Introduction To Materials Science FOR ENGINEERS, Ch. 19
Introduction To Materials Science FOR ENGINEERS, Ch. 19
Electrical Properties
Outline of this Topic
• 1. Basic laws and electrical properties of metals
• 2. Band theory of solids: metals, semiconductors
and insulators
• 3. Electrical properties of semiconductors
• 4. Electrical properties of ceramics and polymers
• 5. Semiconductor devices
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Goals of this topic:
1. Basic laws and electrical properties of metals
• Understand how electrons move in materials: electrical
conduction
• How many moveable electrons are there in a material
(carrier density), how easily do they move (mobility)
• Metals, semiconductors and insulators
• Electrons and holes
• Intrinsic and Extrinsic Carriers
• Semiconductor devices: p-n junctions and transistors
• Ionic conduction
• Electronic Properties of Ceramics: Dielectrics,
Ferroelectrics and Piezoelectrics
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• Ohm’s Law
V = IR
E = V/L
where E is electric field intensity
µ = ν/ E where µ = the mobility
ν = the drift velocity
• Resistivity
ρ = RA / L (Ω.m)
• Conductivity
σ = 1 / ρ (Ω.m)-1
2
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Materials Choices for Metal Conductors
• Electrical conductivity between different materials
varies by over 27 orders of magnitude, the greatest
variation of any physical property
• Most widely used conductor is copper: inexpensive,
abundant, very high σ
• Silver has highest σ of metals, but use restricted due to cost
• Aluminum main material for electronic circuits, transition
to electrodeposited Cu (main problem was chemical
etching, now done by “Chemical-Mechanical Polishing”)
• Remember deformation reduces conductivity, so high
strength generally means lower σ : trade-off. Precipitation
hardening may be best choice: e.g. Cu-Be.
• Heating elements require low σ (high R), and resistance to
high temperature oxidation: nichrome.
Metals: σ > 105 (Ω.m)-1
Semiconductors: 10-6 < σ < 105 (Ω.m)-1
Insulators: σ < 10-6 (Ω.m)-1
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Conductivity / Resistivity of Metals
•
•
• Electric field causes electrons to accelerate in direction opposite
to field
• Velocity very quickly reaches average value, and then remains
constant
• Electron motion is not impeded by periodic crystal lattice
• Scattering occurs from defects, surfaces, and atomic thermal
vibrations
• These scattering events constitute a “frictional force” that
causes the velocity to maintain a constant mean value: vd, the
electron drift velocity
• The drift velocity is proportional to the electric field, the
constant of proportionality is the mobility, µ. This is a measure
of how easily the electron moves in response to an electric field.
• The conductivity depends on how many free electrons there
are, n, and how easily they move
High number of free (valence) electrons
→ high σ
Defects scatter electrons, therefore they
increase ρ (lower σ).
ρtotal = ρthermal+ρimpurity+ρdeformation
ρ
thermal
from thermal vibrations
ρimpurity from impurities
ρdeformation from deformation-induced point defects
•
•
•
Resistivity increases with temperature
(increased thermal vibrations and point
defect densities)
ρT = ρo + aT
Additions of impurities that form solid
sol:
ρI = Aci(1-ci) (increases ρ)
Two phases, α, β:
ρi = ραVα + ρ βV β
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
2. Band theory of solids: metals, semiconductors and
insulators
Scattering
events
Band Theory of Solids
• Schroedinger’s eqn (quantum mechanical equation for
behavior of an electron)
Kψ + V ψ = E ψ
Net electron motion
2
(-h’2/2m) δ ψ + V ψ = ih’ δ ψ
δx2
δt
vd = µeE
σ = n|e| µe
• Solve it for a periodic crystal potential, and you will find
that electrons have allowed ranges of energy (energy
bands) and forbidden ranges of energy (band-gaps).
n : number of “free” or
conduction electrons per
unit volume
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Electrons in an Isolated atom (Bohr Model)
(m) = Metal
(s) = Semicon
Na (m)
Ag (m)
Al (m)
Si (s)
GaAs (s)
InSb (s)
Mobility (RT)
µ (m2V-1s-1)
0.0053
0.0057
0.0013
0.15
0.85
8.00
Carrier Density
Ne (m-3)
2.6 x 1028
5.9 x 1028
1.8 x 1029
1.5 x 1010
1.8 x 106
Electron orbits defined by
requirement that they contain
integral number of wavelengths:
quantize angular momentum,
energy, radius of orbit
σmetal >> σsemi
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•
•
•
Introduction To Materials Science FOR ENGINEERS, Ch. 19
When N atoms in a solid
are relatively far apart, they
do not interact, so electrons
in a given shell in different
atoms have same energy
As atoms come closer
together, they interact,
perturbing electron energy
levels
Electrons from each atom
then have slightly different
energies, producing a
“band” of allowed energies
•
•
•
•
•
•
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Each band can contain certain number of electrons (xN, where N is the
number of the atoms and x is the number of electrons in a given atomic
shell, i.e. 2 for s, 6 for p etc.). Note: it can get more complicated than this!
Electrons in a filled band cannot conduct
In metals, highest occupied band is partially filled or bands overlap
Highest filled state at 0 Kelvin is the Fermi Energy, EF
Semiconductors, insulators: highest occupied band filled at 0 Kelvin:
electronic conduction requires thermal excitation across bandgap; σ↑ T↑
(At 0 Kelvin) highest filled band: valence band; lowest empty band:
conduction band. Ef is in the bandgap
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Metals, Semiconductors, Insulators
Empty
band
Empty
band
Ef
Filled
band
– Remember “Pauli Exclusion Principle” that only two electrons (spin
up, spin down) can occupy a given “state” defined by quantum
numbers n, l, ml
– So to conduct, electrons need empty states to scatter into, i.e. states
above the Fermi energy
Empty
conduction
band
Ef
Band gap
Ef
Empty states
Filled states
Empty
conduction
band
Band gap
Ef
Band gap
• At 0 Kelvin all available electron states below Fermi energy
are filled, all those above are vacant
• Only electrons with energies above the Fermi energy can
conduct:
Semiconductors Eg
< 2 eV
Insulators
Eg > 2 eV
Metals
Filled
valence
band
Filled
valence
band
• When an electron is promoted above the Fermi level (and can
thus conduct) it leaves behind a hole (empty electron state)
– A hole can also move and thus conduct current: it acts as a “positive
electron)
– Holes can and do exist in metals, but are more important in
semiconductors and insulators
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Introduction To Materials Science FOR ENGINEERS, Ch. 19
The Fermi Function
Metals
This equation represents the probability that an energy level, E,
is occupied by an electron and can have values between 0 and 1
. At 0K, the f (E) is equal to 1 up to Ef and equal to 0 above Ef
Energy
Empty
states
f (E) = [1] / [e(E - Ef) / kT +1]
EF
EF
Electron
excitation
Filled
states
(b)
(a)
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Introduction To Materials Science FOR ENGINEERS, Ch. 19
Semiconductors, Insulators
Conduction
band
Conduction
band
EF
(a)
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Free
electron
Electron
excitation
Hole in
valence
band
Valence
band
•
Energy
•
•
•
Band
Gap
•
•
•
In metals, electrons near the Fermi energy see empty states a very small
energy jump away, and can thus be promoted into conducting states above
Ef very easily (temp or electric field)
High conductivity
Atomistically: weak metallic bonding of electrons
In semiconductors, insulators, electrons have to jump across band gap into
conduction band to find conducting states above Ef : requires jump >> kT
No. of electrons in CB decreases with higher band gap, lower T
Relatively low conductivity
An electron in the conduction band leaves a hole in the valence band, that
can also conduct
Atomistically: strong covalent or ionic bonding of electrons
Valence
band
•
(b)
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Introduction To Materials Science FOR ENGINEERS, Ch. 19
Intrinsic Semiconductors: Conductivity
E field
Si
Si
Si
Si
Si
Si
Si
• Both electrons and holes conduct:
σ = n|e|µe + p|e|µh
Si
hole
Si
Si
Si
Si
Si
Si
Si
Si
free electron
Si
Si
Si
Si
Si
Si
Si
Si
(a)
Si
Si
Si
free electron
Si
Si
Si
Si
hole
Si
Si
Si
• In intrinsic semiconductor, n = p:
σ = n|e|(µe + µh) = p|e|(µe + µh)
(b)
E field
Si
n: number of conduction electrons per unit volume
p: number of holes in VB per unit volume
Si
Electrical conduction in intrinsic Si, (a) before
excitation, (b) and (c) after excitation, see the
response of the electron-hole pairs to the external
field. Note: holes generally have lower mobilities
than electrons in a given material (require
cooperative motion of electrons into previous
hole sites)
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
• Number of carriers (n,p) controlled by thermal
excitation across band gap:
n = p = C exp (- Eg /2 kT)
C : Material constant
Eg : Magnitude of the bandgap
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Extrinsic Semiconductors
3. Electrical properties of semiconductors
Semiconductors
•
•
•
Semiconductors are the key materials in the electronics and
telecommunications revolutions: transistors, integrated circuits,
lasers, solar cells….
Intrinsic semiconductors are pure (as few as 1 part in 1010
impurities) with no intentional impurities. Relatively high
resistivities
Extrinsic semiconductors have their electronic properties (electron
and hole concentrations, hence conductivity) tailored by
intentional addition of impurity elements
• Engineer conductivity by controlled addition of
impurity atoms: Doping
Room
Temp
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Introduction To Materials Science FOR ENGINEERS, Ch. 19
n-type semiconductors
E field
• In Si which is a tetravalent lattice, substitution of
pentavalent As (or P, Sb..) atoms produces extra electrons,
as fifth outer As atom is weakly bound (~ 0.01 eV). Each As
atom in the lattice produces one additional electron in the
conduction band.
• So NAs As atoms per unit volume produce n additional
conduction electrons per unit volume
• Impurities which produce extra conduction electrons are
called donors, ND = NAs ~ n
• These additional electrons are in much greater numbers
than intrinsic hole or electron concentrations, σ ~ n|e|µe ~
ND |e|µe
• Typical values of ND ~ 1016 - 1019 cm-3 (Many orders of
magnitude greater than intrinsic carrier concentrations at
RT)
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Si
4+
Si
4+
Si
4+
Si
4+
Si
4+
P
5+
Si
4+
Si
4+
Si
4+
Si
4+
Si
4+
Si
4+
Si
4+
Si
4+
free electron
P
5+
Si
4+
Si
4+
(a)
Si
4+
Si
4+
Si
4+
Si
4+
Si
4+
Si
4+
n-type
(b)
Si
4+
Si
4+
Si
4+
Si
4+
Si
4+
Si
4+
Si
4+
B
3+
Si
4+
Si
4+
Si
4+
Si
4+
Si
4+
Si
4+
B
3+
Si
4+
Si
4+
Si
4+
Si
4+
hole
hole
Si
4+
Si
4+
Si
4+
Si
4+
Si
4+
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p-type
(b)
(a)
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Introduction To Materials Science FOR ENGINEERS, Ch. 19
Semiconductors
n-type “more electrons”
p-type semiconductors
Conduction
band
Donor state
Valence
band
(a)
26
Free
electrons
in the
conduction
band
Band
Gap
Valence
band
Energy
Conduction
band
• Substitution of trivalent B (or Al, Ga...) atoms in Si
produces extra holes as only three outer electrons exist to
fill four bonds. Each B atom in the lattice produces one
hole in the valence band.
• So NB B atoms per unit volume produce p additional holes
per unit volume
• Impurities which produce extra holes are called acceptors,
NA = NB ~ p
• These additional holes are in much greater numbers than
intrinsic hole or electron concentrations, σ ~ p|e|µh ~ NA
|e|µh
• Typical values of NA ~ 1016 - 1019 cm-3 (Many orders of
magnitude greater than intrinsic carrier concentrations at
RT)
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Si
4+
(b)
For an n-type material, excitation occurs from the donor state in which
a free electron is generated in the conduction band.
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Introduction To Materials Science FOR ENGINEERS, Ch. 19
Conduction
band
4. Electrical properties of
ceramics and polymers
Band
Gap
Energy
Conduction
band
Semiconductors
p-type “more holes”
Hole in
the valence
band
Valence
band
Valence
band
Acceptor state
(a)
(b)
For an p-type material, excitation of an electron into the acceptor level, leaving
behind a hole in the valence band.
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
ln p, n
Saturation
Extrinsic
{∆ln p/ [∆(1/T)]}
= Eg / 2 k
1/T
Dielectric Materials
• Our basic equation:
σ = n|e|µe + p|e|µh
• Main temperature variations
are in n,p rather than µe , µh
• Intrinsic carrier concentration
n = p = C exp (- Eg /2 kT)
Extrinsic carrier concentration
• A dielectric material is an insulator which contains electric
dipoles, that is where positive and negative charge are
separated on an atomic or molecular level
– low T (< room temp) Extrinsic
regime: ionization of dopants
– mid T (inc. room temp) Saturated
regime: most dopants ionized
– high T Intrinsic regime: intrinsic
generation dominates
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Temperature Dependence of carrier Concentration and
Conductivity
Intrinsic
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• When an electric field is applied, these dipoles align to the
field, causing a net dipole moment that affects the material
properties.
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Capacitance
Polarization
• Capacitance is the ability to store
charge across a potential difference.
• Examples: parallel conducting plates,
semiconductor p-n junction
• Magnitude of the capacitance, C:
C = Q/V
• Magnitude of electric dipole moment
from one dipole:
p = qd
+++++
----- -
P
D
N
• In electric field, dipole will rotate in
direction of applied field: polarization
- - - + ++
- - - + ++
- - - + ++
Units: Farads
• Parallel- plate capacitor, C depends on
geometry of plates and material
between plates
C = εr εo A / L
• The surface charge density of a
capacitor can be shown to be:
D = εoεrξ
D : Electric Displacement
(units Coulombs / m2)
A : Plate Area; L : Plate Separation
ε o : Permittivity of Free Space (8.85x10-12 F/m2)
ε r : Relative permittivity, εr = ε /εo
Vac, εr = 1
L
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• Increase in capacitance in dielectric
medium compared to vacuum is due
to polarization of electric dipoles in
dielectric.
• In absence of applied field (b), these
are oriented randomly
• In applied field these align according
to field (c)
• Result of this polarization is to create
opposite charge Q’ on material
adjacent to conducting plates
• This induces additional charge (-)Q’
on plates: total plate charge Qt =
|Q+Q’|.
• So, C = Qt / V has increased
• Magnitude of dielectric constant depends upon frequency
of applied alternating voltage (depends on how quickly
charge within molecule can separate under applied field)
• Dielectric strength (breakdown strength): Magnitude of
electric field necessary to produce breakdown
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Electronic
• Surface density charge now
D = εξ = εoεrξ = εoξ + P
• P is the polarization of the material
(units Coulombs/m2). It represents
the total electric dipole moment
per unit volume of dielectric, or the
polarization electric field arising
from alignment of electric dipoles
in the dielectric
Ionic
Orientation
• From equations at top of page
P = εo(εr-1)ξ
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Origins of Polarization
• Where do the electric dipoles come from?
– Electronic Polarization: Displacement of negative
electron “clouds” with respect to positive nucleus.
Requires applied electric field. Occurs in all materials.
– Ionic Polarization: In ionic materials, applied electric
field displaces cations and anions in opposite directions
– Orientation Polarization: Some materials possess
permanent electric dipoles, due to distribution of charge
in their unit cells. In absence of electric field, dipoles
are randomly oriented. Applying electric field aligns
these dipoles, causing net (large) dipole moment.
Barium Titanate, BaTiO3 : Permanent Dipole Moment
for T < 120 C (Curie Temperature, Tc). Above Tc, unit
cell is cubic, no permanent electric dipole moment
Ptptal = Pe + Pi + Po
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Piezoelectricity
• In some ceramic materials, application of external forces
produces an electric (polarization) field and vice-versa
• Applications of piezoelectric materials microphones, strain
gauges, sonar detectors
• Materials include barium titanate, lead titanate, lead
zirconate
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Introduction To Materials Science FOR ENGINEERS, Ch. 19
Ionic Conduction in Ceramics
Electrical Properties of Polymers
• Cations and anions possess electric charge (+,-) and
therefore can also conduct a current if they move.
• Ionic conduction in a ceramic is much less easy than
electron conduction in a metal (“free” electrons can move
far more easily than atoms / ions)
• In ceramics, which are generally insulators and have very
few free electrons, ionic conduction can be a significant
component of the total conductivity
σtotal = σelectronic + σionic
• Overall conductivities, however, remain very low in
ceramics.
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•
•
•
•
•
•
•
42
Most polymeric materials are relatively poor conductors of electrical
current - low number of free electrons
A few polymers have very high electrical conductivity - about one
quarter that of copper, or about twice that of copper per unit weight.
Involves doping with electrically active impurities, similar to
semiconductors: both p- and n-type
Examples: polyacetylene, polyparaphenylene, polypyrrole
Orienting the polymer chains (mechanically, or magnetically) during
synthesis results in high conductivity along oriented direction
Applications: advanced battery electrodes, antistatic coatings,
electronic devices
Polymeric light emitting diodes are also becoming a very important
research field
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Applied Voltage
P
5. Semiconductor Devices and Circuits
N
- - - -
- - - + ++
- - - + ++
- - - + ++
Vb
+
D
P
D
Forward Bias
-
N
- +
- +
- +
-
Vb
+
++
++
++
Reverse Bias
Vo
Vo
Vo-Vb
Ec+
Vo+|Vb|
Ec0
Ec0
EF0
EcEF-
Ev+
Ev0
Ev0
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Lower Barrier , I ↑
Ev-
Higher Barrier, I ↓
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The Semiconductor p-n Junction Diode
P
D
N
- - - + ++
- - - + ++
- - - + ++
ξ
n
p
Vh
• A rectifier or diode allows
current to flow in one
direction only.
• p-n junction diode consists of
adjacent p- and n-doped
semiconductor regions
• Electrons, holes combine at
junction and annihilate:
depletion region containing
ionized dopants
• Electric field, potential barrier
resists further carrier flow
Ve
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
MOSFET (Metal-Oxide-Semiconductor Field Effect
Transistor)
Transistors
• The basic building block of the microelectronic revolution
• Can be made as small as 1 square micron
• A single 8” diameter wafer of silicon can contain as many as
1010 - 1011 transistors in total: enough for several for every
man, woman, and child on the planet
• Cost to consumer ~ 0.00001c each.
• Achieved through sub-micron engineering of semiconductors,
metals, insulators and polymers.
• Requires ~ $2 billion for a state-of-the-art fabrication facility
•
•
•
•
•
•
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Introduction To Materials Science FOR ENGINEERS, Ch. 19
Take Home Messages
•
•
•
•
n-p-n or p-n-p sandwich structures. Emitter-base-collector. Base is very thin (~ 1
micron or less) but greater than depletion region widths at p-n junctions.
Emitter-base junction is forward biased; holes are pushed across junction. Some of
these recombine with electrons in the base, but most cross the base as it so thin. They
are then swept into the collector.
A small change in base-emitter voltage causes a relatively large change in emitterbase-collector current, and hence a large voltage change across output (“load”)
resistor: voltage amplification
The above configuration is called the “common base” configuration (base is common
to both input and output circuits). The “common emitter” configuration can produce
both amplification (V,I) and very fast switching
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Bipolar Junction Transistor
•
Nowadays, the most important type of transistor.
Voltage applied from source to drain encourages carriers (in the above case
holes) to flow from source to drain through narrow channel.
Width (and hence resistance) of channel is controlled by intermediate gate
voltage
Current flowing from source-drain is therefore modulated by gate voltage.
Put input signal onto gate, output signal (source-drain current) is
correspondingly modulated: amplification and switching
State-of-the-art gate lengths: 0.18 micron. Oxide layer thickness < 10 nm
•
•
•
•
•
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Language: Resistivity, conductivity, mobility, drift velocity, electric field
intensity, energy bands, band gap, conduction band, valence band, Fermi
energy, hole, intrinsic semiconductor extrinsic semiconductor, dopant,
donor, acceptor, extrinsic regime, extrinsic regime, saturated regime,
dielectric, capacitance, (relative) permittivity, dielectric strength, (electronic,
ionic, orientational) polarization, electric displacement, piezoelectric, ionic
conduction, p-n junction, rectification, depletion region, (forward, reverse)
bias, transistors, amplification.
Fundamental concepts of electronic motion: Conductivity, drift velocity,
mobility, electric field
Band theory of solids: Energy bands, band gaps, holes, differences between
metals, semiconductors and insulators
Semiconductors: Dependence of intrinsic and extrinsic carrier conc. on
temperature, band gap; dopants - acceptors and donors.
Capacitance: Dielectrics, polarization and its causes, piezoelectricity
Semiconductor devices: basic construction and operation of p-n junctions,
bipolar transistors and MOSFETs
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