Electrical properties
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ELECTRICAL CONDUCTION
ENERGY BAND STRUCTURE IN SOLIDS
INSULATORS AND SEMICONDUCTORS
METALS: ELECTRON MOBILITY
INFLUENCE OF TEMPERATURE
INFLUENCE OF IMPURITY
SEMICONDUCTORS
P-N RECTIFYING JUNCTION
M Medraj / PM Wood-Adams
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/1
ELECTRICAL CONDUCTION
A
• Ohm's Law:
e-
(cross
sect.
area)
R depends on
specimen geometry
I
ΔV
L
ΔV = I R
voltage drop (volts)
resistance (Ohms)
current (amps)
• Resistivity, ρ and Conductivity, σ:
Æ geometry-independent forms of Ohm's Law
E: electric
field intensity
• Resistance:
M Medraj / PM Wood-Adams
ΔV I
= ρ
L
A
resistivity
(Ohm-m)
J: current density
Conductivity:
I
σ=
ρ
ρL
L
=
R=
A Aσ
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/2
CONDUCTIVITY: COMPARISON
• solid materials exhibit a very wide range of electrical conductivity
– widest range compared to other phys. properties.
Æ Materials can be classified according to their electrical conductivity.
Conductivity values (Ohm-m)-1 at room temp.
METALS
Silver
Copper
Iron
conductors
6.8 x 107
6.0 x 107
1.0 x 107
CERAMICS
Soda-lime glass 10-10
Concrete
10-9
Aluminum oxide <10-13
SEMICONDUCTORS
POLYMERS
Polystyrene
Silicon
4 x 10-4
Polyethylene
Germanium 2 x 100
GaAs
10-6
semiconductors
M Medraj / PM Wood-Adams
Mech. Eng. Dept. - Concordia University
Selected values from
Tables 18.1, 18.2, and
18.3, Callister 6e.
<10-14
10-15-10-17
insulators
MECH 221 fall 2008/3
EXAMPLE
A copper wire 100 m long must experience a voltage drop of less than
1.5 V when a current of 2.5 A passes through it. If σ is 6.07 x 107
(Ohm-m)-1, compute the minimum diameter of the wire.
Cu wire -
e-
100m
I = 2.5A
+
ΔV
M Medraj / PM Wood-Adams
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/4
Energy bands in solid materials
The energy required for an electron to jump from the 1s band to
the 2s band is equal to the bad gap.
Band gap
Equilibrium separation in crystal
M Medraj / PM Wood-Adams
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/5
Energy Band Structure in Solids
The electrical properties of a solid material are a consequence of its electronic
band structure: the arrangement of the outermost electron bands and the way
in which they are filled with electrons.
The various possible electron band structures in solids at 0 K:
From Fig. 18.4
Callister 6th. ed.
metals such as
copper, in which
electron states are
available above
and adjacent to
filled states, in the
same band.
M Medraj / PM Wood-Adams
The electron band
structure of metals
such as magnesium,
wherein there is
an overlap of filled
and empty outer
bands.
Insulators: the
filled valence band
is separated from
the empty
conduction band by
a relatively large
band gap (>2 eV).
Mech. Eng. Dept. - Concordia University
Semiconductors:
same as for
insulators except
that the band gap is
relatively narrow
(<2 eV).
MECH 221 fall 2008/6
CONDUCTION & ELECTRON TRANSPORT
• Only electrons with energies greater than the Fermi
energy Ef (i.e. free electrons) may be acted on and
+
accelerated when the electric field is applied.
- Holes have energies less than Ef and also
net e- flow
participate in electronic conduction.
Electron sea in metals
- The electrical conductivity depends on the
numbers of free electrons and holes.
Energy
Energy
electrons into a higher energy state.
• Energy States:
- for metals the nearby energy
states are accessible by thermal
fluctuations.
GAP
partly
filled
valence
band
filled
Free electrons are different from the electron band
sea! They do not become truly free until they
have the required excitation (E>Ef)
M Medraj / PM Wood-Adams
kT
kT
e.g. copper
Mech. Eng. Dept. - Concordia University
empty
band
filled states
- Thermal energy (kT) puts many
empty
band
filled states
• Metals:
filled
valence
band
filled
band
e.g. magnesium
MECH 221 fall 2008/7
INSULATORS AND SEMICONDUCTORS
- Higher energy states not
accessible due to gap.
Energy
empty
band
• Semiconductors:
- Higher energy states
separated by a smaller gap.
Energy
GAP
filled states
kT < Egap
filled
valence
band
filled
band
empty
band
?
GAP
filled states
• Insulators:
filled
valence
band
filled
band
The larger the band gap, the lower is the electrical conductivity at a given temp.
M Medraj / PM Wood-Adams
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/8
METALS: Electron Mobility
• Imperfections increase resistivity
- grain boundaries
- dislocations
- impurity atoms
- vacancies
These act to scatter
electrons so that they
take a less direct path.
From Fig. 18.7
Callister 6th ed.
ρ thermal = ρ o + aT
Matthiessen’s rule
Where ρo and a are constants for each metal.
T ↑ Æ vibration and lattice defects ↑
Æ electron scattering ↑
%CW ↑ Æ dislocation concentration ↑
Æ resistivity ↑
M Medraj / PM Wood-Adams
6
(10-8 Ohm-m)
ρ total = ρ thermal + ρ impurity + ρ def
Resistivity, ρ
• Resistivity increases with temp., impurity concentration and %CW
5
4
3
2
1
0
.3 2
3
+
Ni
%
t
a
Ni %Ni
%
t
16 a .12 at
.
2
+
+1
Cu
u
dC
e
i
N
m
r
%
o
t
a
2
d ef
1
.
+1
Cu
” Cu
e
r
“Pu
Cu
-200 -100
0
T (°C)
Adapted from Fig. 18.8, Callister 6e.
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/9
INFLUENCE OF IMPURITY
ρ total = ρ thermal + ρ impurity + ρ def
ρ impurity = Aci (1 − c )
i
Where ci is impurity concentration
in atomic % and A is constant.
Ni atoms scatter the
electrons Æ ρ ↑
The effect of Ni impurity additions
on the room temp. resistivity of Cu.
For a two phase alloy a rule of mixtures applies and the impurity resistivity can
be estimated as:
ρ impurity = ρ aVa + ρ β Vβ
M Medraj / PM Wood-Adams
V’s and ρ’s are the volume fraction and
individual resistivities for each phase.
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/10
EXAMPLE
180
160
140
120
100
21 wt%Ni
80
60
0 10 20 30 40 50
Resistivity, ρ
(10-8 Ohm-m)
Yield strength (MPa)
Estimate the electrical conductivity of a Cu-Ni alloy that has a yield
strength of 125 MPa.
wt. %Ni, (Concentration C)
Adapted from Fig.
7.14(b), Callister 6e.
50
40
30
20
10
0
0 10 20 30 40 50
wt. %Ni, (Concentration C)
ρ = 30x10 −8 Ohm − m
Adapted from Fig.
18.9, Callister 6e.
1
σ = = 3.3x10 6 (Ohm − m) −1
ρ
M Medraj / PM Wood-Adams
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/11
SEMICONDUCTORS
• Pure “intrinsic” Silicon:
-T↑Æσ↑
- opposite to metals
Energy
(Ohm-m) -1
10 2
10 1
10 0
10 -1
10 -2
50
pure
(undoped)
1 000
Adapted from Fig. 19.15, Callister 5e.T(K)
10 0
M Medraj / PM Wood-Adams
empty
band
?
GAP
filled states
electrical conductivity, σ
For every electron excited into the
conduction band there is left behind a
missing electron - hole
10 4
10 3
σundoped ∝ e
−E gap / kT
electrons
can cross
filled
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 18.2, Callister 6e.
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/12
ELECTRON AND HOLE MIGRATION
• Electrical Conductivity given by:
# holes/m3
σ = n e μe + p e μh
electron mobility
# electrons/m3
valence
electron
hole mobility
In intrinsic semiconductors n|e = p|e
electron hole
pair creation
Si atom
+
no applied
electric field
electron hole
pair migration
- +
applied
electric field
applied
electric field
Adapted from Fig. 18.10, Callister 6e.
M Medraj / PM Wood-Adams
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/13
INTRINSIC VS EXTRINSIC CONDUCTION
Extrinsic:
Intrinsic:
# electrons = # holes (n = p)
- example pure Si or Ge
-n≠p
- occurs when impurities are added with
a different # valence electrons than the
host (e.g., doping Si with P or B)
• N-type Extrinsic: (n >> p)
• P-type Extrinsic: (p >> n)
Boron atom
Phosphorus atom
4+ 4+ 4+ 4+
4+ 5+ 4+ 4+
hole
conduction
electron
4+ 4+ 4+ 4+
valence
electron
4+ 4+ 4+ 4+
4+ 4+ 4+ 4+
no applied
electric field
Si atom
σ ≈ n e μe
M Medraj / PM Wood-Adams
4+ 3+ 4+ 4+
no applied
electric field
σ ≈ p e μh
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/14
INTRINSIC VS EXTRINSIC CONDUCTION
Donor impurities → n-type
(negative) conductivity: by electrons
(a) Donor impurity energy level located just
below the bottom of the conduction band. (b)
Excitation from a donor state in which a free
electron is generated in the conduction band.
Acceptor impurities → p-type
(positive) conductivity: by holes
(a) Acceptor impurity level just above the
top of the valence band. (b) Excitation of
an electron into the acceptor level, leaving
behind a hole in the valence band.
‰ Can control concentration of donors/acceptors ⇒ concentration of charge
•
carriers ⇒ control conductivity
Materials with desired conductivities can be manufactured
M Medraj / PM Wood-Adams
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/15
Semiconductors: Summary
ƒ Intrinsic conductivity (pure materials): electronhole pairs
• Conductivity: Si 4×10-4 (Ωm)-1 vs. Fe 1×107 (Ωm)-1
• Electron has to overcome the energy gap Eg
Intrinsic conductivity strongly depends on
temperature and as-present impurities
• Extrinsic Conductivity
• Doping: substituting a Si atom in the lattice by
an impurity atom (dopant) that has one extra
or one fewer valence electrons
• Donor impurities have one extra electron
(group V: P, As, Sb), donate an electron to Si.
• Acceptor impurities have one fewer electrons
(group III: B, Al, In, Ga), accept electrons from
Si which creates holes.
M Medraj / PM Wood-Adams
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/16
electrical conductivity, σ
(Ohm-m)-1
104
103
102
0.0052at%B
1
intrinsic
2
extrinsic
3
0
0
200 400 600 T(K)
Adapted from Fig. 18.16, Callister 6e.
• Intrinsic vs Extrinsic conduction:
doped
0.0013at%B
- extrinsic doping level:
101
100
10-1
doped
undoped
freeze-out
• Doped Silicon:
- Dopant concentration ↑ - σ ↑
- Reason: imperfection sites
lower the activation energy to
produce mobile electrons.
conduction electron
concentration (1021/m3)
CONDUCTIVITY VS T FOR EXTRINSIC SEMICOND.
pure
(undoped)
10-2
50 100
Adapted from Fig. 19.15, Callister 5e.
M Medraj / PM Wood-Adams
1000
T(K)
1021/m3 of a n-type donor
impurity (such as P).
freeze-out
- for T < 100K: “………….."
thermal energy insufficient to
excite electrons.
extrinsic
- for 150K < T < 450K: “…………"
intrinsic
- for T >> 450K: “…………."
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/17
SUMMARY
• Electrical resistance is:
- a geometry and material dependent parameter.
• Electrical conductivity and resistivity are:
- material parameters and geometry independent.
• Conductors, semiconductors, and insulators...
- different in whether there are accessible energy states for 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).
M Medraj / PM Wood-Adams
Mech. Eng. Dept. - Concordia University
MECH 221 fall 2008/18