3.1 Nearly Free Electron Model Flashcards
How are electrons modelled in this model?
They are considered to be weakly perturbed by the periodic potential due to the positive ions at lattice sites
Describe the wavefunction in this model
A plane wave
From the Bragg/Laue condition, where does the first reflection of the wavefunction occur?
At the BZ boundary
Describe what is observed when a plane wave is reflected at the BZ boundary due to the Laue/Bragg condition
A right travelling wave becomes a left travelling wave and vv due to superposition
Describe the two possible electron wavefunctions from the superposition of the reflected waves at the BZ boundary
Positive wavefunction - electrons on lattice sites
Negative wavefunction - electrons away/between lattice sites
Describe why we observe an energy gap at the BZ boundary
The charge density is proportional to ΨΨ*,so electrons start piling up at the lattice sites (+ve Ψ) and between the lattice sites (-ve Ψ)
- Regions have different potentials so electrons have different energies which open up gaps at the BZ boundary
How are the number of k states and the number of unit cells in a system related in 1D?
Number of k states = number of unit cells
Describe the properties for an electron band in a monovalent primitive unit cell
A primitive unit cell is monovalent ( 1 Electron) and there are two possible spin states
- Band is half full
Describe why a monavalent primitive cell can easily be a metal
The half full band has a number of reachable empty states which electrons can be moved into under the application of an electric field
- Current can flow - Metal
State the number of electrons in the primitive unit cell necessary for a metal, insulator and semiconductor
Monovalent (1 electron) for a metal
Divalent (2 electrons) for a semiconductor or insulator
Describe when we get an insulator or a semiconductor in a divalent primitive unit cell
We have a filled band and an empty band so an energy is required to move an electron from the valence band to the conduction band
- T = 0, Electric field wont move electrons - INSULATOR
- Room temp and if the band gap is < 4eV - SEMICONDUCTOR as electrons can be excitedqq
How does the area/volume of the fermi surface change for a monovalent material in 2D or 3D
Need to consider band filling
Area/volume of the Fermi surface will be half that of the first BZ
What happens to the Fermi surface as the periodic potential is increased?
The surface becomes more distorted
Why does the Fermi surface become more distorted as the periodic potential increases?
The states in the BZ are pushed down in energy and electrons start to preferentially fill these
- States outside the BZ are pushed up in energy
How does the area/volume of the fermi surface change for a divalent material in 2D or 3D
It is the same as the first BZ
