CHAPTER 10 (Rev 03)
Electrical Properties of
Materials
(Callister Ch.19, p680 – 688, pg 693-699 ,p722-723
(Callister 10th ed Ch.18 pg.689-709; pg.729-731)
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10.1 Outcomes

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
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Formulate the relevant definitions of the terms and know and
understand the terms used in this chapter
Relate examples of typical practical applications of conductors,
semiconductors and insulators
Explain why electrical resistance (R) is not a materials property,
whilst resistivity and conductivity are considered as materials
properties
Explain and indicate the similarity between the microscopic and
macroscopic form of Ohm's law .Calculate the resistivity and
conductivity of a material at a specific temperature if its
temperature resistivity coefficient is given
Predict the effect of impurities and alloying elements on the
resistivity and the conductivity of a conductor, by applying
graphs with data
Explain why the resistivity of a conductor like copper is
increased by: (1) a temperature increase, (2) alloy additions and
(3) by cold working of the material
2
Outcomes






apply the energy-band model to explain why some elements are
good conductors and why others are poor conductors of electricity
Explain the mechanism by which electrical conduction occurs in
intrinsic semiconductors like silicon and germanium
Derive an expression for the conductivity of an intrinsic
semiconductor calculate the resistivity and the conductivity of
intrinsic semiconductors.
Explain why the conductivity of semiconductors is increased by an
increase of the temperature, whilst the conductivity of good
conductors is decreased by an increase of the temperature.
Explain by using a typical example of each, the difference in the
mechanism of electrical conduction between n-type extrinsic
semiconductors and p-type extrinsic semiconductors.
Derive expressions for the conductivity of n-type and p-type
extrinsic semiconductors and calculate the resistivity and the
conductivity of extrinsic semiconductors
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10.2 Use of electricity and materials
Which areas are we going to cover?
Materials are vital for conducting, for insulating, for
semi-conductors and for capacitors. Without these,
our civilisation would still be very primitive.
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Use of Electricity and Materials
What is the fundamental difference between these?
The differences lie in the electron arrangements
and in particular in the valence electrons
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Use of Electricity and Materials
Valence electrons and the band gap
The valence electrons need to be lifted into the conduction band to
conduct electricity. In conductors this requires no energy, and in
insulators it requires a very large energy input. In semi-conductors one
can supply enough energy to become a conductor or to “switch it off”.
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10.3 Electrical Conduction in Metals
How do metals become good conductors?
No electric field
Electric field: electrons directed
1. In the absence of an electric potential, the motion of
the valence electrons is at random and there is no net
current or current flow.
2. When an electric potential is applied, the electrons
attain a directed drift velocity, which results in directed
flow of electrons (or current).
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Electrical Conduction and Resistivity
i) Ohm's Law:
∆V = I R
voltage drop (volts = J/C)
resistance (Ohms)
current (amps = C/s)
C = Coulomb
A
(cross
sect.
area)
e-
I
∆V
L
Note the geometry
dependence: A and L
ii) Resistivity ρ and Conductivity σ:
-- geometry-independent forms of Ohm's Law
-- Resistivity is a matl. property, is independent of sample
E: electric
field
intensity
iii) Resistance:
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∆V
I
= ρ
A
L
ρL
L
R=
=
A Aσ
resistivity
(Ohm-m)
J: flux or current density
conductivity
1
σ=
ρ
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Electrical Conduction and Resistivity
Ohm's law can also be expressed in its microscopic
form, which is independent of the shape or size of the
conductor:
J = E/ρ
or
J = Eσ
J = current density [A / m2 ]
E = electrical field [V / m ]
ρ
= electrical resistivity [Ω.m]
σ = electrical conductivity [Ω.m] -1
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Electrical Conduction and Resistivity
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Conductivity: Comparison
Room Temp. values for σ (Ohm-m)-1
METALS
Silver
Copper
Iron
conductors
6.8 x 10 7
6.0 x 10 7
1.0 x 10 7
CERAMICS
-10
Soda-lime glass 10 -10-11
Concrete
10 -9
Aluminum oxide <10-13
POLYMERS
SEMICONDUCTORS
Polystyrene
Silicon
4 x 10 -4
Polyethylene
Germanium 2 x 10 0
GaAs
10 -6
semiconductors
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-14
<10
10 -15-10-17
insulators
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10.4 Factors that affect the Conductivity
Some metals are good conductors and others not?
i)
Pure defect-free metals at low temperatures
have low resistivity or high conductivity, like
pure and annealed Cu at room temperature
ii) In general a metal’s total resistivity is governed
by a combination of:
External component
– a thermal (ρt) component
– an impurity (ρi) component
Material
– a deformation (ρd) component component
ρTot = ρt + ρi + ρd
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Factors that affect the Conductivity
The material and external components
ρTot = ρt + ρi + ρd
The Material component ρr = (ρi+ρr)
is known as the
residual resistivity with the total becoming
ρTot = ρt + ρr
i)
The thermal ρt increases almost linearly with temperature.
ii) At higher temperatures the metal ions at the lattice positions
vibrate more vigorously, by which the conduction (valence)
electrons are increasingly scattered.
iii) Thus, the free path of electrons are decreased and
resistivity
is increased
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Factors that affect the Conductivity
The thermal component of resistivity/conductivity
ρT = ρ0°C (1 + αTT) [Ω.m] or σ = σ0°C (1 + αTT)-1 [Ω.m]-1
ρ0°C = electrical resistivity at 0°C σ0°C= conductivity at 0°C
αT = temperature coefficient of resistivity
(αT is a function of the type of metal)
T = temperature of conductor (°C)
An increase in
temperature as well
as impurities increases
Copper’s resistivity,
in many cases
almost linearly
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Factors that affect the Conductivity
The deformation and alloying components
i) Alloying Elements
Many alloying elements form solid solutions. This strains the
crystal lattice and causes additional scattering of the
conduction electrons. Resistivity increases
ii) Crystal Defects
Crystal defects like dislocations and grain boundaries also
cause additional scattering of the conducting electrons. By
recrystallisation the conductivity may be restored.
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Factors that affect the Conductivity
The resistivity of Cu – Ni alloys
1. Note the very
sharp increase from
adding Ni to Cu
2. Secondly a
temperature increase
has a moderate
effect.
3. Deformation has
the smallest effect.
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10.5 Energy band model of conduction
The fundamental model of conduction and insulation
The energy band model explains conductivity by:
i.
 The number of electrons available for conduction
 The arrangement of these electrons in energy levels
 How these levels are filled with electrons
ii.
According to the Pauli-exclusion principle each electron has
an unique energy and finds itself on energy levels
according to the electron configuration of that element
iii. Electrons fill the lowest energy levels with two electrons
with opposite spins per energy level.
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Energy band model of conduction
The fundamental model of conduction and insulation
1. The electrical properties of any material is governed by
its electron structure, i.e. the arrangement of the outer
electron levels and whether they are filled or not
2. Electrons in lower levels are more strongly bound to the
core of the atom but at higher levels they are more free
to move around and become available for electron
pairing.
.
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Energy band model of conduction
Example of Sodium
Sodium (Na)
Atomic Number: 11
Electron configuration = 1s2 2s2 2p6 3s1
Note that the
3s1 electron is
only loosely
bound and is
available for
conduction
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Energy band structure
Metals (Cu etc)
Magnesium (e.g.)
Insulators
Semi-conductors
Ef = Highest filled level at 0 K (Fermi-energy). E.g. Mg no band gap.
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metals (a) and (b) electrons are easily elevated into the conduction band.20
Energy band structure
 Valence band: – filled by highest energy electrons
 Conduction band: - empty, lowest unfilled energy
band
In Mg the valence band and
conduction band already overlap
Conduction
band
Metals have small band gap
valence band
Adapted from Fig. 18.3, Callister 7e.
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10.6 Energy band model: insulators
All the electrons are strongly bound by covalent or
ionic interactions.
No free electrons are available for conduction
The valence electrons are separated by a large
energy gap Eg from the conduction band
For conduction a very large
amount of energy is required
to “tear loose” an electron
into the conduction band
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10.7 Semi-conductors: band model
 Electrical conduction in semi-conductors is possible by the
activation of conducting electrons over the band gap
 This requires a “switch-on” excitation energy of >Eg but it
can also be “switched off” by removing this activation.
 Semi-conductors are, therefore, basically nothing else
than an “on-off switch”, from there the binary system.
A conducting electron
can be excited at will
into the conduction
band or de-excited to
become a nonconductor
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Semi-conductors: types
 Intrinsic semi-conductors: these consist of pure
Silicon or Germanium and have no additive alloying
elements
 Extrinsic semi-conductors: these consist of Si and
Ge but alloyed with dilute solid solution quantities of
elements from either group 3 or group 5 atoms
3
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10.8 Intrinsic Semi-conductors
i.
ii.
iii.
iv.
v.
Silicon and Germanium are both covalent bonded
These form directional sp3 hybrid covalent bonds
A Diamond cubic structure is formed
Each atom’s four valence electrons are therefore
strongly bound to the core of the atom
These valence electrons therefore need a high exciting
potential Eg to become conductive
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Intrinsic Semi-conductors
Conduction in Si and Ge semi-conductors
1. Where an electron is excited from its energy level, it leaves
a positively charged “hole” behind
2. Conduction will occur under the applied electric field due
to the movements of the positive "holes" to the negative pole
and the negative electrons to the positive pole
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Electric field
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Intrinsic Semi-conductors
Relationships for Conduction in Si and Ge
1. There are two charge carriers contributing to
electrical conduction in intrinsic semiconductors i.e.
free electrons and positive holes
2. The conduction is expressed as:
σ = nqµn + pqµp (Ω.m)-1
n = number of conduction electrons/m3
p = number of conduction holes/m3
q = absolute value of electron or hole charge
(q = 1.6 x 10 -19 coulombs)
µn = mobility of electrons [m2/V.s]
µp = mobility of holes [m2/V.s]
But n = p = ni ni = intrinsic carriers/m3
Thus:
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σ = ni q (µn + µp ) (Ωm)-1
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Conduction in Intrinsic Semi-conductors
 Concept of electrons and holes:
valence
electron
no applied
electric field
electron
hole
pair migration
electron
hole
pair creation
Si atom
+ applied
electric field
+
applied
electric field
n = p = ni
Electrical Conductivity given by:
-1
σ
=
n
q
(µ
+
µ
)
(Ωm)
i
n
p
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Conduction in Intrinsic Semi-conductors
Effect of temperature on intrinsic semi-conductivity
1. At zero Kelvin the valence bands of intrinsic semiconductors are completely filled and their
conduction bands are completely empty.
2. In contrast to metals, the conductivity of
semiconductors increase with an increase in
temperature.
σ = σo e-Eg/2kT
σo = temp. independent constant
k = Boltzman’s constant = 8.620x10-5 (eV/K)
E = Energy gap (eV)
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Conduction in Intrinsic Semi-conductors
Typical values for Si and Ge semiconductors
σ = ni q (µn + µp )
Note the lower band gap Eg for Ge and its higher
mobilities for both electrons and holes, which provides
a much higher electrical conductivity at RT than in Si
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Conduction in Intrinsic Semi-conductors
Effect of temperature on intrinsic semi-conductivity
Room
temperature
σ = ni q (µn + µp )
Increase in ni with
temperature as thermal
activation also assists in
exciting valence electrons
Here Ge has a much larger
number of electrons or holes
at room temperature than Si
Si: ni = 1.5 × 1016 carriers/m3
Ge: ni = 2.4 × 1019 carriers/m3
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Conduction in Intrinsic Semi-conductors
Example 1:
For an intrinsic Ga-As semi-conductor at room temperature
electrical conductivity σ = 10-6 (Ωm)-1. The electron/hole
mobilities are 0.85 en 0.04 m2/V.s respectively. Calculate the
intrinsic carrier concentration at room temperature.
q = 1.6x10-19 Coulomb or Ampere.seconds
σ = ni q (µn + µp )
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10.9 Extrinsic Semi-conductors
1. Highly dilute substitutional solid solution atoms
or solutes, are added to the parent semiconductor Si or Ge by doping
2. The valence electron’s arrangement of these
donor atoms differ from those of the parent or
solvent Si or Ge
There are two types of extrinsic semi-conductors:
• n-type extrinsic semi-conductors: negative
• p-type extrinsic semi-conductors: positive
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n- and p-type Extrinsic semi-conductors
Differences between n- and p- types:
From the periodic table:
Group IIIA
Dopant added
(valence 3+)
B
Al
Ga
In
p-type
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Group IVA
Parent metal
(valence 4+)
Si
Ge
Group VA
Dopant added
(valence 5+)
P
As
Sb
n-type
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10.10 n- Type Extrinsic semi-conductors
1. If a Group 5 atom (P, As or Sb) replaces a Si
atom (Group 4), one extra valence electron is
available on that atom
2. This extra valence atom is loosely bound and
requires only Eg=0.044 eV to cross the energy
gap
3. Compare this with the excitation energy gap of
Eg=1.1 eV that would have been necessary in
the intrinsic semi-conductor of pure Si
4. The donor impurities from Group 5 (P, As, Sb)
therefore provide conducting electrons and
are known as “impurity donor atoms”
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n- Type Extrinsic semi-conductors
Note that the donor electron from the Group 5 atoms require
very little excitation energy to become conducting. For a pure
Si atom in an intrinsic semi-conductor, excitation over the full
E2/7/2022
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g would have been necessary.
n- Type Extrinsic semi-conductors
Free electron
Note the “free” electron at the donor atom that can be
excited to the conducting level relatively easy
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10.11 p-Type Extrinsic semi-conductors
1. If a parent Si or Ge atom with valence 4+ is replaced by a
Group III atom (B, Al or Ga) with a valence of 3+, there is
a shortage of one valence electron.
2. This creates a positive hole at one of the covalent bonds
that surrounds this Group III atom
3. With the introduction of an electric excitation energy on
the Si crystal, one of the electrons from a nearby Si
atom can be excited to neutralise the hole.
4. Note that the hole has now moved to another electron
position which made the hole also mobile
5. Conduction, therefore, talks place by the movement of the
holes to the negative pole of the applied electric field
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p-Type Extrinsic semi-conductors
6. The excitation energy
associated with the
movement of holes from B
in Si, Eg =0.045 eV is
small in comparison to
the Eg = 1.1 eV for
conductivity with only
Si atoms present.
7. These atoms are known as “impurity acceptor donor”
atoms
8. The majority carriers are therefore holes (p-type)
while the minority carriers are electrons
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p-Type Extrinsic semi-conductors
The excitation energy associated with the movement of
holes from B in Si, Eg =0.045 eV is small in comparison to
the Eg = 1.1 eV for conductivity with only Si atoms present.
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10.12 Intrinsic vs Extrinsic Conduction
1. Intrinsic:
# electrons = # holes (n = p)
--case for pure Si
2. Extrinsic:
--n ≠ p
--occurs when impurities are added with a different
# valence electrons than the host (e.g., Si atoms)
2.1 n-type Extrinsic: (n >> p) 2.2 p-type Extrinsic: (p >> n)
Phosphorus atom
4+ 4+ 4+ 4+
σ ≈ n e µe
4+ 5+ 4+ 4+
4+ 4+ 4+ 4+
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no applied
electric field
Boron atom
hole
conduction
electron
4+ 4+ 4+ 4+
valence
electron
4+ 4+ 4+ 4+
Si atom
4+ 3+ 4+ 4+
no applied
electric field
σ ≈ p e µh
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10.13 Finding carrier concentrations in
extrinsic semiconductors
A. n-type semi-conductors σ = ni q (µn + µp )
In n-type semiconductors nn >> pn as electrons are the
majority carriers and holes as minority carriers make only
a minor contribution
1. Therefore σ ≈ nnqµn
as nn >> pn
2. Resistivity ρ ≈ 1 / n qµ
n
n
3. And the concentration of the donor in atoms/m3
(P, As or Sb)
N =n
d
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n
42
Finding carrier concentrations in
extrinsic semiconductors
A. n-type semi-conductors (cont’d)
4. At constant temperature, the product of the negative free
electron concentration (n) and the positive hole
concentration (p) is a constant
where ni is the intrinsic concentration of carriers in a
semiconductor and is constant at a given temperature.
5. Charge densities in extrinsic semiconductors:
 The material must be electrically neutral
 Charge carriers in extrinsic semiconductors are donor ions (Nd),
acceptor ions (Na), free electrons (n) and holes (p).
In general then, sum of negative carriers = sum of positive carriers,
or:
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Na + n = Nd + p
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Finding carrier concentrations in
extrinsic semiconductors
A. n-type semi-conductors (cont’d)
6. Charge carriers in extrinsic semiconductors are donor ions
(Nd), acceptor ions (Na), free electrons (n) and holes (p).
7. In general then, sum of negative carriers = sum of positive
carriers, or:
Na + n = N d + p
8. For n-type extrinsic semiconductors:
Where nn = concentration (-) electrons in n-type semiconductor and pn = concentration (+) holes in n-type semiconductor
Na = 0 and nn>> pn and nn = Nd
Nd = concentration of dopant atoms like P,As and Sb
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Finding carrier concentrations in
extrinsic semiconductors
B. p-type semi-conductors σ = nqµn + pqµp
1. Conductivity: (p-type) where pn >> n
σ ≈ pp q µp
pp ≈ Na
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10.14 Example on extr. semiconductors
Example 1:
A silicon wafer is doped with 1021 phosphorous atoms / m3.
Calculate or find:
(a) the majority carrier concentration.
(b) the minority carrier concentration.
(c) the electrical resistivity of the doped silicon at 300 K. Data
for silicon: ni = 1.5 x 1016 [carriers / m3]
µn = 0.135 [m2 / V s]
µp = 0.048 [m2 / V s]
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10.15 Manufacture of semi-conductors
1. Grow Si or Ge single crystals of about 100 to
120 mm diameter and about 25 cm long by the
Czochralski process
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Manufacture of semi-conductors
2. For Intrinsic semi-conductors, slice wafers off
each single crystal and proceed with the
manufacture of the semi-conducting chips
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Manufacture of semi-conductors
2. For Extrinsic semi-conductors, the Si or Ge single
crystal needs to doped with, for instance P atoms. This
can be done by one of two methods:
i)
By diffusion doping in a suitable P-containing
atmosphere at a high temperature. This produces a Pconcentration gradient through the thickness which
needs careful consideration in the final chips.
ii)
A more elegant process is to lightly radiate them in a
research reactor as in SAFARI 1 where the correct amount
of P atoms are produced by a neutron-Si reaction. Here
no P-concentration gradient is found through the
thickness.
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10.16 Dielectric (insulation) Behaviour
A dielectric material is one that is electrically insulating and
exhibits (or is made to exhibit) an electric dipole structure
High and low voltage electrical insulators.
i) Examples: ceramic materials used as electrical insulators:
electrical porcelain , alumina (Al2O3) , steatite
ii) Examples: polymer materials used as electrical insulators:
phenol formaldehyde (bakelite) , nylon , polyethylene
Inserts for capacitors. (A capacitor is an electrical device
which can store electric charge.)
i) Examples of ceramic materials used in capacitors:
barium titanate (BaTiO3), or BaTiO3 mixed with other ceramics
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10.17 Piezoelectric ceramics
Piezoelectric ceramics: these are some ceramics which can
convert electrical pulses into mechanical vibrations and vice
versa
 Utilized in transducers
 Examples of piezoelectric ceramics:
 barium titanate (BaTiO3)
 mixtures of lead zirconate (PbZrO3) and
 lead titanate (PbTiO3)
Cochlear
implant
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10.18 References
Chapter 10
Slide
no
Reference
4
Thought Co: https://www.thoughtco.com/examples-of-electrical-conductors-and-insulators-608315;
Toshiba: https://www.elprocus.com/different-types-of-transistor-and-their-functions/
5
http://nextgenelectrical.blogspot.com/2015/03/dc-circuits-basics-introduction-circuit.html
6
Quora: https://www.quora.com/How-do-semiconductors-differ-from-conductors-and-insulators
7
Howard Univ:
https://chem.libretexts.org/Courses/Howard_University/General_Chemistry%3A_An_Atoms_First_Approach/Unit_2%3A__Molecular_S
tructure/Chapter_5%3A_Covalent_Bonding/Chapter_5.7%3A_Metallic_Bonding
10
NGK, Japan;
14
Medeliene et al, Galvanotechnik, 103(2012)p942
16
Material Science and Engineering Callister 9e Fig 19.8 pg 689
19
http://www.gcsescience.com/a4-sodium-ion.htm ;
20
Material Science and Engineering Callister 9e Fig 19.4 pg 685
21
Material Science and Engineering Callister 9e Fig 19.3 pg 684
24
Thought Co: https://www.thoughtco.com/examples-of-electrical-conductors-and-insulators-608315
26
Material Science and Engineering Callister 9e Fig 19.11pg 695
31
Material Science and Engineering Callister 9e Fig 19.16 pg 700
36
Material Science and Engineering Callister 9e Fig 19.13 pg 697
37
http://hobbygenius.co.uk/tutorials/physics/1456
39
http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/dope.html
40
Material Science and Engineering Callister 9e Fig 19.15 pg 698
47
http://meroli.web.cern.ch/Lecture_silicon_floatzone_czochralski.html
48
http://maltiel-consulting.com/How_to_Make_a_Semiconductor_Chip.htm
51
CERAMTEC: https://www.ceramtec.com/ceramic-materials/piezo-ceramics/ ;
Shin et al, Sensors: 18(2018)p1694 https://doi.org/10.3390/s18061694
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