1. For review/introduction of Schrodinger equation: http://web. monroecc
Section Homework from this section: 1.5 (We will do a similar problem in class today) Bloch’s Theorem and Krönig-Penney Model

2. Learning Objectives for Today
After today’s class you should be able to:
Apply Bloch’s theorem to the Kronig-Penney model or any other periodic potential
Explain the meaning and origin of “forbidden band gaps”
Begin to understand the Brillouin zone
For another source on today’s topics, see Ch. 7 of Kittel’s Intro to Solid State Physics.
Crystal basics to prepare us for next class

3. “Realistic” Potential in Solids
Multi-electron atomic potentials are complex
Even for hydrogen atom with a “simple” Coulomb potential solutions are quite complex
So we use a model infinite one-dimensional periodic potential to get insight into the problem (last time, looked at 1-6 atoms)

4. Periodic Potential
For one dimensional case where atoms (ions) are separated by distance a, we can write condition of periodicity as a

5. Section 1.3: 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 in the book)
When using this theorem, we still 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):

6. Bloch Wavefunctions a
Bloch’s Theorem states that for a particle moving in the periodic potential, the wavefunctions ψ(x) are of the form
uk(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

7. Main points in the proof of Bloch’s Theorem in 1-D
1. First notice that Bloch’s theorem implies:
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+T for r 1 2 N 3
Built into the ring model is the periodic boundary condition:

8. Proof of Bloch’s Theorem in 1-D: Conclusion
The symmetry of the ring (and last lecture) implies that we can find a solution to the wave equation:
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:
For example, remember when k=0, they were all the same phase.
Where we define the Bloch wavevector:
Now that we know C we can rewrite

9. 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!
Makes sense to talk about a specific x (± n a)
Show this on board (might want to add to PPT in future) si times si(star)
Again we used the independent electron approximation which is great for metals (not as much for insulators)

10. 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.
First done for a chain of finite square well potentials model by Krönig and Penney in 1931 with E<V0 V x a
a+b
2a+b
2(a+b) V0 -b
Each atom is represented by a finite square well of width a and depth V0. The atomic spacing is a+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
I wish the book had selected different letters than K and , but staying consistent
-b < x < 0

11. 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
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 
(3)
(4, deriv.)

12. 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:

13. Gap occurs 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)
Gap occurs at Ka=N or K=N/a Ka
Plotting left side of equation

14. Not really much different
Single Atom
Multiple Atoms
Not really much different from what we discussed last time with multiple atoms. Only difference now is that we have an infinite line (or array) of atoms.

15. Greek Theater Analogy: Energy Gaps

16. What Else Can We Learn From This Model?

17. Exercise 1.4

18. Different Ways to Plot It Extended Zone Scheme
Note that the larger the energy, the larger the band/gap is (until some limit).
y=cos ka
Different x axis
k = Bloch wavevector Ka
The range -<ka< is called the first Brillouin zone.
Until some limit because Vo is not infinite.
Atoms

19. Different Representations of E(k)
Reduced zone scheme
All states with |k| > /a are translated back into 1st BZ
Frequently only one side is shown as they are degenerate.
In 3D, often show one side along with dispersion along two other directions (e.g. 100, 110, 111)

20. Band diagrams can refer to either E vs
Band diagrams can refer to either E vs. real space or E vs momentum space k
Momentum space example
Real space examples
If someone asks you to draw a band diagram, they probably mean real space, but it’s fine to ask.

21. Compare to the free-electron model
Free electron dispersion
...with first Brillouin zone:
–/a ® /a
(a the lattice constant)
–/a /a
Let’s draw it in 3D!
Let’s slowly turn on the periodic potential

22. 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

23. How do X-rays Work?
The soft tissue in your body is composed of smaller atoms, and so does not absorb X-ray photons particularly well. The calcium atoms that make up your bones are much larger, so they are better at absorbing X-ray photons.

24. Consequence of Bloch’s Theorem
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)

30. Crystal Directions Figure shows [111] direction
Choose one lattice point on the line as an origin (point O). Choice of origin is completely arbitrary, since every lattice point is identical.
Then choose the lattice vector joining O to any point on the line, say point T. This vector can be written as;
R = n1 a + n2 b + n3 c
To distinguish a lattice direction from a lattice point, the triplet is enclosed in square brackets [ ...]. Example: [n1n2n3]
[n1n2n3] is the smallest integer of the same relative ratios. Example: [222] would not be used instead of [111].
Negative directions can be written as
Figure shows
[111] direction
Also sometimes [-1-1-1]