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

Electrical Conductivity Copper vs. Aluminum wiring

Electrical Conductivity 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

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 Resistivity, ρ (m) and Conductivity, σ (m)-1 material properties are independent of sample size and geometry

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” ***

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

Team Problem

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

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

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

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

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”

Conductivity of Metals and Semiconductors metal >> semi

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 + 3.32 at%Ni • Resistivity increases with  = t -- temperature T = o + aT thermal deformed Cu + 1.12 at%Ni i -- % impurity + impurity Cu + 1.12 at% Ni d -- % cold work + deformation “Pure” Cu

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

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

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.

Team Problem