Part 3 (Beyond Free Electron Theory)

Question 1

Which metals are poor free electrons?

Answer 1

Any transition metals
e.g. bcc Mo, fcc Rh

Mn: the fermi level lies within the d band (the free electron model is a very poor description)

Question 2

How is the idea of the particle in a box ammended?

Answer 2

The box needs some internal structure: the periodic potential of the crystal lattice

Question 3

What is the difference between widely spaced atoms and closely spaced atoms?

Answer 3

Widely spaced: discrete energy levels (orbitals)

Closely spaced: overlap of wave functions which leads to splitting (number of split levels = number of atoms)

Question 4

Where do d orbitals extend to?

Answer 4

They do not extend far outside the atom (little broadening, narrow band)

Question 5

Where do s and p orbitals extend to?

Answer 5

They extend far outside the atom (wide band)

Question 6

What does adding electrons do?

Answer 6

It moves the Fermi level up

Question 7

What happens when free-electron plane waves interact with a periodic potential?

Answer 7

This causes the Bragg condition: at this value of k, we have standing waves and group velocity is zero

Question 8

What is the group velocity at zone boundaries?

Answer 8

Zero

Question 9

Why are there two possible values of k for the standing wave?

Answer 9

Due to bonding and anti-bonding of the travelling free electron

Question 10

Why are the gaps open at zone boundaries?

Answer 10

There is a difference in energy between bonding and anti-bonding wave functions which means that a gap opens at the zone boundaries

Question 11

Where do energy gaps appear?

Answer 11

At the corresponding periodicity of k (+- pi/a, +- 2pi/a ,+-3pi/a)

Question 12

When does the parabola of electron dispersion relation change?

Answer 12

If the mass of electron is changed

Question 13

What is the effective mass of a particle given by?

Answer 13

The inverse curvature of its dispersion relation

Question 14

What does effective mass account for?

Answer 14

The interactions with the lattice

real mass does not change

Question 15

What happens to the effective mass near the top of the band?

Answer 15

It becomes negative

Question 16

What happens at the top of the band?

Answer 16

The complex interactions with the lattice mean that an electric field accelerates
electrons in the opposite direction to the one
they would move if they were free particles.

This explains the negative hall coefficients in Mg, Al etc

Question 17

What do narrow energy bands tend to have?

Answer 17

Less curvature and so larger effective masses (electron band masses)

Question 18

In which band are electrons “heavier”?

Answer 18

in d-band compared to in s-bands

Question 19

Which metals have lower conductivity, transition or free electron metals?

Answer 19

Transition metals

Question 20

What happens to filled bounds?

Answer 20

They cannot conduct

Question 21

What is the Kronig-Penny model?

Answer 21

A highly simplified 1D model where the periodic potential is a square wave (square well). It is not very
realistic, but it can be solved analytically.

Question 22

What are Bloch waves?

Answer 22

Periodic wave functions

Question 23

What is Bloch’s theorem?

Answer 23

The energy eigenstates for an electron in a crystal can be written as Bloch waves

Question 24

Why does scattering occur when the lattice periodicity is broken?

Answer 24

Either by static defect or by dynamic deviation such as phonon

Question 25

What happens as a result of the LHS of the Bloch wave solution only taking values between -1 and 1?

Answer 25

There are not always solutions and there are energy gaps at the Brillouin zone boundaries

Question 26

In metals, where is the fermi energy located?

Answer 26

in a band

Question 27

In semiconductors, where is the fermi energy located?

Answer 27

in a (small) gap

Question 28

In insulators, where is the fermi energy located?

Answer 28

in a gap