5. Band theory Flashcards

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1
Q

What is band theory

A

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.

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2
Q

Give the band theory Schrodinger equation

A

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

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3
Q

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

A

The periodicity of the crystal lattice

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4
Q

What is Bloch’s theorem?

A

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).

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5
Q

State the equation for the Bloch function

A

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

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6
Q

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

A

The Bloch form

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7
Q

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

A

Plane waves
Free electron model

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8
Q

What are forbidden energies?

A

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

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9
Q

Can there be several energies for the same wavevector?

A

Yes

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10
Q

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

A

Periodic
Periodic zone

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11
Q

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

A

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.

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12
Q

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

A

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

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13
Q

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

A

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

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14
Q

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

A

N allowed k-states for a crystal containing N atoms

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15
Q

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

A

2N for a crystal containing N atoms

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16
Q

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

A

An energy band will be completely filled

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17
Q

Why can’t insulating crystals become conducting?

A

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.

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18
Q

What makes a crystal insulating?

A

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

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19
Q

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

A

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

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20
Q

What can increase the conductivity of a semiconductor?

A

Increasing temperature so that more electrons can transition.

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21
Q

How are metals and semiconductors different?

A

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

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22
Q

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

A

Group velocity

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23
Q

Give the equation for group velocity

A

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

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24
Q

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

A

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.

25
Q

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

A

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

26
Q

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

A

The Bragg condition for diffraction

k = ± nπ / a

27
Q

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

A

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

28
Q

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

A

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

29
Q

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

A

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

30
Q

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

A
31
Q

Give the equation for the probability density of a particle

A

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

32
Q

What is the probability density for a pure plane wave?

A
33
Q

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

A
34
Q

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

A

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)

35
Q

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

A

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

36
Q

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

A

Schrodinger equation

37
Q

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

A

L = crystal length
k = wave number
x = position

38
Q

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

A

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

39
Q

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

A

π / a

40
Q

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

A

PE = potential energy
U0 = initial potential energy

41
Q

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

A

U0 = bandgap
Gradient = 0 at the bandgap

42
Q

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.

A

Unaffected
Increased
Lowered

43
Q

What is the tight binding approximation?

A

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

44
Q

What is the tight binding approximation similar to?

A

The linear combination of orbitals (LCAO)

45
Q

Describe the shape of the tight binding approximation energy levels

A

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

46
Q

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

A
47
Q

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

A

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

48
Q

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

A

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

49
Q

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

A

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

50
Q

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

A

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

51
Q

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

A

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

52
Q

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

A

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

53
Q

What is the result of the tight binding approximation?

A

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

54
Q

Give the equation for α in the tight binding approximation

A
55
Q

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

A
56
Q

How do Fermi surfaces change as the Fermi energy increases?

A

k also increases with increasing Fermi energy

57
Q

Give the equation for the effective mass of an electron

A

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

58
Q

What is a hole?

A

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.