ENGR-1600Materials Science for Engineers

Lecture 24: Electrical Conductivity

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• Copper vs. Aluminum wiring

Electrical Conductivity

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• 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

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Macroscopic Ohm’s Law

Resistivity, ρ (m) and Conductivity, σ (m)-1

material properties are independent of sample size and geometry

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AoV: voltage (volts = joule/coulomb) VI: current (ampere = coulomb/sec) AR: resistance (ohm = volt/amp) Ω

Ohm’s Law

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Macroscopic Ohm’s Law

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AoV: voltage (volts = joule/coulomb) VI: current (ampere = coulomb/sec) AR: resistance (ohm = volt/amp) Ω

electric field

current density

(amp/m2)

*** Don’t confuse A for “ampere” with Ao for “cross-sectional area” ***

Ohm’s Law

Ohm’s Law

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Team Problem

What’s the difference between resistance and resistivity?

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Team Problem

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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

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Relating Energy Band Structures to Bonding

MetalsSemiconductors

Eg < 2 eVInsulators 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

Conduction & Electron TransportMetals: • Empty energy states are adjacent to filled states• Thermal energy excites electrons into empty higher energy states• Hence, these electrons conduct electricity

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Energy Band StructuresSemiconductors / insulators: • highest occupied band is filled at 0 K • electronic conduction requires thermal excitation across a bandgap, T• EF is in the bandgap

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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

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metal >> semi

Conductivity of Metals and Semiconductors

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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

=

deformed Cu + 1.12 at%Ni

T (°C)-200 -100 0

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Resi

stivi

ty, ρ

(10

-8 O

hm-m

)

0

Cu + 1.12 at% Ni

“Pure” Cu

d -- % cold work

+ deformation

i

-- % impurity

+ impurityt

-- temperature T = o + aT

thermal

Cu + 3.32 at%Ni

Imperfections:

Team Problem

Cold working Copper alloy causes an increase in resistivity.

Explain why.

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• Solid solution: i = A ci(1-ci)

• Two phases (+): i = V + V

Influence of Impurities

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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.

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Team Problem