Electronic- Band Theory of Solids Flashcards

1
Q

What does the Feynman model start from and consider?

A

Starts from atomic orbitals of atoms. Considers what happens as they are brought together in a 3D array

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2
Q

What happens where two atoms come together?

A

The orbitals in each have a wavefunction. They overlap. They can sum together to make stable molecular orbitals (electrons concentrate between nuclei) or one can subtract to give high energy molecular orbitals. The energy levels must split because electrons cannot occupy the same orbitals (Pauli)

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3
Q

How does interatomic separation affect orbital splitting?

A

If atoms far away, the electrons in a given state in each atom can have the same energy. When one state begins to overlap, the orbitals will split to create as many new electron bands as there are atoms overlapping. Splitting occurs first in the outermost orbitals from the nucleus

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4
Q

Why is it rare to see splitting of 1s orbitals?

A

This orbital is very close the the nucleus and atoms rarely get close enough to each other for these to overlap

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5
Q

What happens when there is interaction between outer electrons of atoms in a bulk material at equilibrium separation?

A

The orbitals that overlap between two atoms split. There is a band between the high and low energy molecular orbitals produced. There is a band gap between bands produced by other types of overlapping orbitals

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6
Q

Wavefunction of free electron

A

ψ=sin(πnx/L)=sin(kx)

k=πn/L

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7
Q

Energy states of a free electron

A

E=h^2n^2/8mL^2
=(hbar)^2k^2/2m
hbar=h/2π

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8
Q

Formula for wave vector, k

A

k=p/hbar=mv/hbar
p is momentum
In 3D k is vector quantity and measures quantum state number and electron momentum

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9
Q

Formula for kinetic energy involving k

A

KE=1/2 mv^2= (hbar)^2k^2/2m

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10
Q

What type of energy does a free electron have?

A

All energy is kinetic as the potential energy is set to 0.

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11
Q

E-k curves

A

For free electrons E proportional to k^2.

E vs k is like y=x^2 curve

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12
Q

What does the Ziman model consider?

A

Weak interactions between the nuclei and electrons. When interatomic spacing, a, is multiple of half wavelength of EM waves, they reflect. Electrons can be waves so there are multiple Bragg reflections at each plane of atoms creating a standing wave. Corresponds to wave vectors k=nπ/a

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13
Q

Why are the atoms in a metal positively charged ions?

A

Their valence electrons are itinerant (not localised) so each is associated with an electrostatic potential

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14
Q

What is the result of electrons being close to positively charged ions in metals?

A

The coulombic interaction leads to a reduction in potential energy, V. The electron will therefore experience a potential at each ion meaning its potential is not 0 (it would be if free).

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15
Q

Describe the standing waves and potential in the Ziman model

A

The electron can either be a forward (ψ(+)) or backwards (ψ(-)) standing wave. The potential experienced is either +V or -V depending on the direction of the electron standing wave. The maxima in the probability distributions are above the atoms [ψ(+)]^2 or above the midpoint between atoms [ψ(-)]^2. Other wave vectors k are travelling waves with constant probability distribution.

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16
Q

Formation of band gaps from Ziman model

A

ψ(+) is decreased in energy. ψ(-) is increased in energy. Other wavefunction have no potential energy term. Above each ion an energy gap appears because
E=(hbar)^2k^2/2m +-V.
Results in allowed and forbidden bands of electron energies. Allowed bands separated by band gap of energy Eg. k still continuous but has abrupt jumps in E

17
Q

E-k curve for electron in metal (not free)

A

Follows sin^2 shape until k=π/a (a maximum) then discontinuous jump up Eg until new sin^2 shape starts. First allowed band then forbidden band then second allowed band

18
Q

When do energy breaks occur?

A

When k=nπ/a
Or n=L/a or 2L/a etc
Or when wavelengths are na
This is when electron wavelengths are related to ions separations

19
Q

What factors determine the conductivity of a material?

A

Conduction only uses electrons near the Fermi surface. It is a function of electronic states near the surface and not total free electrons. Must consider how many states exist near the surface and where the Fermi level lies in relation to the band gaps. Mg poorer conductor than Na even with more free electrons

20
Q

Arrangement of bands for metals, semi-metals, insulators and semiconductors

A

Metals: filled states, empty states (Ef at interface), band gap, empty band.
Semi-metals: filled band overlaps with empty band (Ef at top of filled band).
Insulators: filled valence band, large band gap, empty conduction band.
Semiconductors: filled valence band, small band gap, empty conduction band

21
Q

What are the highest filled and lowest empty bands called for insulators and semiconductors?

A

Highest filled= valence band

Lowest empty= conduction band

22
Q

How does electrical conduction happen in a metal?

A

Electrons from filled states can be excited into empty states just above Ef. Both excited electron and vacant hole contribute to the electrical conduction

23
Q

Formula for electrical current density

A

J=nve
n is number of charge carriers
v is charge carrier velocity
Eg is charge on electron

24
Q

How does n vary across metals, insulators and semiconductors?

A

In metals at reasonable temperature, n is very large.
In insulators, n is very low as Eg is too large (>2ev) for thermal excitation of electrons.
Semiconductors have small Eg so thermal excitation possible and n highly temperature dependent