Electronic- Band Theory of Solids Flashcards
What does the Feynman model start from and consider?
Starts from atomic orbitals of atoms. Considers what happens as they are brought together in a 3D array
What happens where two atoms come together?
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)
How does interatomic separation affect orbital splitting?
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
Why is it rare to see splitting of 1s orbitals?
This orbital is very close the the nucleus and atoms rarely get close enough to each other for these to overlap
What happens when there is interaction between outer electrons of atoms in a bulk material at equilibrium separation?
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
Wavefunction of free electron
ψ=sin(πnx/L)=sin(kx)
k=πn/L
Energy states of a free electron
E=h^2n^2/8mL^2
=(hbar)^2k^2/2m
hbar=h/2π
Formula for wave vector, k
k=p/hbar=mv/hbar
p is momentum
In 3D k is vector quantity and measures quantum state number and electron momentum
Formula for kinetic energy involving k
KE=1/2 mv^2= (hbar)^2k^2/2m
What type of energy does a free electron have?
All energy is kinetic as the potential energy is set to 0.
E-k curves
For free electrons E proportional to k^2.
E vs k is like y=x^2 curve
What does the Ziman model consider?
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
Why are the atoms in a metal positively charged ions?
Their valence electrons are itinerant (not localised) so each is associated with an electrostatic potential
What is the result of electrons being close to positively charged ions in metals?
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).
Describe the standing waves and potential in the Ziman model
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.
