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Chapter 18: Electrical Properties
ISSUES TO ADDRESS...
• How are electrical conductance and resistance
characterized?
• What are the physical phenomena that distinguish
conductors, semiconductors, and insulators?
• For metals, how is conductivity affected by
imperfections, T, and deformation?
• For semiconductors, how is conductivity affected
by impurities (doping) and T?
Chapter 18 - 1
View of an Integrated Circuit
• Scanning electron microscope images of an IC:
Al
Si
(doped)
(d)
(d)
(a)
45 mm
0.5 mm
• A dot map showing location of Si (a semiconductor):
-- Si shows up as light regions.
(b)
• A dot map showing location of Al (a conductor):
-- Al shows up as light regions.
Fig. (d) from Fig. 18.27 (a), Callister 7e. (Fig. 18.27 is
courtesy Nick Gonzales, National Semiconductor Corp.,
West Jordan, UT.)
(c)
Fig. (a), (b), (c) from Fig. 18.0,
Callister 7e.
Chapter 18 - 2
Electrical Conduction
• 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
• Resistivity, r and Conductivity, s:
-- geometry-independent forms of Ohm's Law
-- Resistivity is a material property & is independent of sample
E: electric
field
intensity
• Resistance:
V
I
 r
L
A
rL
L
R

A As
resistivity
(Ohm-m)
J: current density
conductivity
1
s
r
Chapter 18 - 3
Electrical Conduction
When an electrical potential V [volts, J/C] is applied across a piece of
material, a current of magnitude I [amperes, C/s] flows. In most metals,
at low values of V, the current is proportional to V, and can be
described by Ohm's law:
I = V/R
where R is the electrical resistance [ohms, Ω]. R depends on the
intrinsic resistivity ρ of the material [Ω-m] and onthe geometry (length l
and area A through which the current passes): R = ρl/A
In most materials (e.g. metals), the
current is carried by electrons (electronic
conduction). In ionic crystals, the charge
carriers are ions (ionic conduction).
Chapter 18 - 4
Electrical Properties
• Which will conduct more electricity?
D
2D
RA
VA
r


I
• Analogous to flow of water in a pipe
• So resistance depends on sample
geometry, etc.
Chapter 18 - 5
Definitions
Further definitions
J=s
<= another way to state Ohm’s law
J  current density
current
I


surface area A
like a flux
  electric field potential = V/ or (V/ )
J = s (V/ )
Electron flux
conductivity
voltage gradient
Current carriers
• electrons in most solids
• ions can also carry (particularly in liquid solutions)
Chapter 18 - 6
Conductivity: Comparison
• Room T values (Ohm-m)-1 = ( - m)-1
METALS
CERAMICS
conductors
-10
Silver
6.8 x 10 7
Soda-lime glass 10 -10-11
Copper
6.0 x 10 7
Concrete
10 -9
Iron
1.0 x 10 7
Aluminum oxide <10-13
Up to 27 orders of magnitude, possibly
widest variation in materials properties
SEMICONDUCTORS
POLYMERS
Polystyrene
Silicon
4 x 10 -4
Polyethylene
Germanium 2 x 10 0
GaAs
10 -6
semiconductors
-14
<10
10 -15-10-17
insulators
Selected values from Tables 18.1, 18.3, and 18.4, Callister 7e.
Chapter 18 - 7
Conductivity / Resistivity
Chapter 18 - 8
Example: Conductivity Problem
What is the minimum diameter (D) of the wire so that V < 1.5 V?
e-
Cu wire -
100m
I = 2.5A
+
V
100m
D 2
4
Solve to get
L
V
R

As
I
< 1.5V
2.5A
6.07 x 107 (Ohm-m)-1
D > 1.87 mm
Chapter 18 - 9
Energy Band Structures in Solids
•
Electrical conductivity in materials are strongly related to # of electrons available
for conduction
– Not all electrons will move in a material under an applied potential
difference !
– More detail in Quantum Mechanics !
•
In an isolated atom electrons occupy well defined energy states, as discussed in
Chapter 2.
– Shells (1,2,3….), subshells (s,p,d,f…..), states in subshells, spin states
When atoms come together to form a solid (N =12 atoms), a bonded regular
atomic arrangement, their valence electrons interact with each other and with
nuclei due to Coulombic forces.
In addition, two specific quantum mechanical effects happen.
– First, by Heisenberg's uncertainty principle, constraining the electrons to a
small volume raises their energy, this is called promotion.
– The second effect, due to the Pauli exclusion principle, limits the number of
electrons that can have the same energy.
As a result of these effects, the valence electrons of atoms at distinct energy
states get split into closely spaced electron states and form wide electron
energy bands when they form a solid. These bands are separated by gaps,
where electrons cannot exist.
•
•
•
Chapter 18 - 10
Energy Band Structures in Solids
The extend of energy states splitting
depends on inter-atomic distance
Outermost shells start forming
bands as atoms get closer,
however core shells may not form
bands as still far from each other.
Band gaps, depending upon the
inter-atomic distances also start
forming
Chapter 18 - 11
Electronic Band Structures
In solids gap between
states could be 10-10 eV
apart
Adapted from Fig. 18.2, Callister 7e.
Chapter 18 - 12
Band Structure
•
•
Valence band – filled – highest occupied energy levels
Conduction band – empty – lowest unoccupied energy levels
Conduction
band
valence band
Adapted from Fig. 18.3, Callister 7e.
Chapter 18 - 13
Band Structure defines Electrical Properties
conductor
insulator
>2eV
Metals, single s valence
electron N atoms => 4s band
capable of 2N electrons =>
but only one s electron
means half filled band
structure.
Metals with overlaping
bands. Eg. Both s valence
electrons exists in s
subshell. In x’tal structure
s and p subshells overlap,
but p shell is empty.
<2eV
the valence band is filled, and no more electrons
can be added (Pauli's principle). Electrical
conduction requires that electrons be able to gain
energy in an electric field. This is not possible in
these materials because that would imply that the
electrons are promoted into the forbidden band
gap.
Chapter 18 - 14
Energy States: Insulators &
Semiconductors
• Insulators:
• Semiconductors:
-- Higher energy states not
-- Higher energy states separated
accessible due to gap (> 2 eV). by smaller gap (< 2 eV).
Energy
Energy
empty
band
filled
valence
band
filled
band
?
GAP
filled states
filled states
GAP
empty
band
filled
valence
band
filled
band
Chapter 18 - 15
Semiconductors and Insulators
•
•
•
•
•
•
In semiconductors and insulators, electrons have to jump/move across the
band gap into conduction band to find conducting states above Ef
The energy needed for the jump may originate from heat, or from irradiation
at sufficiently small wavelength.
The difference between semiconductors and insulators is that in
semiconductors electrons can reach the conduction band at ordinary
temperatures, where in insulators they cannot. Eg is too large for insulators
to have thermally or optically exited electrons promote to the conduction
band.
The probability that an electron reaches the conduction band is about exp(Eg/2kT) where Eg is the band gap. If this probability is < 10-24 one would not
find a single electron in the conduction band in a solid of 1 cm3. How come ?
Remember NAv ~ 1024
This requires Eg/2kT > 55. At room temperature, 2kT = 0.05 eV, Eg > 2.8 eV
corresponds to an insulator.
An electron promoted into the conduction band leaves a hole (positive
charge) in the valence band, that can also participate in conduction. Holes
exist in metals as well, but are more important in semiconductors and
insulators.
Chapter 18 - 16
Charge Carriers
Adapted from Fig. 18.6 (b), Callister 7e.
Two charge carrying mechanisms
Electron – negative charge
Hole
– equal & opposite
positive charge
Move at different speeds - drift
velocity
Higher temp. promotes more electrons into the conduction band

s as T
Electrons scattered by impurities, grain boundaries, etc.
Chapter 18 - 17
Conduction & Electron Transport
• Metals (Conductors):
-- Thermal energy puts
many electrons into
a higher energy state.
-
• Energy States:
Energy
-- for metals nearby
energy states
are accessible
by thermal
fluctuations.
empty
band
+
-
Energy
empty
band
filled
band
filled states
partly
filled
valence
band
filled states
GAP
filled
valence
band
filled
band
Chapter 18 - 18
Conduction in Metals
• In metals, highest occupied band is partially
filled or bands overlap.
• Conduction occurs by promoting electrons
into conducting states, that starts right above
the Fermi level. The conducting states are
separated from the valence band by an
infinitesimal amount (~10-10 eV).
• Energy provided by an electric field is
sufficient to excite many electrons into
conducting states. => High conductivity.
Chapter 18 - 19
Summary
Energy Band Structures and Bonding
(metals, semiconductors, insulators)
• Relation to atomic bonding:
– Insulators – valence electrons are tightly bound
to (or shared with) the individual atoms – strongest
ionic (partially covalent) bonding. Remember
electro-negativity.
– Semiconductors - mostly covalent bonding
somewhat weaker bonding. Sharing of electrons.
– Metals – valence electrons form an “electron gas”
that are not bound to any particular ion.
Chapter 18 - 20
Electron Mobility
•
The force acting on the electron is -eE, where e is the electric
charge and F=ma. As long as the electric field is present, in the
absence of obstacles the electron is expected to speed up
continuously in an electric field.
– So, is the case in vacuum (e.g. inside a TV tube) or in a
perfect crystal .
– In a real solid, electrons motion are hindered by defects
(dislocations, impurities, vacancies, etc), and even thermal
vibrations of atoms. Electrons scatter by collisions with
imperfections and due to atomic thermal vibrations.
• “frictional forces” => resistance => a net drift velocity of
electron motion is established:
where μe – electron mobility [m2/V-s]. The “friction”
transfers part of the energy supplied by the electric field
into the lattice as heat. That is how electric heaters work.
Chapter 18 - 21
Electron Mobility
Electrical conductivity is proportional to number of free electrons and
electron mobility:
σ = Ne |e| μe
Ne - number of “free” or conduction electrons per unit volume
e is the absolute charge on a free electron 1.609 x 10-19 C
Chapter 18 - 22
Metals: Resistivity vs T, Impurities
Read pages 621- 623
• Imperfections increase resistivity
These act to scatter
electrons so that they
take a less direct path.
6
(10 -8 Ohm-m)
Resistivity, r
-- grain boundaries
-- dislocations
-- impurity atoms
-- vacancies
• Resistivity
5
increases with:
4
-- temperature
-- wt% impurity
-- %CW
3
2
1
0
-200
-100
0
T (°C)
Adapted from Fig. 18.8, Callister 7e. (Fig. 18.8 adapted from J.O.
Linde, Ann. Physik 5, p. 219 (1932); and C.A. Wert and R.M.
Thomson, Physics of Solids, 2nd ed., McGraw-Hill Book Company,
New York, 1970.)
r = rthermal
+ rimpurity
+ rdeformation
Chapter 18 - 23
Estimating Conductivity
• Question:
180
160
140
125
120
100
21 wt%Ni
80
60
0 10 20 30 40 50
Resistivity, r
(10 -8 Ohm-m)
Yield strength (MPa)
-- Estimate the electrical conductivity s of a Cu-Ni alloy
that has a yield strength of 125 MPa.
wt. %Ni, (Concentration C)
Adapted from Fig. 7.16(b), Callister 7e.
From step 1:
CNi = 21 wt%Ni
Adapted from Fig.
18.9, Callister 7e.
50
40
30
20
10
0
0 10 20 30 40 50
wt. %Ni, (Concentration C)
r  30x108 Ohm  m
1
s   3.3x106 (Ohm  m)1
r
Chapter 18 - 24
Pure Semiconductors: Conductivity vs T
• Data for Pure Silicon:
-- s increases with T
-- opposite to metals
electrical conductivity, s
(Ohm-m) -1
10 4
10 2
10 1
10 0
10 -1
10 -2
pure
(undoped)
50 10 0
Energy
empty
band
?
GAP
filled states
10 3
sundoped  e
1000
T(K)
Adapted from Fig. 19.15, Callister 5e. (Fig. 19.15
adapted from G.L. Pearson and J. Bardeen, Phys. Rev.
75, p. 865, 1949.)
 Egap / kT
electrons
filled
can cross
valence gap at
band
higher T
filled
band
material
Si
Ge
GaP
CdS
band gap (eV)
1.11
0.67
2.25
2.40
Selected values from Table
18.3, Callister 7e.
Chapter 18 - 25
Semiconductivity
Semiconductor do have a lower conductivity than metals but
unique properties of semiconductors make them very
useful materials.
Electrical properties of semiconductors are very sensitive to
the presence of impurities:
• Intrinsic semiconductors - electrical conductivity is based on
the electronic structure of pure material.
• Extrinsic semiconductors - electrical conductivity is dictated
by impurity atoms.
WHY ?
Chapter 18 - 26
Conduction in Terms of Electron and
Hole Migration
• Concept of electrons and holes:
valence
electron
electron
hole
pair creation
Si atom
+ -
no applied
electric field
electron
hole
pair migration
applied
electric field
• Electrical Conductivity given by:
applied
electric field
# holes/m 3
s  n e me  p e m h
# electrons/m3
+
Adapted from Fig. 18.11,
Callister 7e.
hole mobility
electron mobility
Chapter 18 - 27
Intrinsic Semiconductors
Number of electrons in conduction band increases
exponentially with temperature:
C is a material constant
Eg is the bandgap width
Thermally excited
conductions
electrons
Eg
Holes left behind
0K
RT ~ 300 K
An electron promoted into the conduction band leaves a hole (positive charge) in the valence band. In an
electric field, electrons and holes move in opposite direction and participate in conduction
In Si (Eg = 1.1 eV) one out of every 1013 atoms contributes an electron to the conduction band
Chapter 18 - 28
at room temperature.
Intrinsic Semiconductors
Since both electrons and holes conduct the conductivity of an intrinsic
semiconductor is
s  n e me  p e m h
Electrons are more mobile than holes, μe > μh
In an intrinsic semiconductor, a hole is produced by the promotion of each
electron to the conduction band. Therefore, n = p and
s  n e me  mh or s  p e me  mh 
n (and p) increase exponentially with temperature,
whereas μe and μh decrease (about linearly) with
temperature.
Conductivity of intrinsic semiconductors increase
with temperature (different from metals!)
Chapter 18 - 29
Intrinsic Semiconductors
Chapter 18 - 30
Extrinsic Semiconductors
• Extrinsic semiconductors - electrical properties (conductivity)
is dictated by impurity atoms.
Example: Si is considered to be extrinsic at room T if impurity
concentration is one atom per 1012 (remember our estimation of
the number of electrons promoted to the conduction band by
thermal fluctuations at 300 K)
– Unlike intrinsic semiconductors, an extrinsic semiconductor
may have different concentrations of holes and electrons. It
is called p-type if p > n and n-type if n > p.
– One can engineer conductivity of extrinsic semiconductors
by controlled addition of impurity atoms – doping (addition
of a very small concentration of impurity atoms).
– Two common methods of doping are diffusion and ion
implantation.
Chapter 18 - 31
n-type Extrinsic SMs
• When excess electron carriers are produced by
substitutional impurities that have more valence
electron per atom than the semiconductor matrix.
Example: phosphorus (or As, Sb..) with 5 valence
electrons, is an electron donor in Si since only 4
electrons are used to bond to the Si lattice when it
substitutes for a Si atom. Fifth outer electron of P
atom is weakly bound in a donor state (~ 0.01 eV)
and can be easily thermally promoted to the
conduction band.
• Impurities which produce extra conduction electrons
are called donors, ND = NPhosphorus ~ n
• Elements in columns V and VI of the periodic table
are donors for semiconductors in the IV column such
as Si and Ge.
Chapter 18 - 32
n-type Extrinsic SMs
The hole created in donor state is far from the
valence band and is immobile upon the promotion
of the donated electron to the conduction band.
n >>p
Conduction occurs mainly by the donated electrons
(thus n-type).
σ ~ ND |e| μe
Chapter 18 - 33
Intrinsic vs Extrinsic Conduction
• Intrinsic:
# electrons = # holes (n = p)
--case for pure Si
• Extrinsic:
--n ≠ p
--occurs when impurities are added with a different
# valence electrons than the host (e.g., Si atoms)
• n-type Extrinsic: (n >> p)
• p-type Extrinsic: (p >> n)
Phosphorus atom
4+ 4+ 4+ 4+
s  n e me
4+ 5+ 4+ 4+
4+ 4+ 4+ 4+
Adapted from Figs. 18.12(a)
& 18.14(a), Callister 7e.
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
s  p e mh
Chapter 18 - 34
Intrinsic Semiconductors
• Pure material semiconductors: e.g., silicon &
germanium
– Group IVA materials
• Compound semiconductors
– III-V compounds
• Ex: GaAs & InSb
– II-VI compounds
• Ex: CdS & ZnTe
– The wider the electronegativity difference between
the elements the wider the energy gap.
Chapter 18 - 35
Doped Semiconductor: Conductivity vs. T
10 4
0.0052at%B
10 3
10 2
doped
0.0013at%B
-- extrinsic doping level:
1021/m3 of a n-type donor
impurity (such as P).
-- for T < 100 K: "freeze-out“,
thermal energy insufficient to
excite electrons.
-- for 150 K < T < 450 K: "extrinsic"
-- for T >> 450 K: "intrinsic"
10 1
10 -1
pure
(undoped)
10 -2
50 100
1000
T(K)
Adapted from Fig. 19.15, Callister 5e. (Fig. 19.15
adapted from G.L. Pearson and J. Bardeen, Phys. Rev.
75, p. 865, 1949.)
doped
undoped
3
freeze-out
10 0
conduction electron
concentration (1021/m3)
electrical conductivity, s
(Ohm-m) -1
lower the activation energy to
produce mobile electrons.
extrinsic conduction...
2
1
0
0
intrinsic
-- s increases doping
-- reason: imperfection sites
• Comparison: intrinsic vs
extrinsic
• Data for Doped Silicon:
Adapted from Fig.
18.17, Callister 7e.
(Fig. 18.17 from S.M.
Sze, Semiconductor
Devices, Physics, and
Technology, Bell
Telephone
Laboratories, Inc.,
1985.)
200 400 600 T(K)
Chapter 18 - 36

Number of Charge Carriers
Intrinsic Conductivity
s = n|e|me + p|e|me
• for intrinsic semiconductor n = p

s = n|e|(me + mn)
• Ex: GaAs
s
106 ( m)1
n

e me  mn
(1.6x1019 C)(0.85  0.45 m2/V  s)

For GaAs
For Si

n = 4.8 x 1024 m-3
n = 1.3 x 1016 m-3
Chapter 18 - 37
p-n Rectifying Junction
• Allows flow of electrons in one direction only (e.g., useful
to convert alternating current to direct current.
• Processing: diffuse P into one side of a B-doped crystal.
Adapted from Fig. 18.21,
• Results:
p-type
n-type
+ + +
+ +
Callister 7e.
--No applied potential:
no net current flow.
--Forward bias: carrier
flow through p-type and
n-type regions; holes and
electrons recombine at
p-n junction; current flows.
--Reverse bias: carrier
flow away from p-n junction;
carrier conc. greatly reduced
at junction; little current flow.
-
-
-
-
-
p-type
+
-
+ - n-type
+
++- - + -
+ p-type
+ +
+ +
n-type
-
-
-
-
+
-
Chapter 18 - 38
Properties of Rectifying Junction
Fig. 18.22, Callister 7e.
Fig. 18.23, Callister 7e.
Chapter 18 - 39
Transistor MOSFET
• MOSFET (metal oxide semiconductor field effect
transistor)
Fig. 18.24,
Callister 7e.
Chapter 18 - 40
Integrated Circuit Devices
Fig. 18.26, Callister 6e.
• Integrated circuits - state of the art ca. 50 nm line
width
– 1 Mbyte cache on board
– > 100,000,000 components on chip
– chip formed layer by layer
• Al is the “wire”
Chapter 18 - 41
Ferroelectric Ceramics
Ferroelectric Ceramics are dipolar below Curie TC = 120ºC
• cooled below Tc in strong electric field - make material
with strong dipole moment
Fig. 18.35, Callister 7e.
Chapter 18 - 42
Piezoelectric Materials
Piezoelectricity – application of pressure produces current
at
rest
compression
induces
voltage
applied voltage
induces
expansion
Adapted from Fig. 18.36,
Callister 7e.
Chapter 18 - 43
Summary
• Electrical conductivity and resistivity are:
-- material parameters.
-- geometry independent.
• Electrical resistance is:
-- a geometry and material dependent parameter.
• Conductors, semiconductors, and insulators...
-- differ in accessibility of energy states for
conductance electrons.
• For metals, conductivity is increased by
-- reducing deformation
-- reducing imperfections
-- decreasing temperature.
• For pure semiconductors, conductivity is increased by
-- increasing temperature
-- doping (e.g., adding B to Si (p-type) or P to Si (n-type).
Chapter 18 - 44
ANNOUNCEMENTS
Reading:
Core Problems:
Self-help Problems:
Chapter 18 - 45