Electronic Structure of Materials Flashcards
Magnitude of electron spin
h(bar)/2
General Hamiltonian expression.
H = T + V
What the Hamiltonian for an electron in a hydrogen atom
Check
Write the momentum operator
Check
Use the momentum operator to give an expression for the Hamiltonian operator.
Check
State the time dependent Schrödinger equation.
Check
State the time independent Schrödinger equation,
Check
Show that the position and momentum operators do not commute.
Show
Show that the components of the angular momentum operator do not commute.
Check
Show that the angular momentum squared commutes with an angular momentum operator in a single dimension.
Check
State the eigenvalue equations with the spherical harmonics for the angular momentum operators.
Check
What values can the quantum numbers l and m take?
0<=l
How do you get from the Hamiltonian for hydrogen to the Hamiltonian operator for hydrogen?
Insert the p operator.
When solving the Hamiltonian operator for the hydrogen atom, do the angular parts of the wavefunction depend on the potential?
No.
The solutions of the TISE for any spherically symmetric potential have the same angular dependence.
Expression for energy of each level in a hydrogen atom.
Check
Define a Ry in terms of constants
Check
Define Bohr radius in terms of constants
Check
l=0,1,2,3 can be denoted by which letters?
s,p,d,f
Give the Hamiltonian for the electrons in a He atom.
Check
What is the mean field approach to the problem of many electrons in a system?
All the atoms are said to have the same potential which is an average of all the interactions in the system.
What is a central field approxiamtion?
The use of a mean field however it is spherically symmetric.
What does the Aufbau principle state?
That electrons will occupy the lowest energy states available.
What are the empirical rules that the Aufbau principle follows?
Fill the states with the lowest value of n + l first.
If there are multiple states with equal n + l then fill the states with the lowest n first.
State the Hamiltonian operator for a single electron in a hydrogen molecule.
Check
Derive the bonding and antibonding solutions for a single electron in a hydrogen molecule.
Check derivation
Sketch the probability densities for bonding and antibonding in a hydrogen molecule.
Sketch and check
How is the bond integral usually expressed?
In a modulus sign.
What do HOMO and LUMO stand for?
Highest occupied molecular orbital
Lowest unoccupied molecular orbital
What is the HOMO-LUMO gap?
Minimum energy required to promote an electron to a higher energy level.
What is a basis function for molecular orbitals?
It is a building block function that we can make molecular orbitals from.
The number of molecular orbitals calculated equals the number of basis functions (due to bonding and antibonding).
When extending the LCAO for a general molecule, what does the Hamiltonian become?
A matrix.
How can we use density of states and energy levels to show degeneracy?
Plot DoS against energy and the relative heights of the peaks show the relative degeneracies.
What is the area under a single peak on a plot of DoS against energy equal to?
1
What is a perturbation?
A stimulus provided by experiment to the system.
State the Kronecker delta function.
= 1 where I=j
=0 otherwise
State the Dirac delta function
delta(x) = 0 for all x except when x=0 int(delta(x)) = 1 between -inf and +inf
State the Lorentzian function that approximates the Dirac delta function,
Check
Write the TISE in Dirac notation
Check
Write the Hamiltonian equation for a perturbed electron
Check
What is the big idea in perturbation theory?
That the full Hamiltonian operator can be split into a simpler part that can be solved exactly and a correctional term (V).
The nearly free electron model uses which type of perturbation theory?
Time independent perturbation theory.
State Fermi’s golden rule.
Check
What are the three parts of Fermi’s golden rule expressing?
A constant
Transition matrix element (V is the perturbation)
Conservation of energy in a Dirac delta.
Fermi’s golden rule allows transitions between states |phi(i)> and |phi(f)> if what criteria are met?
Transition is not forbidden by the matrix element
Energy is conserved.
What is the selection rule part of Fermi’s golden rule?
Any transition where l does not change is forbidden (eg 1s to 2s)
Give the Hamiltonian operator for the FEG in 3D
Check
Give the energy eigenvalue for the FEG model as a function of k.
Check
What two boundary conditions can be used for the FEG?
Hard wall
Periodic
What are the hard wall boundary coniditions?
psi(x=0)=psi(x=L)=0
What is the solution to the Schrödinger eqn with hard wall boundary conditions in 3D?
Check
What are the periodic boundary conditions for the FEG?
psi(x)=psi(x+L)
Solve the Schrödinger eqn in 3D for the FEG and periodic BCs.
Check
What are the differences in solutions of the FEG for the different boundary conditions?
With hard wall nx, ny and nz are positive integers but for periodic, they are just integers.
In the periodic case, psik is an energy eigenstate of the Schrödinger equation but also of the momentum operator unlike the hard wall case.
What is the advantage of the periodic BC case for the FEG?
Translational symmetry is preserved.
What is a sphere in k-space?
A constant energy surface.
Derive the Fermi wavevector.
Check
Derive the Fermi energy.
Check
What is the density of states?
Number of states with energy between E and E+dE = DoS x dE
Calculate the DoS for the 3D FEG.
Check
Derive the bulk modulus from density of states in terms of the Fermi energy.
Check
For what atoms does the FEG work well for and not?
It works well for simple atoms with no d electrons
Reasonably well for metals with filled d shells
Does not work well for partially filled d shells
What does the FEG predict about electrical conductivity?
It predicts that when subject to an electric field every material will conduct.
Derive the 2D and 1D density of states for the FEG.
Check
On a 1D chain of atom, what does the nearly free electron model cause (due to the added potential term)?
A gap in the dispersion relation at ±π/a
Perturbation theory has what effect on the dispersion relation?
It shows that two electrons cannot have the same magnitude of wavevector or be 2π/a apart. Thus gaps appear in the dispersion relation at ±π/a
Write the degenerate perturbation theory Hamiltonian matrix.
Check
Show how degenerate perturbation theory gives a bandgap width of V0.
Check
If the dispersion states are filled to E- what kind of material is it?
An insulator.
What letter represents any real space vector in crystallography?
T
What letter represents any reciprocal space vector in crystallography?
G
What is the difference between a primitive unit cell and a conventional unit cell?
A primitive unit cell is the smallest possible cell
A conventional unit cell may contain more than one lattice point but illustrates the symmetry of the whole lattice.
In electronic structures, which type of unit cell is preferred?
Primitive
What must you remember to do when calculating the reciprocal lattice vector in electronic structures?
To divide by the scalar 2π.
How can we calculate the volume of the BZ using reciprocal lattice vectors?
The scalar triple product of the reciprocal lattice vectors.
Define the 1st Brillion Zone crystallographically.
The smallest possible unit cell of the reciprocal lattice centred on a reciprocal lattice point.
What is Wigner-Seitz construction?
A method of construction the 1st BZ by bisecting lines between a reciprocal lattice points and its neighbours.
Derive the dispersion relation for the tight binding model.
Check
Give a general expression for dispersion using the 3D tight binding model
Check
What structure do semiconductors generally have?
Tetrahedral (diamond or zincblende)
What is the hybridized state in the tetrahedral structure?
sp3
Sketch a plot of electron energy against lattice spacing for sp3 hybrization.
Check
What are direct and indirect semiconductors?
If the top of the valence band has the same Bloch wavevector to the bottom of the conduction band it is a direct semiconductor.
If different then it is an indirect.
Sketch the absorption of energy for direct and indirect absorption in a semiconductor.
Check
Why do we only notice indirect absorption when direct transitions are forbidden by conservation of energy?
Fermi’s golden rule.
Equation for the group velocity of a free electron
Check
Describe Bloch oscillations.
For a electron in a crystal, when a the k0 value has changed so much that it is at the edge of the BZ (-π/a), the wavepacket in k-space jumps to the other edge of the BZ as all wavevectors can be represented in the 1st BZ.
Derive an expression for effective mass.
Check
A particle with a negative effective mass is what?
A hole.
Give an expression for the effective mass of an electric in the conduction band and its dispersion.
Check
Give an expression for the effective mass of an electron in the valence band and its dispersion.
Check
