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
How does the velocity of an electron change as it moves through the Brillouin zone?
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.
What does it mean when the velocity of an electron is 0 in band theory?
Motion is equally probable in the + and the - directions.
What condition do electrons satisfy at the edge of the Brillouin zone?
The Bragg condition for diffraction
k = ± nπ / a
What types of waves do electrons form at the edge of the Brillouin zone?
Standing waves (as they satisfy the Bragg condition for diffraction so are reflected back on themselves)
What happens to electrons at the edges of the Brillouin zone in band theory?
They are Bragg reflected if they meet the diffraction condition, causing standing waves
What are the standing waves at the Brillouin zone edges made of?
Equal amounts of waves travelling to the left and to the right.
State the two wavefunctions that make up standing waves at the zone edge
Give the equation for the probability density of a particle
ρ = probability density
ψ = wavefunction
ψ* = complex conjugate wavefunction
What is the probability density for a pure plane wave?
What are the probability densities for the ψ+ and ψ- states of a wavefunction?
Graphically describe the locations of the peaks of the ψ+ and ψ- states of a wavefunction within a crystal
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)
Give the equation for the lattice potential in the nearly free electron model
U(x) = potential
U0 = initial potential
x = position
a = unit cell length
Give the equation for the expectation value for the energy of an electron using the nearly free electron model
Schrodinger equation
State the value of the normalised wavefunctions ψ+ and ψ-
L = crystal length
k = wave number
x = position
Give the equation for the kinetic energy in the nearly free electron model
KE = kinetic energy
k = wave number
m = electron mass
The potential energy is equal to zero for all values of k in the nearly free electron model except if k is equal to ___.
π / a
Give the equation for the potential energy in the nearly free electron model when k = π / a
PE = potential energy
U0 = initial potential energy
Describe the nearly free electron model graphically and using the two equations for energy
U0 = bandgap
Gradient = 0 at the bandgap
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.
Unaffected
Increased
Lowered
What is the tight binding approximation?
A model of the crystal lattice that considers the electrons to be tightly bound to the atoms but with some overlap with neighbouring atoms.
What is the tight binding approximation similar to?
The linear combination of orbitals (LCAO)
Describe the shape of the tight binding approximation energy levels
Overlap between neighbouring atoms smears discrete energies into narrow bands separated by large bandgaps.
Describe how the 4 electron/crystal models differ in terms of energy levels
Give the tight binding approximation Schrodinger equation for an atomic orbital in the isolated atom
U = potential
ε = energy of isolated atom
φ = atomic orbital wavefunction
r = position
Give the equation for the approximate wavefunction for one electron in a whole crystal (tight binding approximation)
ψ = molecular orbital wavefunction
φ = atomic orbital wavefunction
N = number of atoms
T = Lattice translation vector
r = position
Give the tight binding approximation Schrodinger equation for an electron in a whole crystal
ψ = molecular orbital wavefunction
U = crystal potential
ε = energy of electron in the crystal
What components does the energy of an electron in a crystal consist of in the tight binding approximation?
The sum of a kinetic energy term and an atomic potential term
Give the equation for the energy of an electron in a crystal (tight binding approximation)
ε = energy of electron in the crystal
ε_atomic = energy of isolated atom
U = potential energy
ψ = molecular wavefunction
What happens to the energy of an electron when it is in a crystal?
It is impacted by a change in potential due to being in the crystal
What is the result of the tight binding approximation?
Term 1 = discrete energy
Term 2 = constant lowering discrete energy
Term 3 = overlap integral
Give the equation for α in the tight binding approximation
Give the equation for A(T) in the tight binding approximation
How do Fermi surfaces change as the Fermi energy increases?
k also increases with increasing Fermi energy
Give the equation for the effective mass of an electron
m* = effective mass
ε = energy of electron
k = wave number
What is a hole?
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.
