Electrical- Conductivity and Resistance of Metals

Question 1

What is resistivity a function of?

Answer 1

Temperature, impurity concentration, defect concentration

The resistivity due to each add together to get the total

Question 2

Describe current flow and resistance in conventional metals

Answer 2

Current is result of movement of electrons under influence of applied electric field. Their flow is impeded by resistance which results from the scattering of electrons. Resistance can also be caused by defects in the metal such as impurities and grain boundaries. Scattering from such defects results in extrinsic resistance

Question 3

What are phonons?

Answer 3

Interactions of electrons with atoms or defects

Question 4

How does resistivity change with temperature?

Answer 4

Increases with temperature because atomic vibrations increase with temperature hence more electron-phonon interactions.
ρT=ρ25(1+αΔT)
α is associated with thermal vibrations of atoms and also thermally generated defects
T and 25 subscript

Question 5

Resistivity due to impurities formula

Answer 5

In simple homogeneous alloy system where impurities present at low concentrations c:
ρ impurities =Ac(1-c)
A depends on host metal and impurity
Independent of temperature
ρ vs c plot a dome
If heterogeneous solution then overall resistivity sum of two fractional components
ρ=ρaVa+ρbVb

Question 6

How does resistivity due to deformation compare with that due to temperature and impurities?

Answer 6

It’s effect is smaller than them

Question 7

What do metals nee for high conductivity σ?

Answer 7

They need to be pure and O2 free

Question 8

Ways to improve the problem of pure metals being soft and prone to deformation

Answer 8

Use solid solution alloy. Use dispersed second phase (little effect). Use cold working (can reduce σ)

Question 9

What is the mean free path λ of conduction electrons?

Answer 9

The average distance between collisions with atoms

Question 10

What does more frequent collisions of conduction electrons mean?

Answer 10

Shorter λ

Lower electron mobility

Question 11

Difference between ideal metal and in practice when T=0K

Answer 11

Ideal λ tends to infinity
In practice: atoms vibrate about equilibrium position, impurities may be present, crystal lattice may contain lattice imperfections

Question 12

Typical λ in good metal

Answer 12

5-20nm

Question 13

What happens if λ is extremely long with only a few collisions?

Answer 13

Ohm’s law breaks down

New equation is I=rt(V)

Question 14

How does conductivity and λ change for homogeneous metal alloys, age hardened or dispersion strengthened alloys and work hardened metals compared to the pure metal?

Answer 14

In homogeneous metal alloys, λ is very short as the scattering centres are relatively close together. The σ is much less than pure.
In age hardened or dispersion strengthened alloys, λ is longer than above as the second phase is concentrated in discrete positions rather than being randomly distributed. σ is only reduced a small amount from pure.
In work hardened metals, λ even longer than above as dislocations are well separated as their stress field act to repel each other. σ almost the same as pure.

Question 15

Which electrons do what in a metal?

Answer 15

The inner orbitals are discrete and localised so don’t take part in conductivity. These are the core electrons. Only the outer (valence) electrons which overlap orbitals to form bands of energy levels do.

Question 16

The two types of band structures in metals

Answer 16

Partly full bands such as Cu

Overlapping bands where both are partly occupied, e.g Mg where 3s and 3p overlap

Question 17

Bad theory with metallic bonding

Answer 17

Above the highest occupied levels, there are empty levels of only slightly higher energy so electrons can hop from one to the other. There is a partly filled valence band.

Question 18

Band theory for insulators

Answer 18

There is a large forbidden energy gap between the highest occupied levels and the next lowest unoccupied level. The band gap is high (>3eV). There is a filled valence band and empty conduction band

Question 19

Band theory for semiconductors

Answer 19

The band gap is small (<3eV). Electrons can be promoted across the gap either through intrinsic conduction (usually thermally) or by adding dopants that have energy states within the band gap (extrinsic conduction)

Question 20

Where is the Fermi energy level Ef?

Answer 20

At approximately half the band gap Eg

Question 21

General points about band theory

Answer 21

The VB is the series of electron energy levels at which all valence electrons normally reside. Unless there are fancies the electrons will not move. The CB is the band level at which electrons are free to move. Theory ignores e-e repulsion effects within the bands (works well for wide bands not for narrow). Width of band related to relative energy of orbitals involved and on degree of overlap between orbitals. Actual position if Ef in intrinsic semi and insulators depends on relative density of states in VB and CB but roughly 0.5Eg

Question 22

Why is Al a metal?

Answer 22

Due to its structure and bonding. It is hcp so has 12 close neighbours but only 3 valence electrons so not enough electrons to form covalent bonds. Means metallic bonding with the valence electrons delocalised in 3s and 3p.

Question 23

Why isn’t Si a metal?

Answer 23

Forms solid where all atoms linked by covalent bonds. Structure of solid shows coordination number and geometry similar to that found in molecules where the atom exhibits its normal valency. Group IV solids are tetrahedrally bonded. Covalent bonds in the solid are similar to those in small molecules.

Question 24

How does sp3 hybridisation work?

Answer 24

Mixing of 3s and 3p atomic orbitals on the same atom to produce 4 sp3 hybridised orbitals that have a definite direction in space. Happens in Si. One electron in 3s and one in each of 3p orbitals goes to one in each of 4sp3 hybridised orbitals. Has tetrahedral geometry.

Question 25

Band picture in Si solid

Answer 25

Two sp3 orbitals are the AOs. Makes σ and σ* with the 2 electrons in the σ. Band gap about 1.1eV. This is one Si-Si bond.
In pure Si a small number of electrons have sufficient thermal energy to be promoted to CB. This number increases with temperature