1. Lecture 24: Electrical Conductivity
ENGR-1600
Materials Science for Engineers
Lecture 24: Electrical Conductivity

2. Electrical Conductivity
Copper vs. Aluminum wiring

3. Electrical Conductivity
Electrical conductivity between different materials varies by over 20 orders of magnitude
The greatest variation of any physical property
Metals:  > (m)-1
Semiconductors: 10-6 <  < (m)-1
Insulators:  < (m)-1

4. Macroscopic Ohm’s Law I VAo
V: voltage (volts = joule/coulomb) V
I: current (ampere = coulomb/sec) A
R: resistance (ohm = volt/amp) Ω I V
Resistivity, ρ (m) and Conductivity, σ (m)-1
material properties are independent of sample size and geometry

5. Macroscopic Ohm’s Law I V Ao V: voltage (volts = joule/coulomb) V
I: current (ampere = coulomb/sec) A
R: resistance (ohm = volt/amp) Ω I V
electric field
current density
(amp/m2)
Ohm’s Law
*** Don’t confuse A for “ampere” with Ao for “cross-sectional area” ***

6. What’s the difference between resistance and resistivity?
Team Problem
What’s the difference between resistance and resistivity?

7. Team Problem

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. Relating Energy Band Structures to Bonding Semiconductors Eg < 2 eV
Insulators Eg > 2 eV
Semiconductors Eg < 2 eV
Metals
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

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. Microscopic Electric Conductivity
due to imperfections in the crystal
vd = eE
 = n e e
vd drift velocity [m/s]
μ e- mobility [m2/Vs]
n # of free electrons
|e| charge of an e- [C]
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.
The vd is proportional to E by the factor μ, the “electron mobility”

13. Conductivity of Metals and Semiconductors
metal >> semi

14. Resistivity of Metals grain boundaries dislocations impurities
vacancies
Imperfections:
these all scatter electrons so that they take a less direct path  lower σ
T (°C)
-200
-100 1 2 3 4 5 6
Resistivity, ρ
(10 -8
Ohm-m)
Cu at%Ni
• Resistivity increases with
 = t
-- temperature T = o + aT
thermal
deformed Cu at%Ni i
-- % impurity
+ impurity
Cu at% Ni d
-- % cold work
+ deformation
“Pure” Cu

15. Cold working Copper alloy causes an increase in resistivity.
Team Problem
Cold working Copper alloy causes an increase in resistivity.
Explain why.

16. Influence of Impurities
Solid solution: i = A ci(1-ci)
Two phases (+): i = V +  V 

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.

18. Team Problem