3. 3
• Electrical conductivity between different materials varies by over 20
orders of magnitude
• The greatest variation of any physical property
Metals: > 105 (m)-1
Semiconductors: 10-6 < < 105 (m)-1
Insulators: < 10-6 (m)-1
Electrical Conductivity
4. 4
Macroscopic Ohm’s Law
R = r
ℓ
A
s =
1
r
Resistivity, ρ (m) and Conductivity, σ (m)-1
material properties are independent of sample size and geometry
V
I
V = IR Ao
V: voltage (volts = joule/coulomb) V
I: current (ampere = coulomb/sec) A
R: resistance (ohm = volt/amp) Ω
Ohm’s Law
5. 5
Macroscopic Ohm’s Law
V
I
R = r
ℓ
Ao
V = IR Ao
s =
1
r
V: voltage (volts = joule/coulomb) V
I: current (ampere = coulomb/sec) A
R: resistance (ohm = volt/amp) Ω
r =
VAo
Iℓ
=
E
J
electric field
current
density
(amp/m2)
*** Don’t confuse A for “ampere” with Ao for “cross-sectional area” ***
J =sE Ohm’s Law
Ohm’s Law
8. 8
Electron Energy Band Structures
• Pauli Exclusion Principle: no
two e- in an interacting
system can have exactly same
energy
• When N atoms are far apart,
they do not interact, so
electrons in a given shell in
different atoms have same
energy
• As atoms come closer
together, they do interact,
perturbing electron energy
levels
• Electrons from each atom
then have slightly different
energies, producing a “band”
of allowed energies
9. 9
Relating Energy Band Structures to Bonding
Metals
Semiconductors
Eg < 2 eV
Insulators
Eg > 2 eV
• In metals, highest occupied band is partially filled or bands overlap
• Highest filled state at 0 K is the Fermi Energy, EF
• at 0 K, all e- states below EF are filled, all above are vacant
• Electrons in a filled band cannot conduct
• Only e- with energies above EF can conduct
10. Conduction & Electron Transport
Metals:
• Empty energy states are adjacent to filled states
• Thermal energy excites electrons into empty higher energy states
• Hence, these electrons conduct electricity
10
11. 11
Energy Band Structures
Semiconductors / insulators:
• highest occupied band is filled at 0 K
• electronic conduction requires thermal excitation across a bandgap, T
• EF is in the bandgap
12. 12
vd = eE
= n e e
Microscopic Electric Conductivity
• When an electric field E is applied, e- experience a force. Hence, they accelerate.
• This force is counteracted by scattering events (analogy to friction).
• When the forces balance out, there is a constant mean value of e- velocity vd.
vd drift velocity [m/s]
μ e- mobility [m2/Vs]
n # of free electrons
|e| charge of an e- [C]
• The vd is proportional to E by the factor μ, the “electron mobility”
due to
imperfections in
the crystal
14. 14
Resistivity of Metals
- grain boundaries
- dislocations
- impurities
- vacancies
these all scatter
electrons so that they
take a less direct path
lower σ
• Resistivity increases with
=
T (°C)
-200 -100 0
1
2
3
4
5
6
Resistivity,
ρ
(10
-8
Ohm-m)
0
d -- % cold work
+ deformation
i
-- % impurity
+ impurity
t
-- temperature T = o + aT
thermal
Imperfections:
16. 16
• Solid solution: i = A ci(1-ci)
• Two phases (+): i = V + V
Influence of Impurities
17. 17
Materials Choices for Metal Conductors
• Most widely used conductor is copper: inexpensive,
abundant, very high
• Silver has highest of metals at RT, but use restricted due to
cost
• Aluminum used to be main material for electronic circuits,
transition to electrodeposited Cu
• Remember deformation reduces conductivity, so high
strength generally means lower : trade-off.
• Heating elements require low (high R), and resistance to
high temperature oxidation.
