1. Project topics due today. Next HW due in one week
Project topics due today! Next HW due in one week. Test corrections due in 9 days.
Discuss Drude and Sommerfield and said we have to consider periodicity. So, we looked at several crystal types, but actually we can learn a lot from just knowing there is periodicity, regardless of its exact form.
Chapter 8 and 9 lectures run together a little Krönig-Penney Model and the Origin of Bands and Band Gaps
For review of Schrodinger equation:

2. Learning Objectives for Today
After today’s class you should be able to:
Explain the meaning and origin of energy bands and “forbidden band gaps”
Finally understand difference between metals, semiconductors and insulators!
(If time) Relate DOS to energy bands
Another source on today’s topics, see Ch. 7 of Kittel or search Kronig-Penney model
Gaps may happen in the next lecture

3. Using Bloch’s Theorem: The Krönig-Penney Model
Bloch’s theorem allows us to calculate the energy bands of electrons in a crystal if we know the potential energy function.
Each atom is represented by a finite square well of width a and depth V0. The atomic spacing is a+b. V x a
a+b
2a+b
2(a+b) V0 -b
Last time we would have used sine and cosines in region 0 to a. What is an alternative approach? Exponents will be easier for derivatives as we already have exponents using our Bloch approach.
In a past class, I tried switching one of these letters to Q, but students complained about it being too confusing, so I’m sticking with the book.
We can solve the SE in each region of space:
0 < x < a
-b < x < 0

4. Boundary Conditions and Bloch’s Theorem
The solutions of the SE require that the wavefunction and its derivative be continuous across the potential boundaries. Thus, at the two boundaries (which are infinitely repeated):
Boundary Conditions and Bloch’s Theorem
(2)
x = 0
(1)
x = a
Now using Bloch’s theorem for a periodic potential with period a+b:
Don’t get enough info on K and kappa if don’t use bloch’s theorem.
Doesn’t matter if use Bloch’s theorem on si1 or si2 for equation 3
On equation 4, we use bloch’s theorem on both si’s
Have A, B, C and D in each equation, so use matrix form
Have them figure out first two lines of matrix.
k = Bloch wavevector
Now we can write the boundary conditions at x = a:
The four simultaneous equations (1-4) can be written compactly in matrix form  Let’s start it!
(3)
(4)

5. Results of the Krönig-Penney Model
Since the values of a and b are inputs to the model, and  depends on V0 and the energy E, we can solve this system of equations to find the energy E at any specified value of the Bloch wavevector k. What is the easiest way to do this?
Taking the determinant, setting it equal to zero and lots of algebra gives:
By reducing the barrier width b (small b), this can be simplified to:

6. Problems occur at Ka=N or K=N/a
Graphical Approach
small b
Right hand side cannot exceed 1, so values exceeding will mean that there is no wavelike solutions of the Schrodinger eq. (forbidden band gap)
Problems occur at Ka=N or K=N/a Ka
Plotting left side of equation

7. Turning the last graph on it’s side
This equation determines the energy bands.
For values of K where the left side of the equation has a magnitude < 1, then k is real and energy bands are allowed.
BAND 2
Forbidden band gap
Energy in terms of E0
BAND 1
ka/

8. Greek Theater Analogy: Energy Gaps
There are certain radial distances away from the stage where there are no seats

9. Energy Levels of Single vs Multiple Atoms
Single Atom
Multiple Atoms
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.

10. Ideal Double Quantum Wells
How do we start?
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 different (fourier theorem of adding sine waves, quick changes require higher frequency terms)

11. The two solutions have different energies
The two solutions have different energies
Symmetric (Bonding) and Antisymmetric (Antibonding)
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.

12. Energy vs. Barrier Width
What happens as make b go to 0?
Energy vs. Barrier Width
Spins not coupled
What would 3 wells look like?

13. Which has the lowest energy? Any relation between nodes and energy?
Triple Quantum Wells
Which has the lowest energy?
Any relation between nodes and energy?

14. Quadruple Quantum Wells

15. Five Quantum Wells

16. Figure 1.7: Coupled Well Energies
What happens to these levels as the atoms get closer (b smaller)?
How would the energy levels look for multiple wells?

17. Band Overlap
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
Also partly allows hybridization, like in carbon
Reminder: Why do bands get wider? Bring atoms closer together.
For reminder of hybridization from energy level perspective:

18. Energy Band Overlap 14Si: 3s23p2
Mixing of bands known as hybridization (Si=sp3)
Conduction Band
The bottom unfilled energy band
Valence Band
Typically the last filled energy band
4 states to bond with
14Si: 3s23p2
Out of 8 possible n=3 electrons (2s and 6p)

19. This is at T=0. What happens at higher T?
Energy Band Formation
Diagram (flat or with momentum k) showing energy levels is a band diagram.
This is at T=0. What happens at higher T?
Conduction Band
bottom unfilled
Valence Band
last filled
Semiconductor
Small gap (<~1eV)
Insulator
Big gap (>~1eV)
Metal
No gap

20. Semiconductor Flat Band Diagram (Quantum Well)
1.43 eV
InAs bandgap is .36 at room temp, 1.43 is room temp for GaAs In
What do I mean by flat?
Before any movement of charge, could cause bands to bend
At a single point in the crystal (changes with momentum)

21. What happens as you approach the gap?
Classically E = ½ m v2
What happens to v as k gets close to Brillouin zone edge?
BAND 2
Forbidden band gap
Energy in terms of E0
It’s stalling at the BZ edges. Why might it do that?
BAND 1
Find v for the free electron energy.
ka/

22. Compare to the free-electron model
Free electron dispersion
–/a /a
Let’s slowly turn on the periodic potential

23. Electron Wavefunctions in a Periodic Potential (Another way to understand the energy gap)
Consider the following cases:
Wavefunctions are plane waves and energy bands are parabolic:
–/a /a
Electrons wavelengths much larger than atomic spacing a, so wavefunctions and energy bands are nearly the same as above V x a
a+b
2a+b
2(a+b) V1 -b

24. Wavelength much greater than atomic spacing
Energy of wave
What happens as I lower this energy?
Similar to how radio waves pass through us without affecting

25. Electron Wavefunctions in a Periodic Potential U=barrier potential
Consider the following cases:
Wavefunctions are plane waves and energy bands are parabolic:
–/a /a
Electrons wavelengths much larger than a, so wavefunctions and energy bands are nearly the same as above
Electrons wavelengths approach a, so waves begin to be strongly back-scattered by the potential:
Electrons waves are strongly back-scattered (Bragg scattering) so standing waves are formed:

26. The nearly-free-electron model (Standing Waves)
Due to the ±, there are two such standing waves possible:
These two approximate solutions to the S. E. at have very different potential energies has its peaks at x = a, 2a, 3a, …at the positions of the atoms, where V is at its minimum (low energy wavefunction). The other solution, has its peaks at x = a/2, 3a/2, 5a/2,… at positions in between atoms, where V is at its maximum (high energy wavefunction).
Either:
Nodes at ions
Or:
Nodes midway between ions a

27. Symmetric and Antisymmetric Solutions
The nearly-free-electron model
Strictly speaking we should have looked at the probabilities
before coming to this conclusion:
Symmetric and Antisymmetric Solutions
Different energies for electron standing waves a

28. Summary: The nearly-free-electron model
BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE
The periodic potential V(x) splits the free-electron E(k) into “energy bands” separated by gaps at each BZ boundary.
-2π/a –π/a π/a 2π/a
In between the two energies there are no allowed energies; i.e., wavelike solutions of the Schrodinger equation do not exist.
Forbidden energy bands form called band gaps. E- Eg E+

29. BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE
Approximating the Band Gap
BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE
-2π/a –π/a π/a 2π/a E- Eg E+
For square potential: V(x) =Vo for specific values of x (changes integration limits)