration of the energy barrier between lattice sites,
Q = lattice energy barrier or activation energy
(J mol -1 or, often, eV atom -1 ; 1 eV = 1.6×10 -19 J)
Arrhenius equation:
D = D 0
exp −Q RT
( ) or D = D exp −Q 0 ( kT )
Mobility, or diffusivity, depends upon the probability
that an atom / ion can exceed this energy barrier, Q.
depending on units used for Q
R = gas constant, 8.314 J K -1 mol -1 ; k = Boltzmann constant, 1.38×10 -23 J K -1 atom -1
pre-exponential factor, D 0 , is related to the atomic vibrational frequency & lattice spacing

BH46 Course B: Materials for Devices BH46
(b) Drift current (due to an electric field)
energy
Addition of an electric field
field, E →
(E, direction of positive charge flow)
⇒ reduction of Q in one direction (&
equivalent increase in the opposite direction)
energy barrier = (Q – y)
€
The current density is governed by the conductivity (the ease of motion) & the electric field (the
driving force), and is given by the microscopic form of Ohm’s law.
If σ = conductivity:
drift current density,
€
j = −σ ∂V
∂x
E = − ∂V
∂x ,
current flow (+ve charge)
V
j →
from high to low potential
E →
and so
The number € of diffusing ions in an electric potential follows Boltzmann’s statistics:
Therefore
€
j drift
= σE
⎛⎛
n = n o
exp⎜⎜
−qV ⎞⎞
⎟⎟ and the differential gives the concentration gradient:
⎝⎝ kT⎠⎠
∂n
∂x = −q ⎛⎛ ⎛⎛
n 0
exp −qV ⎞⎞⎞⎞
⎜⎜ ⎜⎜ ⎟⎟⎟⎟ × ∂V
kT ⎝⎝ ⎝⎝ kT ⎠⎠⎠⎠
∂x
∂n
∂x = −nq ∂V
kT ∂x = nqE
kT
Voltage,
i.e. the electric field leads to a concentration gradient
Distance,
x
At equilibrium the diffusion current density (BH45) j diffusion
= − qD ∂n , due to this resultant
∂x
concentration gradient, must be equal to the drift current density, i.e. - j diffusion = j drift :
qD ∂n
∂x = σE ⇒
⎛⎛
qD nqE ⎞⎞
⎜⎜ ⎟⎟ = σE
⎝⎝ kT ⎠⎠
€
σ
€
D = nq2
kT
Nernst-Einstein equation
€

BH47 Course B: Materials for Devices BH47
since
D = D o
exp( −Q kT) (BH45) and
⎛⎛
n = n o
exp⎜⎜
−qV ⎞⎞
⎝⎝ kT⎠⎠
⎟⎟ (BH46)
Nernst-Einstein (BH46): σ = Dnq2
kT
€
Therefore,
⎛⎛
lnσ ≈ lnσ 0
− Q ⎞⎞
⎜⎜ ⎟⎟ 1 ⎝⎝ k ⎠⎠ T
≈ D n o o q2
exp −Q
kT kT
€
( ) for small E, where
€
where σ 0
= D n 0 0 q2
kT
qV

1 / 4
, 3 / 4 1 / 4
, 3 / 4
BH48 Course B: Materials for Devices BH48
Solid State Ionic Conductors
For high levels of ionic conductivity a structure should include vacancies, or disordered regions.
e.g. anion conductors
yttrium stabilised zirconia (YSZ): (ZrO 2 + x mol.% Y 2 O 3 ); doped forms of
CeO 2 , Bi 2 O 3 (O 2- ions), and CaF 2 , PbF 2 (F - ions)
cation conductors
doped-Al 2 O 3 (with Na + or Ca 2+ ions), AgI (Ag + ions) – see BP4
Yttrium Stabilized Zirconia, YSZ: formation of oxygen vacancies by doping
Start with zirconia, ZrO 2 (fluorite structure: cations in an fcc arrangement & anions in all tetrahedral
interstices – see Data Book & AH31).
UN-DOPED
1 / 2
O 2-
1 / 4
, 3 / 4 1 / 4
, 3 / 4
Zr 4+
1 / 2 1 / 2
1 / 2
- dope with yttria, Y 2 O 3 : Y 3+ ions substitute for Zr 4+
- to maintain charge neutrality, oxygen vacancies are created:
(cation substitution using different valence ions)
for every 2 Zr 4+ ions replaced with 2 Y 3+ ions there is a loss of charge of 2+
⇒ one oxygen vacancy is formed (with an effective charge of 2+)
Oxygen ions, O 2- , are the carriers, & oxygen vacancies mediate the conduction.
Pure, undoped ZrO 2 is monoclinic at room temperature. The addition of Y 2 O 3 to ZrO 2 is also
beneficial because it allows the cubic form of fluorite (required for high ionic conductivity) to be
stabilised over a wide temperature range (room temp. → ~2500°C).
δ-Bi 2 O 3
Fluorite structure, with two formula units per cell.
⇒ in each unit cell:
⇒
8 tetrahedral oxygen sites
6 oxygen ions (on average)
2 oxygen vacancies per cell (on average)

BH49 Course B: Materials for Devices BH49
Applications of Ionic Conductors
Solid ionic conductors are analogous to electrolytic conductors, in that they can transport ions.
1. Oxygen Concentration Cell
anode
cathode
(for monitoring oxygen levels)
pO 2 (I)
YSZ solid
electrolyte
2O 2 -
pO 2 (II)
porous Pt electrodes (catalyse the
reactions & conduct electrons)
4e -
If there are different O 2 concentrations (partial pressures) at each electrode, oxygen is transported
from the high to the low concentration:
Pt(s) | O 2 (g)(I) | YSZ | O 2 (g)(II) | Pt(s)
If partial pressure of oxygen, pO 2 (I) < pO 2 (II):
Half cell reactions: O 2 (g)(II) + 4e - → 2O 2- reduction (cathode)
2O 2- → O 2 (g)(I) + 4e -
oxidation (anode)
Overall: O 2 (g)(II) → O 2 (g)(I)
Ions are transported across the electrolyte / Electrons through the external circuit.
Electrochemistry tells us that
E = − RT
4F ln pO 2(I)
pO 2
(II)
E = electrochemical cell potential (Volts)
[Note: not field in this case]
F = Faraday’s constant (96,500 C mol -1 ) R = gas constant (8.314 J K -1 mol -1 )
- if pO 2 (I) < pO 2 (II) E will be +ve, O 2- from (II) → (I)
- if pO 2 (I) > pO 2 (II) E will be -ve, O 2- from (I) → (II)

BH50 Course B: Materials for Devices BH50
Use as an O 2 sensor, e.g. for
combustion control in car engines.
Lambda Sensor: fitted into
vehicle exhaust system to
measure oxygen level, and
hence adjust fuel supply to
increase efficiency and
decrease emissions.
permeable
platinum
electrodes
YSZ
electrolyte
reference gas
chamber at
atmosphere
exhaust gas
↓
exhaust manifold
heater
protective
porous
ceramic
heater
connection
voltage
measurement
(see BP4)
The self generated voltage shows the difference in oxygen partial pressure between the exhaust and
the atmosphere. The sensor is linked via a computer to the fuel injection system to allow control of
the air / fuel ratio. The aim is to achieve complete stoichiometric combustion of petrol in order to
minimise emissions (unburned fuel, CO, NO x ):
C 8 H 18 + 25
2 O 2 → 8CO 2 +9H 2 O
From this equation, taking account of the relative molecular masses of the components & that the
atmosphere is ~4 N 2
/ 1 O 2
), for stoichiometric combustion the air-to-fuel-ratio (by weight) = 14.6.
measured ratio
Define λ = ( ) i.e. want λ = 1
14.6
€
1.0
cell potential oxygen partial pressure
emf (V)
emf
pO 2
0.5
10 -1
10 -11
pO 2 (atm)
Fuel-rich
Lean-burn
|
0.0
0.9
λ=1
1.1
10 -21
pO 2 < 10 -20 atm ← fuel-rich ← .0 → lean-burn → pO 2 > 10 -2 atm
high potential ⇒ add oxygen
low potential ⇒ add fuel

BH51 Course B: Materials for Devices BH51
2. Oxygen Pump
O 2- can be driven across a membrane using an metal / metal oxide mix
applied potential. This may be used for removing
oxygen from molten metals during processing.
molten
metal e -
3. Fuel Cell - produce electricity by direct oxidation of fuel.
In normal combustion (e.g. in flames, or internal combustion engines), fuels react with oxygen: the
fuel donates electrons to the oxygen atoms (fuel is oxidised; oxygen is reduced).
A fuel cell is an electrochemical device that converts the chemical energy in a fuel directly into
electrical energy (like a battery, but with the fuel continually supplied, instead of there being a fixed
source which is eventually used up). The fuel is typically CH 4 or H 2 , and the oxidation and reduction
reactions are separated by an electrolyte.
e.g. solid oxide fuel cell (SOFC)
anode: oxidation
H 2 + O 2- → H 2 O + 2e -
current
loop
e.g. YSZ
cathode: reduction
O 2 + 4e - →2O 2-
• high efficiency (e.g. up to ~60%; around twice that of an internal combustion engine)
• no polluting emissions if C is absent in the fuel
• no noise

BH52 Course B: Materials for Devices BH52
For effective ion conduction, the electrolyte (typically YSZ) needs to run at ~600 – 1000°C.
Need high ion conductivity & low electron conductivity.
Anode: electrically conducting; porous (to fuel, & water)
Cathode: electrically conducting; permeable to oxygen
(must not oxidise)
⇒ conducting ceramic: “cer-met”
⇒ doped porous manganite
stack many together, in series to give a useful voltage
cross-section
of real cell
Using H 2 fuel
anode | H 2 (g) | YSZ | O 2 (g) | cathode
Using CH 4 fuel
anode | CH 4 (g) | YSZ | O 2 (g) | cathode
Overall cell reaction:
2H 2 (g) + O 2 (g) → 2H 2 O(g)
CH 4 (g) + 2O 2 (g) → CO 2 (g) + 2H 2 O(g)
Half cell reactions:
2H 2 (g) + 2O 2- → 2H 2 O(g) + 4e - (anode)
CH 4 (g) + 4O 2- → CO 2 (g) + 2H 2 O(g) + 8e -
O 2 (g) + 4e - → 2O 2-
(cathode)
2O 2 (g) + 8e - → 4O 2-
Major materials challenges:
High operating temperature – need stability; no inter-reaction, or inter-diffusion; matched thermal
coefficients of expansion.
Anode operates in a highly reducing environment; Cathode in an oxidising environment.

BH53 Course B: Materials for Devices BH53
Fuel cell cars often use polymer electrolyte membrane fuel cells (PEMFC). Instead of YSZ, these
use thin, flexible polymer membranes (operating at ~80°C), which conduct protons (H + ). The overall
reaction is the same, but the half cell reactions are different, since H + , rather than O 2- , is transported
through the electrolyte:
Overall cell reaction: 2H 2 (g) + O 2 (g) → 2H 2 O(g)
Half cell reactions: 2H 2 (g) → 4H + (g) + 4e - (anode) transport of protons
across membrane,
rather than oxygen
O 2 (g) + 4H + + 4e - → 2H 2 O(g) (cathode) ions across electrolyte
sulphonated fluoropolymer membrane:
There’s a great advantage in the lower running temperature, but currently these require expensive Ptbased
catalysts.
The Hydrogen Economy
This requires hydrogen storage by compressing or liquefying the gas, which is energy intensive. It’s
also extremely flammable! Alternatively, we can produce H 2 where it’s required, by electrolysing
water:
H 2 O(l) → H 2 (g) + 1 / 2 O 2 (g) which requires energy input
or, by separating H from C in hydrocarbons (reforming): pass fuel & H 2 O over a heated catalyst
→ CO / CO 2 & H 2 (still polluting, & relies on fossil fuels)

BH54 Course B: Materials for Devices BH54
Glossary of Terms
Activation energy, Q: in diffusion, this is related to the energy required to move an atom from one
lattice site to another (J mol -1 , or eV atom -1 )
Anode: the electrode at which oxidation occurs.
Anisotropic: having different properties in different directions.
Antiferromagnetism: opposition of adjacent magnetic dipoles causing zero net magnetisation.
Arrhenius plot: plot of ln(rate) for a process or reaction, versus 1/T. The slope is proportional to
the activation energy.
Atactic polymer: a polymer with configurational base units in a random sequence.
Backbone: the main structure of a polymer onto which substituents are attached.
Birefringence, Δn: the difference in refractive index between two permitted vibration directions.
Bloch walls: the boundaries between magnetic domains.
Capacitance, C: charge per unit voltage (Farads, or C V -1 )
Capacitor: electrical device consisting of alternating layers of dielectric and conductor, which is
capable of storing charge.
Cathode: the electrode at which reduction occurs.
Centre of symmetry: a point through which an object can be inverted (i.e. all x, y, z are transformed
to –x, -y, -z) to bring the object into coincidence with itself.
Centrosymmetric: possessing a centre of symmetry.
Ceramics: compounds of metallic anions with non-metallic cations, which commonly have very
high melting temperatures
Chiral nematic, or Cholesteric: molecules in adjacent layers are oriented at a slight angle relative
to each other (rather than parallel, as in the nematic). Therefore the Director rotates helically, around
an axis perpendicular to the layers.
Compensator: sample of an optically anisotropic crystal with a known birefringence (often quartz).
Coercive field, E C
/ H C
: the electric / magnetic field that is required to depolarise / demagnetise a
ferroelectric / magnetic material (V m -1 / A m -1 )
Concentration cell: an electrochemical cell whose electromotive force is driven by differences in
concentrations across the electrolyte.
Concentration gradient: the rate of change of composition with distance (m -4 ).
Conformation: the structure of a polymer chain that arises from rotation about the single bonds.
Constructive interference: the combination of rays which are in phase and give an intense beam.
Coordination number: the number of atoms forming a polyhedron around a central atom in a
structure.
Coordination polyhedron: the polyhedron that can be constructed around a cation with the
surrounding anions forming the vertices.

BH55 Course B: Materials for Devices BH55
Cross-linking: a process in which bonds are formed joining adjacent molecules. At low density
these bonds add to the elasticity of a polymer, at higher densities they produce rigidity.
Crossed polars: two sheets of polarizing material (polarizer and analyzer), oriented at 90° to each
other, between which a sample is placed for optical examination.
Curie temperature: the temperature above which ferroelectric / ferromagnetic behaviour is lost
(°C or K).
Current density, j: the current flowing through unit cross-sectional area (A m -2 ).
Diamagnetism: the effect caused by the magnetic moment due to orbiting electrons which produces
a slight opposition to imposed magnetic field.
Dielectric: an electrical insulator which is polarisable.
Dielectric breakdown: the passage of current through a dielectric material upon application of a
large electric field. The maximum field that a dielectric can sustain without breakdown is the
dielectric strength.
Dielectric constant, κ : a measure of the rate of change of polarisation with electric field; the ratio
of the permittivity of a material to the permittivity of free space.
Diffusion: transport of atoms in which the diffusing atom moves relative to its neighbours.
Diffusion coefficient (or diffusivity), D: a temperature dependent coefficient related to the rate at
which atoms diffuse, which also depends on the activation energy (m 2 s -1 )
Dipole: two equal charges of opposite sign, separated by a small distance.
Dipole moment, µ: the electric field generated by a spatial separation of positive and negative
charges (C m) or the magnetic moment generated by electron orbitals and spin.
Director, D: the direction of preferred orientation of molecules in liquid crystals.
Disclination: point at which the Director in a liquid crystal undergoes an abrupt change in
orientation.
Displacement field, D: resultant total polarisation field created in a material by application of an
electric field (C m -2 )
Domains: regions within a material in which all the electric / magnetic dipoles are aligned.
Domain wall: the boundary between ferroelectric / magnetic domains.
Drift velocity: the average rate at which charge carriers move through a material under the
application of an electric field (m s -1 )
Exchange interaction / exchange energy: a quantum mechanical electron – electron interaction
which favours parallel alignment of magnetic moments. A negative exchange interaction favours antiparallel
alignment of moments.
Extinction positions: orientations in which an anisotropic material appears black between crossed
polars. (They occur every 90°, when the permitted vibration directions of the sample are parallel
to the polarizer / analyzer.)
Fatigue: accumulation of defects in a ferroelectric material which gradually degrades the amount of
switched charge.

BH56 Course B: Materials for Devices BH56
Ferrimagnetism: magnetic behaviour obtained when two types of dipole, with different strengths,
oppose one another, leaving a net magnetization.
Ferroelectric: a material possessing a spontaneous net electrical polarisation which can be switched
by an external electric field. A ferroelectric material must be polar (and switchable).
Ferroelectric phase transition: transition from a high temperature paraelectric phase to a low
temperature ferroelectric phase. Characterised by onset of spontaneous polarisation.
Ferroelectric / magnetic hysteresis: difference in polarisation / magnetisation behaviour upon
reversing electric / magnetic field.
Ferromagnetism: the existence of a spontaneous magnetic dipole; alignment of magnetic domains
can result in a net magnetisation after removal of an imposed magnetic field.
Flux density, J: the number of atoms / ions crossing a unit cross sectional area in unit time (m -2 s -1 ).
Freedericksz Transition: reorientation of liquid crystal molecules on the application of an electric
field.
Frenkel defect: a crystalline defect consisting of an interstitial and a vacancy.
Fuel cell: an energy conversion device that produces electricity by the electrochemical oxidation of a
fuel.
Hard magnet: a ferromagnetic material that has a wide hysteresis loop and high remanence.
Hysteresis loop: the loop traced out by polarisation / magnetisation as an applied electric / magnetic
field is cycled.
Inversion centre: same as centre of symmetry.
Isotactic polymer: a polymer with configurational base units in a single sequential arrangement.
Isotropic: having uniform physical properties in all directions.
Kuhn length / segment: a theoretical treatment of a polymer chain, divided into n Kuhn segments,
with Kuhn length, l, so that each segment can be thought of as freely jointed. Complex polymers may
be simply modeled as a random walk.
Lambda: the air / fuel weight ratio for a petrol engine normalised by the corresponding ratio for
complete stoichiometric combustion of petrol.
Liquid Crystal: a phase characterised by anisotropy of properties without the existence of a 3-
dimensional crystal lattice, i.e. orientational order, without positional order.
Magnetic Flux Density, or Inductance, B: total magnetic flux per unit area (Tesla)
Magnetic moment, m: strength of the magnetic field associated with electron orbitals / spin (A m 2 ).
Magnetic permeability, µ: the ratio between magnetic flux density, B, & applied magnetic field, H
(H m -1 ).
Magnetic susceptibility, χ : a measure of the effect of a magnetic field on the magnetisation of a
material; the ratio between magnetisation, M, and the applied field, H (dimensionless).
Magnetisation, M: the sum of all magnetic moments per unit volume (A m -1 ).
Magnetocrystalline anisotropy energy, K: extra energy, per unit volume, required to magnetise a
material along the ‘hard’ crystallographic axis (J m -3 ).

BH57 Course B: Materials for Devices BH57
Magnetostriction coefficient, Λ : fractional change in length on changing magnetisation from zero
to saturation (dimensionless).
Monomer: the simple chemical unit which, when many are joined together, form a polymer.
Nematic: spontaneously anisotropic liquid, composed of rod-shaped molecules with parallel
alignment.
Optic Axis: for a birefringent sample, a direction along which no birefringence is observed, i.e. a
direction perpendicular to an isotropic section. The optic axis of a nematic liquid crystal is along
the Director.
Optical Path Difference: the extra distance traveled by the fast ray outside an anisotropic material
in the time it takes for the slow ray to reach the edge of the sample (m).
Order parameter: describes the orientational order of a liquid crystalline material, taking account of
the deviation of orientation of individual molecules from the Director (zero for complete disorder;
unity for perfect order).
Paramagnetism: the net magnetic moment caused by alignment of the electron spins when a
magnetic field is applied.
Permitted Vibration Directions: two perpendicular directions in an anisotropic material along
which the transverse electric field of light can vibrate.
Permittivity, ε : describes how an electric field is affected by a dielectric (F m -1 ).
Perovskite: family of ABX 3
compounds which commonly show displacive phase transitions by
octahedral tilts or atomic displacements.
Phase: a part of a system, homogeneous in composition and structure.
Phase diagram: a diagram showing the equilibrium phases and phase compositions at each
combination of temperature (and/or pressure) and overall composition.
Photoelasticity: stress-induced birefringence. In a polymer, this is caused by stress-induced
alignment of polymer chains.
Piezoelectricity: the application of stress produces a change in the dielectric polarisation of a
material or, conversely, application of an electric field results in strain. Piezoelectric materials must be
non-centrosymmetric.
Piezoelectric coefficient, d: measure of the rate of change of dielectric polarisation with applied
stress (C N -1 ).
Pitch: the distance it takes for the Director in a cholesteric / chiral nematic liquid crystal to go
through one complete 360° rotation (m).
Polar material: a material with crystal symmetry such that it contains one (or more) unique
direction.
Polarization, P: the dipole moment per unit volume in a medium, or charge per unit area (C m -2 )
Polarization (of light): the property of electromagnetic waves that describes the direction of the
transverse electric field.
Polarized light: light that has passed through a sheet of polarizer, and has a transverse electric field
that vibrates in one direction only.

BH58 Course B: Materials for Devices BH58
Poled ferroelectric: a ferroelectric material that has been brought to saturation by a strong electric
field and then allowed to relax back to the remanent condition.
Polymer: a long chain of covalently bonded atoms.
Pyroelectricity: change in electrical polarisation due to a change in temperature. Pyroelectric
materials must be polar.
Pyroelectric coefficient, p: measure of the rate of change of dielectric polarisation with temperature
(C m -2 K -1 ).
Refractive Index, n: a property that describes the speed of light in a material relative to that in a
vacuum. The higher the refractive index, the slower the light travels.
Remanence: the polarisation / magnetisation that remains in a material after it has been removed
from an applied electric / magnetic field, due to permanent alignment of dipoles (C m -2 / A m -1 ).
Saturation polarisation / magnetisation: the state in which all electric / magnetic dipoles have
been aligned by an applied electric / magnetic field, producing maximum polarisation / magnetisation
(C m -2 / A m -1 ).
Sawyer-Tower circuit: test circuit for measuring the hysteresis loop of ferroelectric materials.
Schottky defect: an anion and cation vacancy pair.
Smectic: liquid crystal phase in which the molecules organise themselves into layers.
SOFC: Solid Oxide Fuel Cell.
Soft magnet: ferromagnetic material that has a narrow hysteresis loop and little energy loss on
cycling the field.
Spontaneous polarisation / magnetisation: the co-operative phenomenon of electric / magnetic
dipole ordering below the Curie temperature (C m -2 / A m -1 ).
Stoichiometric: obeying a strict chemical formula, e.g. anion / cation ratio.
Susceptibility: see Magnetic susceptibility.
Syndiotactic polymer: a polymer with configurational base units in a regular, alternating sequence.
Tacticity: the arrangement of side-groups on a polymer backbone.
Transducer: a device that receives one type of input (such as strain) and provides a different type of
output (such as an electrical signal).
Transformation (or transition): these terms are used interchangeably to denote a change in phase.
Twisted nematic: liquid crystals physically forced to adopt a chiral structure.
Unique Direction: a lattice vector in a crystal which is not repeated by any of the symmetry
elements present.

Qu. Sheet 5 MATERIALS SCIENCE BQ1
Course B: Materials for Devices
Question Sheet 5
1. A nematic liquid crystal sample, which includes various different director orientation patterns
in different regions [as illustrated in the diagrams (a) – (e)], is viewed between crossed polars.
The appearance of the region of the sample shown in pattern (a) is illustrated. Sketch what
patterns you would observe for the other examples, if the polarizer / analyzer remain in the
same orientation as for (a).
(a) Orientation of liquid crystal
molecules:
Analyzer
Appearance:
Polarizer
(b) (c) (d) (e)
2. For this question you will need to work on-line. Access the following URL*:
http://plc.cwru.edu/tutorial/enhanced/lab/lab.htm
Select Colors from Birefringence on the left hand menu, read through Introduction and
Instructions, and then open the Experiment page. You should see a Virtual Laboratory,
offering you control of Sample Parameters: Angle, thickness (called Length) and birefringence
(called Delta n). White light is shown split into 3 components, each passing
through the sample, and adding together at the right hand side to show a colour on a screen.
Set the Sample Material as General, the Angle = 45 Degrees (or as close as you can get to it),
and Length (i.e. sample thickness) to be about 1000 nm (the precise value is unimportant).
Now vary Delta n (Δn, or birefringence), and observe what happens.
Record the smallest (non-zero) values of Δn that yield zero transmission for each of blue,
green and red light, and thus deduce each of their wavelengths. Note the transmitted colour
when each component is completely blocked.
Can you achieve zero blue transmission at a higher value of Δn
Relate your results to the Michel-Levy chart printed in your handout (BH10).
Why does this exercise work best with the Angle set to 45° Observe what happens as you
change this value.
*Courtesy of the PLC Tutorial (http://plc.cwru.edu), Macromolecular Science and Physics
Departments, CWRU, and the ALCOM S&T Center
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BQ1 - 2 - Question Sheet 5
3. A wedge of quartz, with a birefringence of 0.009 and an angle of 1°, is cut with a permitted
vibration direction along its length. Describe and explain what you will see when the wedge
is viewed between crossed polars, with its length at 45° to the polarizer / analyzer, in sodium
light (λ = 590 nm). What is the distance between dark bands Describe and explain what
happens if the wedge is then rotated so that its length lies parallel with (a) the polarizer, and
(b) the analyzer.
The quartz wedge is now viewed in white light, and a uniform nematic liquid crystal sample
is added to the light path (i.e. the quartz wedge is placed on top of the liquid crystal sample
and viewed between crossed polars). The long axes of the liquid crystal molecules in this
sample (known to correspond to the slow vibration direction of the sample) are oriented
parallel to the length of the wedge. A black band is observed on the wedge, 13.6 mm from its
tip. Does the length of the wedge correspond to the fast or the slow vibration direction in the
quartz What colour would the liquid crystal sample show if the wedge were removed
The liquid crystal sample is measured to have a thickness of 0.05 mm. What is its
birefringence
4. A cholesteric, or twisted nematic, liquid crystal can selectively reflect light of a specific
wavelength, related to its pitch (due to constructive interference of light reflecting from
parallel equivalent planes). What colour light will be
reflected from a twisted nematic of pitch 792 nm illuminated
and viewed at 45° (see diagram).
(For a hint on the geometry for constructive interference,
refer to AH76.)
On increasing the temperature of this sample, the
reflected light becomes blue. Explain this observation.
Pitch
5. (a) Given that the C-C bond length is 0.154 nm and the C-H bond length is 0.114 nm, estimate
the length and width of a linear polyethylene molecule consisting of 1000 carbon atoms (the
tetrahedral bond angle is 109.5°).
(b)
(c)
(d)
(e)
What is the molecular weight of this molecule
What is the root mean squared end to end distance, assuming a completely flexible chain,
made up of C-C segments which can twist and rotate completely independently
What is the root mean squared end to end distance using the (more reasonable) assumption
that the Kuhn length is 3.5 × one C-C segment
If this chain were scaled up to the dimensions of a piece of string (e.g. about 2 mm wide),
how long would it be In a scaled-up polymer model, if this piece of string represents a
linear polyethylene molecule, what would be the average end to end distance in the model
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Qu. Sheet 6 MATERIALS SCIENCE BQ2
Course B: Materials for Devices
Question Sheet 6
1. What is the capacitance of an empty parallel plate capacitor with plates, each with a
surface area of 500 mm 2 , separated by a 2.5 mm gap What is the magnitude of the
charge stored on each plate, and hence the surface charge density, if a potential
difference of 10 V is applied across them
If SiO 2 (κ = 4.52) is introduced between the plates (and a potential difference of 10 V
is applied), calculate the capacitance, the magnitude of the charge stored on each plate
and the total surface charge density on the plates. What is the polarisation of the SiO 2
With reference to the 1 st part of this question (i.e. the empty parallel plate capacitor),
explain why the total charge density on the plates is not the same as the polarisation
of the dielectric.
What would be the surface area of a new capacitor of the same capacitance and
thickness as the SiO 2 one, but made from a ferroelectric material with a dielectric
constant one hundred times greater than SiO 2
2. (a) Sketch a plan, on (001), of one unit cell of the cubic ZnS sphalerite structure
(putting a sulphur ion at the origin of the cell). Describe the shape of the coordination
polyhedra of S around Zn. Are these filled polyhedra all in the same orientation Do
they share corners, edges, or faces
(b) Describe the asymmetry along the directions. Is [111] a unique direction
Does sphalerite have any centres of symmetry
(c) The unit cell that you have drawn is now subject to the shear stress pictured below
(equivalent to a tensile stress along [ 110] and ⎡⎡
⎣⎣110⎤⎤
⎦⎦, and a compressive stress along
⎡⎡110 ⎣⎣
⎤⎤
⎦⎦ and ⎡⎡
⎣⎣110⎤⎤
⎦⎦). Sketch a plan, on (001), of one (filled) coordination polyhedron,
before and after application of the stress, and describe how the ion positions have
changed. Do the centre-of-mass coordinates for the S coordination polyhedron
change What about its centre-of-charge
τ
τ
y
x
(d) If the Zn remains equidistant from its four coordinating sulphur ions, in what
direction does it move Is it still at the centre-of-charge of the S polyhedron In what
direction is the resultant dipole moment Is this the same direction for all filled
polyhedra What property does this lead to
τ
τ
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Qu. Sheet 6 MATERIALS SCIENCE BQ2
Course B: Materials for Devices
3. A ceramic material has a piezoelectric coefficient of 250 pC N -1 and a dielectric
constant of 500. A compressive stress of 5 MPa is applied across a 1 cm thick sample of
the material. Calculate the voltage that will develop across the sample and determine
whether this is sufficient to create a spark across a 1 mm air gap. [Breakdown voltage of
air is ~ 3 MV m -1 ]
4. Pyroelectric infra-red detectors sense temperature change through a change in surface
charge on a sheet of a pyroelectric crystal, and hence a change in measured voltage
across the sheet. Write down an equation for the sensitivity of the detector (ΔV/ΔT) in
terms of the pyroelectric coefficient, p, and the dielectric constant, κ. Hence explain
briefly why BaTiO 3 is not an ideal material to be used in a pyroelectric detector. Using
information provided in the databook, estimate the ΔV/ΔT response achieved from a
polyvinylidene fluoride (PVDF) detector of 1 mm thickness. Comment on the value
obtained.
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Qu. Sheet 7 MATERIALS SCIENCE BQ3
Course B: Materials for Devices
Question Sheet 7
1. (a) Draw a plan of the perovskite BaTiO 3
structure projected onto (010), i.e. the x-z plane.
Indicate mirror planes normal to the paper, and rotation axes parallel to [010]. What are the
fractional coordinates of the mirror planes parallel to (010) What is the crystal system Is
there a centre of symmetry and, if so, where What is the shape of the coordination
polyhedra around Ti, and how are they linked
(b) Re-draw the crystal structure with all the Ti cations displaced slightly along [001],
towards the neighbouring oxygen (illustrate a displacement of about 1 / 5 th of the cell
parameter). Which symmetry elements are lost, and which survive What is the new crystal
system Is there now a centre of symmetry
(c) If the Ti 4+ cation moves 0.2 Å along [001] with respect to other ions as a result of the
ferroelectric phase transition in BaTiO 3
, what will be the dipole moment formed in a single
unit cell (Note that this displacement is much less than that depicted in (b).) If the unit cell
parameter of the cubic phase is 4.0 Å, and the volume of the tetragonal phase is almost the
same as that of the cubic phase, what is the polarisation If the polarisation can be thought
of as an effective charge density accumulating on either side of the sample, what is the
potential difference across a sample of BaTiO 3
of thickness 0.1 mm and of dielectric
constant 1000
2. By referring to the phase diagram of PZT in your Handout (BH35), answer the following:
(a) What is the Curie temperature of PbTi 0.4
Zr 0.6
O 3
(b) What is the crystal structure of PbTi 0.6
Zr 0.4
O 3
at room temperature
(c) What conditions of polarisation would result in a specimen of PbTi 0.6
Zr 0.4
O 3
after each
of the following treatments:
(i) Strong electric field applied at 750 K, held for some hours, & switched off.
Specimen cooled to 300 K.
(ii) Strong electric field applied at 620 K, held for some hours, specimen cooled to
300K, then field switched off.
3. The Bloch wall energy for Ni is about 10 -3 J m -2 . If a spherical Ni particle consists of two
equal and opposite magnetic domains, write down an equation for the total domain wall
energy of the particle, in terms of its radius, r.
The stray field energy (given in Joules) of a small, single domain Ni particle of radius, r, is
approximately 2×10 5 r 3 . Estimate the critical size of a Ni particle, below which magnetic domains
will not form, stating your assumptions.
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BQ3 - 2 - Question Sheet 7
4. A cubic transformer core is made from Supermalloy, and its power dissipation is found to
be ~2 W at 50 Hz. What is the energy required to switch the magnetisation direction of the
transformer core
[Hint: a power dissipation of 2 W means 2 J s -1 , and there are 50 complete switching cycles
– plus, to minus, and back again – each second.]
By comparing the hysteresis loops shown in the DoITPoMS TLP Ferromagnetic Materials,
estimate the energy required to switch the magnetisation direction of a similar sized piece of the
hard magnet, Alcomax II.
[Hint: you can make a direct comparison of hysteresis loop areas without worrying about the
units used, as long as they’re the same!]
5. (a) Using the data given in the table below, calculate the saturation magnetisation of elemental
Fe (α).
(b) Magnetite, Fe 3
O 4
, has the inverse spinel structure, with half of the Fe 3+ ions in octahedral sites
(with spin moments aligned parallel to each other), and the other half in tetrahedral sites (again,
with spin moments parallel to each other, but anti-parallel to those in the octahedral sites). Fe 2+
ions are in octahedral sites, with aligned spin moments. Each cubic unit cell (a = 8.39 Å) contains
8 Fe 2+ ions and 16 Fe 3+ ions. Calculate the saturation magnetisation of magnetite.
Net magnetic moment (×10 -23 A m 2 )
Fe atom 2.06
Fe 2+ cation 3.71
Fe 3+ cation 4.64
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Qu.Sheet 8 MATERIALS SCIENCE BQ4
Course B: Materials for Devices
Question Sheet 8
1. (a) Sketch a unit cell of CaF 2 viewed down [001].
(b) Describe the coordination of Ca by F, and of F by Ca.
(c) Comment on the packing of Ca ions and the location of the F ions. How does this
compare with the zinc blende structure
(d) In δ-Bi 2 O 3 , the Bi sub-lattice is the same as that of Ca in CaF 2 and S in ZnS, but the
stoichiometry means that there are vacant anion sites, randomly distributed. Sketch a
possible unit cell of δ-Bi 2 O 3 .
(e) Comment on why δ-Bi 2 O 3 is a fast ionic conductor whilst stoichiometric CaF 2 and ZnS
are not.
(f) Calculate the composition of YSZ (i.e. Y 2 O 3 doped ZrO 2 , Zr 1-x Y x O [2-(x/2)] ) which would
give just one quarter of the oxygen vacancy concentration as in δ-Bi 2 O 3 .
2. (Tripos Question; 2008)
How is a high-temperature cubic zirconia phase stabilised at room temperature and what
are the consequences for its oxygen mobility Describe how such a material might be
used in a hydrogen fuel cell. In your answer refer to the chemical reactions involved and
explain how a voltage is generated. [40%]
Different amounts of yttria (Y 2 O 3 ) are added to zirconia (ZrO 2 ) to produce two ceramic
materials: Y 0.1 Zr 0.9 O x and Y 0.15 Zr 0.85 O y . The molar cation compositions are readily
determined, but what are the molar oxygen compositions x and y [10%]
Both ceramic materials are cubic and give an X-ray diffraction pattern characteristic of
the fluorite structure, in which the first and largest peak is a reflection from the {111}
planes. Using CuKα radiation this peak occurs at 2θ = 30.147° for Y 0.1 Zr 0.9 O x whereas it
is shifted to 2θ = 29.732° for Y 0.15 Zr 0.85 O y . Give a reason for this change in the
diffraction pattern between the two materials. [15%]
(i) What are the oxygen vacancy concentrations, n, in Y 0.1 Zr 0.9 O x and in Y 0.15 Zr 0.85 O y
(ii) At 1000°C the ionic conductivity of Y 0.1 Zr 0.9 O x is 2.5 × 10 -4 Ω -1 m -1 . What is the
ionic conductivity of Y 0.15 Zr 0.85 O y at the same temperature
(iii) Determine the diffusivity of oxygen in Y 0.15 Zr 0.85 O y at 1000°C. [35%]
[ionic radii: Y 3+ = 1.159 Å, Zr 4+ = 0.98 Å, O 2- = 1.28 Å]
3. (a) From the Arrhenius plot in your handout (BH47), showing ionic conductivity versus
inverse temperature for a number of ionic conductors, estimate the activation energy for
ion transport in yttria-stabilised zirconia (YSZ).
(b) In Zr 0.8 Y 0.2 O 1.9 how many oxygen vacancies are there per unit cell If the lattice
parameter of (cubic) YSZ is 0.54 nm, calculate the number density of vacancies.
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Qu.Sheet 8 MATERIALS SCIENCE BQ4
Course B: Materials for Devices
(c) The Nernst-Einstein equation indicates that the ratio σ /D for a given material varies
only with temperature. Calculate σ/D for Zr 0.8 Y 0.2 O 1.9 at 800°C.
(d) It is often found that the experimental values differ from those predicted by the Nernst-
Einstein equation. Suggest why this might be the case.
4. The α phase of silver iodide (AgI) has a cubic I lattice with a = 5.0855 Å. It is a Ag +
ionic conductor at 150ºC with a diffusivity of 4.5 × 10 -11 m 2 s -1 . A potential difference is
applied across a sample of AgI, using Ag for both electrodes, and current is allowed to
flow:
e - ↑
Ag + is reduced to Ag at the cathode: Ag + + e → Ag
Ag is oxidised to Ag + at the anode: Ag → Ag + + e
What is the number density of charge carriers in AgI
What is the conductivity of the AgI at 150°C
AgI
anode cathode
What is the mass of silver deposited at the cathode if a current of 5 mA flows through
the circuit for 5 minutes
5. (Tripos Question; 2009)
Describe the phenomenon of pyroelectricity, giving an example of a widespread
application. [15%]
ZnO can have the wurtzite crystal structure. Draw a plan of a 2 × 2 array of unit cells of
this structure, projected down the z-axis and with an oxygen atom at the origin. Describe
this structure in terms of coordination polyhedra. What fraction of tetrahedral interstices
in the anion lattice are occupied by Zn cations [30%]
Assuming that the magnitude of the charge on each ion is 2e, and taking into account the
sharing of oxygen ions between filled interstices, determine the charge due to oxygen
which acts at the centre of charge of one of these anion tetrahedra. Give the fractional
coordinates of this centre of charge. [Hint: the geometric centre of a tetrahedron is 1 / 4 of
the distance from its base to its apex.] [20%]
With the zinc ion positioned at ( 2 / 3 , 1 / 3 , u), show that the net dipole moment of one
tetrahedron is 2ec ( 1 / 8 – u), where c is the lattice parameter (along z). Hence, calculate
the electric polarisation, P, of a single crystal of ZnO at room temperature. What further
information would be required to determine the magnitude of the pyroelectric coefficient
of ZnO [35%]
[For ZnO: a = 0.325 nm; c = 0.521 nm; u = 0.118]
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