5. Band theory

Question 1

What is band theory

Answer 1

A way of finding solutions to the one-electron Schrodinger equation. It includes an extra potential energy term to represent the positive ion cores of the crystal due to the lattice.

Question 2

Give the band theory Schrodinger equation

Answer 2

m = mass
ψ = wavelength
U(x) = potential energy
ε = energy

Question 3

What is the periodicity of the potential energy term in the Schrodinger equation equal to?

Answer 3

The periodicity of the crystal lattice

Question 4

What is Bloch’s theorem?

Answer 4

Proof that the solutions of the Schrodinger equation for a periodic potential must be in the form of a Bloch function (where uₖ(r) is some function with the same periodicity as the crystal lattice).

Question 5

State the equation for the Bloch function

Answer 5

ψₖ(r) = wavelength
uₖ(r) = uₖ(r + T) = potential
k = wave vector
r = position
T = lattice translation vector

Question 6

What form are solutions to the band theory Schrodinger equation (electron wavefunction) found in?

Answer 6

The Bloch form

Question 7

In the limit of constant potential energy, the Bloch solutions are ______ ______ with an energy-wavevector relationship given by the _____ _________ ______.

Answer 7

Plane waves
Free electron model

Question 8

What are forbidden energies?

Answer 8

Energies with no wavelike solutions to the Schrodinger equation. This gives bands of allowed energies separated by bandgaps of forbidden energies.

Question 9

Can there be several energies for the same wavevector?

Answer 9

Yes

Question 10

The band structure is ________ (it is a ________ ______ representation).

Answer 10

Periodic
Periodic zone

Question 11

What is the extended zone representation of an energy vs. wavevector graph (Bloch’s theorem)?

Answer 11

When the energy level diagram is split into Brillouin zones. It shows that as k increases, so does energy and that because of periodicity, changing k by a reciprocal lattice vector results in the same observable state.

Question 12

State the alternative statement of Bloch’s theorem that proves it has the same periodicity as the lattice

Answer 12

ψ = wavelength
r = position
T = lattice translation vector
k = wavevector

Question 13

What is the reduced zone representation of an energy vs. wavevector graph (Bloch’s theorem)?

Answer 13

A more compact energy scheme that translates all wavevectors, confining them to the first Brillouin zone.

Question 14

How many allowed k-states are there in each zone of the reduced zone representation of Bloch’s theorem?

Answer 14

N allowed k-states for a crystal containing N atoms

Question 15

How many electron states are there in each zone of the reduced zone representation of Bloch’s theorem?

Answer 15

2N for a crystal containing N atoms

Question 16

What happens when there is an even number of electrons per atom (Bloch’s theorem)?

Answer 16

An energy band will be completely filled

Question 17

Why can’t insulating crystals become conducting?

Answer 17

A great enough electric field must be applied for the electrons to jump the band gap which is generally not possible if the gap is a few eV.

Question 18

What makes a crystal insulating?

Answer 18

It has a band gap between energy bands that cannot be crossed by exciting electrons.

Question 19

Why does a small current flow in semiconductors at room temperature?

Answer 19

The energy gap is generally small (e.g. 1 eV) so a few electrons can be excited across the band gap at room temperature.

Question 20

What can increase the conductivity of a semiconductor?

Answer 20

Increasing temperature so that more electrons can transition.

Question 21

How are metals and semiconductors different?

Answer 21

Semiconductors increase in conductivity with increasing temperature whereas metals decrease in conductivity.

Question 22

In quantum mechanics, the velocity of a particle is given by the ______ ________.

Answer 22

Group velocity

Question 23

Give the equation for group velocity

Answer 23

v_g = group velocity
ω = angular frequency
k = wave number
ε = energy

Question 24

How does the velocity of an electron change as it moves through the Brillouin zone?

Answer 24

It starts at 0 at the middle of the Brillouin zone and increases with increasing k for a while then starts to decrease up to the Brillouin zone edge where the gradient (and therefore the velocity) is equal to 0 again.

Question 25

What does it mean when the velocity of an electron is 0 in band theory?

Answer 25

Motion is equally probable in the + and the - directions.

Question 26

What condition do electrons satisfy at the edge of the Brillouin zone?

Answer 26

The Bragg condition for diffraction k = ± nπ / a

Question 27

What types of waves do electrons form at the edge of the Brillouin zone?

Answer 27

Standing waves (as they satisfy the Bragg condition for diffraction so are reflected back on themselves)

Question 28

What happens to electrons at the edges of the Brillouin zone in band theory?

Answer 28

They are Bragg reflected if they meet the diffraction condition, causing standing waves

Question 29

What are the standing waves at the Brillouin zone edges made of?

Answer 29

Equal amounts of waves travelling to the left and to the right.

Question 30

State the two wavefunctions that make up standing waves at the zone edge

Answer 30

Question 31

Give the equation for the probability density of a particle

Answer 31

ρ = probability density ψ = wavefunction ψ* = complex conjugate wavefunction

Question 32

What is the probability density for a pure plane wave?

Answer 32

Question 33

What are the probability densities for the ψ+ and ψ- states of a wavefunction?

Answer 33

Question 34

Graphically describe the locations of the peaks of the ψ+ and ψ- states of a wavefunction within a crystal

Answer 34

ψ+ is the cos term and peaks exactly where the ion cores are ψ- is the sin term and peaks exactly halfway between ion cores (where the potential is minimum)

Question 35

Give the equation for the lattice potential in the nearly free electron model

Answer 35

U(x) = potential U0 = initial potential x = position a = unit cell length

Question 36

Give the equation for the expectation value for the energy of an electron using the nearly free electron model

Answer 36

Schrodinger equation

Question 37

State the value of the normalised wavefunctions ψ+ and ψ-

Answer 37

L = crystal length k = wave number x = position

Question 38

Give the equation for the kinetic energy in the nearly free electron model

Answer 38

KE = kinetic energy k = wave number m = electron mass

Question 39

The potential energy is equal to zero for all values of k in the nearly free electron model except if k is equal to ___.

Answer 39

π / a

Question 40

Give the equation for the potential energy in the nearly free electron model when k = π / a

Answer 40

PE = potential energy U0 = initial potential energy

Question 41

Describe the nearly free electron model graphically and using the two equations for energy

Answer 41

U0 = bandgap Gradient = 0 at the bandgap

Question 42

The nearly free electron model shows that the kinetic energy of the electrons is ___________ by the periodic lattice, but the potential energy of electrons at the edges of the Brillouin zone are ________ or ________ depending on the phase of the wavefunction.

Answer 42

Unaffected Increased Lowered

Question 43

What is the tight binding approximation?

Answer 43

A model of the crystal lattice that considers the electrons to be tightly bound to the atoms but with some overlap with neighbouring atoms.

Question 44

What is the tight binding approximation similar to?

Answer 44

The linear combination of orbitals (LCAO)

Question 45

Describe the shape of the tight binding approximation energy levels

Answer 45

Overlap between neighbouring atoms smears discrete energies into narrow bands separated by large bandgaps.

Question 46

Describe how the 4 electron/crystal models differ in terms of energy levels

Answer 46

Question 47

Give the tight binding approximation Schrodinger equation for an atomic orbital in the isolated atom

Answer 47

U = potential ε = energy of isolated atom φ = atomic orbital wavefunction r = position

Question 48

Give the equation for the approximate wavefunction for one electron in a whole crystal (tight binding approximation)

Answer 48

ψ = molecular orbital wavefunction φ = atomic orbital wavefunction N = number of atoms T = Lattice translation vector r = position

Question 49

Give the tight binding approximation Schrodinger equation for an electron in a whole crystal

Answer 49

ψ = molecular orbital wavefunction U = crystal potential ε = energy of electron in the crystal

Question 50

What components does the energy of an electron in a crystal consist of in the tight binding approximation?

Answer 50

The sum of a kinetic energy term and an atomic potential term

Question 51

Give the equation for the energy of an electron in a crystal (tight binding approximation)

Answer 51

ε = energy of electron in the crystal ε_atomic = energy of isolated atom U = potential energy ψ = molecular wavefunction

Question 52

What happens to the energy of an electron when it is in a crystal?

Answer 52

It is impacted by a change in potential due to being in the crystal

Question 53

What is the result of the tight binding approximation?

Answer 53

Term 1 = discrete energy Term 2 = constant lowering discrete energy Term 3 = overlap integral

Question 54

Give the equation for α in the tight binding approximation

Answer 54

Question 55

Give the equation for A(T) in the tight binding approximation

Answer 55

Question 56

How do Fermi surfaces change as the Fermi energy increases?

Answer 56

k also increases with increasing Fermi energy

Question 57

Give the equation for the effective mass of an electron

Answer 57

m* = effective mass ε = energy of electron k = wave number

Question 58

What is a hole?

Answer 58

The absence of electrons at the top of a band that normally behave like they have negative mass and negative charge. This absence can be considered as the presence of particles with 'positive' mass and 'positive' charge.