5. Band theory Flashcards
What is band theory
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.
Give the band theory Schrodinger equation
m = mass
ψ = wavelength
U(x) = potential energy
ε = energy
What is the periodicity of the potential energy term in the Schrodinger equation equal to?
The periodicity of the crystal lattice
What is Bloch’s theorem?
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).
State the equation for the Bloch function
ψₖ(r) = wavelength
uₖ(r) = uₖ(r + T) = potential
k = wave vector
r = position
T = lattice translation vector
What form are solutions to the band theory Schrodinger equation (electron wavefunction) found in?
The Bloch form
In the limit of constant potential energy, the Bloch solutions are ______ ______ with an energy-wavevector relationship given by the _____ _________ ______.
Plane waves
Free electron model
What are forbidden energies?
Energies with no wavelike solutions to the Schrodinger equation. This gives bands of allowed energies separated by bandgaps of forbidden energies.
Can there be several energies for the same wavevector?
Yes
The band structure is ________ (it is a ________ ______ representation).
Periodic
Periodic zone
What is the extended zone representation of an energy vs. wavevector graph (Bloch’s theorem)?
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.
State the alternative statement of Bloch’s theorem that proves it has the same periodicity as the lattice
ψ = wavelength
r = position
T = lattice translation vector
k = wavevector
What is the reduced zone representation of an energy vs. wavevector graph (Bloch’s theorem)?
A more compact energy scheme that translates all wavevectors, confining them to the first Brillouin zone.
How many allowed k-states are there in each zone of the reduced zone representation of Bloch’s theorem?
N allowed k-states for a crystal containing N atoms
How many electron states are there in each zone of the reduced zone representation of Bloch’s theorem?
2N for a crystal containing N atoms
What happens when there is an even number of electrons per atom (Bloch’s theorem)?
An energy band will be completely filled
Why can’t insulating crystals become conducting?
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.
What makes a crystal insulating?
It has a band gap between energy bands that cannot be crossed by exciting electrons.
Why does a small current flow in semiconductors at room temperature?
The energy gap is generally small (e.g. 1 eV) so a few electrons can be excited across the band gap at room temperature.
What can increase the conductivity of a semiconductor?
Increasing temperature so that more electrons can transition.
How are metals and semiconductors different?
Semiconductors increase in conductivity with increasing temperature whereas metals decrease in conductivity.
In quantum mechanics, the velocity of a particle is given by the ______ ________.
Group velocity
Give the equation for group velocity
v_g = group velocity
ω = angular frequency
k = wave number
ε = energy