1. By the end of this section you should be able to:
Today’s Objectives
Previously: Simple models of metals. For improvements, we need the band structure.
By the end of this section you should be able to:
Use/define Bloch’s Theorem
Conceptually explain why energy bands and band gaps develop (two different approaches)
Coming up soon: Calculating energy bands
If time, we’ll use what we’ve learn to apply this to predict the energy band diagram for a chain of F atoms

2. Where do energy bands and gaps come from?
GaAs
Real materials have gaps in the available levels
Metals have available states just above the Fermi level
Analogy: Not every distance away from the center of the stage has a seat. Not every potential energy has a seat.
We’ll talk directly about semiconductors and insulators in a few classes. Today our goal is to understand where energy bands and gaps come from.
Technically gaps come from bringing atoms closer together. Yet, the square well model would not make it clear why this happens because we ignore the potential of the atoms kz ky kx
Fermi surface kF
Texts for Today’s Discussion:
Kittel: pages
Ashcroft & Mermin: pages and
Snoke: pages 1-14

3. Simplest Periodic Potential
Quantum Well for Each Atom Rather than a Giant Quantum Well for the Metal
Single Atom
Quantum/Atomic Physics
Simplest Periodic Potential
So, instead of having single atomic energy levels, we get bands were many energy levels exist.
Let’s see how these bands develop as we go from one atom to multiple atoms.
While square wells are a simplification, it turns out we still learn a lot from the exercise. The general trend is still the same in the more complicated potentials.
Multiple Atoms (Condensed Matter Physics)

4. Approach 1: Get atoms close (conceptual) Ideal Double Quantum Wells
How would it change if non-ideal well?
How do we start?
Which more likely to find electron in middle?
Symmetric has higher chance of finding electron between the wells (states also known as bonding and antibonding)
Reference about double well:
If we didn’t want an ideal well, we could use periodic boundary conditions.
Didn’t need the derivative for the single well, redundant information
What’s the difference between these two wavefunctions? Energy. Talk about probably of being found in between and that’s relation to bonding.
Can you see mathematically, why energy would be a little different (fourier theorem of adding sine waves, quick changes require higher frequency terms)

5. A Small Barrier Adjusts the Energies and Wavefunctions
Non-ideal double well
Non-ideal single well

6. Increasing the Barrier Moves the Symmetric and Antisymmetric Closer
What would 3 wells look like?
Larger barrier
Smaller barrier
What happens as make b go to 0?
It depends on the shape of the orbital as to which is lower energy. An s orbital will prefer symmetric alignment. You can imagine this by overlapping to circles. They share the electrons to some degree. It will be easier if the spin doesn’t have to change.

7. Which has the lowest energy? Any relation between nodes and energy?
Triple Quantum Wells
Which has the lowest energy?
@ Nodes, ampitude=0
Any relation between nodes and energy?

8. Quadruple Quantum Square Wells

9. Five Quantum Square Wells

10. What happens to these levels as the atoms get closer (b smaller)?
How Energy Bands Form
What happens to these levels as the atoms get closer (b smaller)?
How would the energy levels look for multiple wells?

11. Band Overlap
Reminder: Why do bands get wider? Bring atoms closer together.
For reminder of hybridization from energy level perspective:
Often the higher energy bands become so wide that they overlap with the lower bands
Many materials are conductors (metals) due to the “band overlap” phenomenon

12. Approach 2: Include U(x) in Sch. Eq
Approach 2: Include U(x) in Sch. Eq. “Realistic” Atomic Potentials in Solids
Multi-electron atomic potentials are complex
Even for hydrogen atom with a “simple” Coulomb potential solutions are quite complex L U
U=0

13. Bringing Atoms Close Together in a Periodic Fashion
For one dimensional case where atoms (ions) are separated by distance a, we can write the condition of periodicity as
Even in a more complicated multi-atomic chain, we can use the same formula
Note: Band gaps occur even without periodicity, but periodic examples are simpler!
Even if there are more than one atom type in the solid (different looking Coulomb potentials) those same Coulomb potentials will repeat themselves in the same pattern over and over again.
Bloch electrons are electrons that obey the 1D Schrodinger equation with a periodic potential.
They reduce to free electrons if you take U(x) = 0 = U(x+an)

14. (3D proof on page 134 of Ashcroft)
Bloch’s Theorem
This theorem gives the electron wavefunction in the presence of a periodic potential energy.
We will prove 1-D version, AKA Floquet’s theorem.
(3D proof on page 134 of Ashcroft)
When using either theorem, we use the time-indep. Schrodinger equation for an electron in a periodic potential
I like to discuss section 1.3 before 1.2. I think it makes more sense that way.
1D version is easier to follow and the concepts are all the same either way
Use time independent SE when making independent electron approximation
where the potential energy is invariant under a lattice translation of a
In 3D (vector):

15. Bloch Wavefunctions a
Bloch’s Theorem states that for a particle moving in the periodic potential, the wavefunctions ψnk(x) are of the form
unk(x) has the periodicity of the atomic potential
The exact form of u(x) depends on the potential associated with atoms (ions) that form the solid
Write Y on board

16. Main points in the proof of Bloch’s Theorem in 1-D
1. First notice that Bloch’s theorem implies (3D version):
Or just:
Can show that this formally implies Bloch’s theorem, so if we can prove it we will have proven Bloch’s theorem.
2. To prove the statement shown above in 1-D:
Consider N identical lattice points around a circular ring, each separated by a distance a. Our task is to prove:
Plug in r+R for r in the top equation. Then use the second equation to set u’s equal. 1 2 N 3
Built into the ring model is the periodic boundary condition:

17. Proof of Bloch’s Theorem in 1-D: Conclusion
The symmetry of the ring implies that we can find a solution to the wave equation (QM reason too):
If we apply this translation N times we will return to the initial atom position:
This requires
And has the most general solution: 1 2 N 3
Or:
There is a quantum mechanical reason is due to commuting operators.
For example, remember when k=0, they were all the same phase.
TR is a translation operator that shifts the position by vector R
Since the Hamiltonian/energy is periodic, it doesn’t matter if we translate the wavefunction before or after the Hamiltonian. T and H commute
Where we define the Bloch wavevector:
Now that we know C we can rewrite

18. What periodic potential do we want to try?
To better understand how band gaps form, let’s model a 1D crystal, i.e. a lattice with a periodic potential.
The exact shape of the periodic potential will not matter, and these potentials could be complicated. a
Ion core x
U(x)

19. One Common Approach
But the exact shape doesn’t matter, so let’s try something easier! What’s Easy?
A Physicist Thinks Quantum Wells are Easy (Kroniq-Penney Model or Nearly Free Electron Approx.)
U(x) x

20. Consequence of Bloch’s Theorem Probability * of finding the electron
Each electron in a crystalline solid “belongs” to each and every atom forming the solid
Very accurate for metals where electrons are free to move around the crystal!
Show math on board (might want to add to PPT in future)
Again we used the independent electron approximation which is great for metals (not as much for insulators)

21. Last Time: Understanding Conduction Using a Quantum Square Well
Sommerfield model
works pretty well for metals L U
End:45 (g)
Compare to the Drude model that treated electrons similar to a gas
Potential handout: the short chapter 3 in Ashcroft
Could move Bloch to just before Kroniq Penney, but very mathy for late in the class, so maybe keep where it is 
Important difference: distribution of energy levels
Important similarity: empty/free L
metal