1. For review of Schrodinger equation: http://web. monroecc
Chapter 8 (also read over problem 8.1 in book) Free energy bands and Bloch’s Theorem
8.1 is another Kroniq Penney example. You do not need to solve it, but having some familiarity with it will be helpful when we go over a similar example in class.
Before we discussed 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 structure.

2. Learning Objectives for Today
After today’s class you should be able to:
Calculate free electron energy bands and the number of k states in the bands
Apply Bloch’s theorem to the Kronig-Penney model or any other periodic potential
Explain the meaning and origin of bands and “forbidden band gaps”
Another source on today’s topics, see Ch. 7 of Kittel or search Bloch theorem in others
Gaps may happen in the next lecture

3. Revisiting the Quantum Metal
Energy of “free” electrons
Will the band fill up?
The# of allowed k states (dots) is equal to the number of primitive cells in the crystal
Consider examples
If I told you I had a simple cubic crystal with one atom in the basis with N atoms in the crystal, how many k states would there be in the lowest energy band? N How about if it’s BCC? Still N (It won’t always be equal to the number of atoms, but that is the case for sc, bcc and fcc if they only have one atom in the basis.) What about the perovskite structure which is simple cubic with 5 atoms in the basis? If there are N atoms in the crystal, there are N/5 k states in the band.
Go back to sc case: Will we fill up this band if that atom in the basis is monovalent? No, because each k state can hold two different spins. If it’s divalent? Yes, in simple cases. (we don’t know enough to see where this can go wrong yet, such as band overlap due to multiple orbitals) What happens if it’s trivalent? More than one band has electrons in it. How would we show that?
Note we fill the lowest energies first (which in this band means the magnitude of k is close to zero is filled first).
We didn’t talk much about the number of k states before because we didn’t yet know what a primitive cell was.
(No time now, but later: Pull the hbar off of one of the later slides, currently slide 34)
–/a  /a
Inside U=0 L U
Fill lowest energies first
Cube V=L3

4. Looking at the energy distribution of multiple electrons
Why have I drawn this?
Looking at the energy distribution of multiple electrons
More generically: E
Does the generic form remind you of anything? It’s just a translation of the reciprocal lattice vector.
K= h b1+k b2+l b3 k
–/a  /a

5. Free Electron Brillouin Zone
Common Methods of Representing the Dispersion Relation E(k)
k is NOT the momentum of the particle. k acts on the wavefunction does not give an eigenvalue (like E). It’s often referred to as the crystal momentum. Like the classical momentum, it’s frequently conserved. What the difference? The rate of change of a free electron's momentum is proportional to the total force, but the rate of change of an electron's crystal momentum is given only by external forces (applied electric and magnetic field) and does not include the periodic field given by the crystal lattice
Repeated Zone Scheme contains a lot of redundant info
First Brillouin Zone

6. Why does this dispersion relation look different?
fcc in real space
BCC
FCC
# of nearest neighbors 8 12
Nearest-neighbor distance
½ a 3
a/2
# of second neighbors 6
Second neighbor distance a
Could you have predicted which direction would be lowest?
First of all, in 3D.
k values along different directions have different magnitudes.
Predict based on nearest neighbor direction.

7. Empty Lattice Bands for bcc Lattice
For the bcc lattice, let’s plot the empty lattice bands along the [100] direction in reciprocal space.
General reciprocal lattice translation vector:
Let’s use a simple cubic lattice, for which the reciprocal lattice is also simple cubic:
Could also use BCC primitive lattice, but would be a little more complex to understand.
And thus the general reciprocal lattice translation vector is:

8. Energy Bands in BCC
We write the reciprocal lattice vectors that lie in the 1st BZ as:
The maximum value(s) of x, y, and z depend on the reciprocal lattice type and the direction within the 1st BZ. For example:
[100]
0 < x < 1
[110]
0 < x < ½, 0 < y < ½ ky kx  H N
Remember that the reciprocal lattice for a bcc direct lattice is fcc!
Here is a top view, from the + kz direction:

9. Group: Plot the Empty Lattice Bands for bcc Lattice
Thus the empty lattice energy bands are given by:
Along [100], we can enumerate the lowest few bands for the y = z = 0 case, using only G vectors that have nonzero structure factors (h + k + l = even, otherwise S=0):
{G} = {000}
{G} = {110}
{G} = {200}

10. Empty Lattice Bands for bcc Lattice: Results
Thus the lowest energy empty lattice energy bands along the [100] direction for the bcc lattice are:

11. “Realistic” Potential 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

12. 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 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)

13. 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):

14. 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

15. 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+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:

16. 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 too (see slide at end of lecture), 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

17. 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)

18. Understanding the notation of the second proof of Bloch’s theorem
Another proof unnecessary, but the notation present will come up again
Known as the Born-von Karman boundary condition
You can always expand a wave packet in terms of plane waves
The BvK BC came from second to last slide.
You can expand any function obeying this condition as the set of plane waves that satisfy the B.C.s
Ψ 𝑟 =𝑞 𝑐 𝑞 𝑒 𝑖 𝑞𝑟

19. Adding plane waves together
What does the bottom figure remind you of? Scattering from an atom.
An example where the plane waves have the same wavelength but different directions
Illustration of wave vectors of plane waves which might be added together

20. Understanding the notation of the second proof of Bloch’s theorem
The potential U(r)=U(r+Na) obeys the Born-von Karman condition
U 𝑟 =𝐾 𝑈𝐾𝑒 𝑖 𝐾𝑟
Ψ 𝑟 =𝑞 𝑐𝑞 𝑒 𝑖 𝑞𝑟
UK are the Fourier coefficients of U
UK= U-K if the crystal has inverse symmetry
Thus you can write U in a similar was to writing psi
Just pulls out a negative q squared
Plugging this form of  into Schrodinger’s Equation gives a kinetic energy term of:

21. Understanding the notation of the second proof of Bloch’s theorem
Ψ 𝑥,𝑡 =𝑞 𝑐 𝑞 𝑒 𝑖 𝑞𝑟
Combining sums and then defining q=q’-K
The BvK BC came from second to last slide.
Just pulls out a negative q squared
Some more manipulation (subbing q=k-K) gives an alternate form of S.E.

22. It looks more complicated than it is.
Schrodinger’s equation in momentum space simplified by condition of periodicity
In the free electron case, all UK are 0, so this simplifies:
Note that if U is not zero, then essentially, you still get free electron like equations with a correction based on the Fourier coefficient as you approach the surface of the Brillouin zone (where there would be Bragg reflection at the surface).
We will look at more complicated examples in the next chapter.

23. Propagation on a crystal lattice
To understand some generic features of electron conduction on a crystal lattice, let’s model a 1D crystal, i.e. a lattice with a periodic potential.
The exact shape of the periodic potential will not matter. a
Ion core x
V(x)

24. But the exact shape doesn’t matter, so let’s try something easier!
Propagation on a crystal lattice
But the exact shape doesn’t matter, so let’s try something easier! .
V(x)
V(x) x d

25. Before we do a whole crystal, let’s remind ourselves how to deal with step𝐸
𝑈(𝑥) 𝑥
𝑈 0
How to determine transmission probability? I II
Solve time-independent Schrodinger equation to find 𝜓 𝑥 .
− ℏ 2 2𝑚 𝑑 2 𝜓 𝑥 𝑑 𝑥 2 +𝑈 𝑥 𝜓 𝑥 =𝐸𝜓 𝑥
Where Ψ 𝑥,𝑡 =𝜓 𝑥 𝑒 −𝑖 𝐸𝑡 ℏ
The next scattering slides are review, so we’ll step through it pretty quickly.
𝑑 2 𝜓 𝑥 𝑑 𝑥 2 =− 2𝑚 ℏ 2 𝐸−𝑈 𝑥 𝜓 𝑥
Region I:
Region II:
𝑑 2 𝜓 𝑥 𝑑 𝑥 2 =− 2𝑚𝐸 ℏ 2 𝜓 𝑥
𝑑 2 𝜓 𝑥 𝑑 𝑥 2 =− 2𝑚 ℏ 2 𝐸− 𝑈 0 𝜓 𝑥

26. Transmission Probability𝐸
𝑈(𝑥) 𝑥
𝑈 0
How to determine transmission probability?
Solve time-independent Schrodinger equation to find 𝜓 𝑥 . I II
Region I:
Region II:
𝑑 2 𝜓 𝑥 𝑑 𝑥 2 =− 2𝑚𝐸 ℏ 2 𝜓 𝑥
𝑑 2 𝜓 𝑥 𝑑 𝑥 2 =− 2𝑚 ℏ 2 𝐸− 𝑈 0 𝜓 𝑥
𝑘 1 2
𝑘 2 2
𝜓 1 𝑥 =𝐴 𝑒 𝑖 𝑘 1 𝑥 +𝐵 𝑒 −𝑖 𝑘 1 𝑥
𝜓 2 𝑥 =𝐶 𝑒 𝑖 𝑘 2 𝑥 +𝐷 𝑒 −𝑖 𝑘 2 𝑥
Ψ 1 𝑥,𝑡 =𝐴 𝑒 𝑖 𝑘 1 𝑥−𝜔𝑡 +𝐵 𝑒 −𝑖 𝑘 1 𝑥+𝜔𝑡
Ψ 2 𝑥,𝑡 =𝐶 𝑒 𝑖 𝑘 2 𝑥−𝜔𝑡 +𝐷 𝑒 −𝑖 𝑘 2 𝑥+𝜔𝑡

27. Region I I II The general solution is:
Ψ 1 𝑥,𝑡 =𝐴 𝑒 𝑖 𝑘 1 𝑥−𝜔𝑡 +𝐵 𝑒 −𝑖 𝑘 1 𝑥+𝜔𝑡
Concept Test:
What do the terms starting with 𝐴 and 𝐵 represent, physically?
𝐴 is kinetic energy, 𝐵 is the potential energy.
𝐴 is a wave traveling to the right, 𝐵 is a wave traveling to the left.
𝐴 is a wave traveling to the left, 𝐵 is a wave traveling to the right.
𝐴 is a standing wave peak, 𝐵 is a standing wave trough.
These terms have no physical meaning. 𝐸
𝑈(𝑥) 𝑥
𝑈 0 I II
I stole this scattering section and questions directly from Professor Flagg. I really like the questions and I think the reminder is helpful.

28. Coefficients I II Region I: Region II: 𝑑 2 𝜓 𝑥 𝑑 𝑥 2 =− 2𝑚𝐸 ℏ 2 𝜓 𝑥
𝑈(𝑥) 𝑥
𝑈 0 I II
Coefficients 𝐴 𝐶 𝐵
𝑒 − 𝐷
Region I:
Region II:
𝑑 2 𝜓 𝑥 𝑑 𝑥 2 =− 2𝑚𝐸 ℏ 2 𝜓 𝑥
𝑑 2 𝜓 𝑥 𝑑 𝑥 2 =− 2𝑚 ℏ 2 𝐸− 𝑈 0 𝜓 𝑥
𝑘 1 2
𝑘 2 2
Ψ 1 𝑥,𝑡 =𝐴 𝑒 𝑖 𝑘 1 𝑥−𝜔𝑡 +𝐵 𝑒 −𝑖 𝑘 1 𝑥+𝜔𝑡
Ψ 2 𝑥,𝑡 =𝐶 𝑒 𝑖 𝑘 2 𝑥−𝜔𝑡 +𝐷 𝑒 −𝑖 𝑘 2 𝑥+𝜔𝑡
Right-going
Left-going
Right-going
Left-going
What do each of these waves represent?
𝐴= reflected
𝐵= transmitted
𝐶= incoming from left
𝐷= incoming from right
𝐴= transmitted
𝐵= reflected
𝐴= incoming from left
𝐶= transmitted
𝐶= reflected A. B. C. D.

29. Coefficients I II Region I: Region II: 𝑑 2 𝜓 𝑥 𝑑 𝑥 2 =− 2𝑚𝐸 ℏ 2 𝜓 𝑥
𝑈(𝑥) 𝑥
𝑈 0 I II
Coefficients 𝐴 𝐶 𝐵 𝐷
Region I:
Region II:
𝑑 2 𝜓 𝑥 𝑑 𝑥 2 =− 2𝑚𝐸 ℏ 2 𝜓 𝑥
𝑑 2 𝜓 𝑥 𝑑 𝑥 2 =− 2𝑚 ℏ 2 𝐸− 𝑈 0 𝜓 𝑥
𝑘 1 2
𝑘 2 2
Ψ 1 𝑥,𝑡 =𝐴 𝑒 𝑖 𝑘 1 𝑥−𝜔𝑡 +𝐵 𝑒 −𝑖 𝑘 1 𝑥+𝜔𝑡
Ψ 2 𝑥,𝑡 =𝐶 𝑒 𝑖 𝑘 2 𝑥−𝜔𝑡 +𝐷 𝑒 −𝑖 𝑘 2 𝑥+𝜔𝑡
Incoming
from left
Reflected
Transmitted
Incoming
from right
Boundary and initial conditions determine values of coefficients.
Initial conditions: electron incoming from left
𝐷=0
Boundary conditions:
Continuity 𝜓 1 0 = 𝜓 2 0
𝐴+𝐵=𝐶
Smoothness 𝑑𝜓 1 𝑑𝑥 𝑥=0 = 𝑑𝜓 2 𝑑𝑥 𝑥=0
𝑖 𝑘 1 (𝐴−𝐵)=𝑖 𝑘 2 𝐶

30. Transmission I II Region I: Region II:𝐸
𝑈(𝑥) 𝑥
𝑈 0 I II
Transmission 𝐴 𝐶 𝐵
Region I:
Region II:
Ψ 1 𝑥,𝑡 =𝐴 𝑒 𝑖 𝑘 1 𝑥−𝜔𝑡 +𝐵 𝑒 −𝑖 𝑘 1 𝑥+𝜔𝑡
Ψ 2 𝑥,𝑡 =𝐶 𝑒 𝑖 𝑘 2 𝑥−𝜔𝑡
Reflected
Transmitted
Incoming
from left
Concept Test: If an electron comes in with an amplitude 𝐴, what is the probability that it is reflected or transmitted?
Reflected Transmitted
𝐵 1−𝐵
𝐵/𝐴 1−𝐵/𝐴
𝐵 2 1− 𝐵 2
𝐵 2 / 𝐴 2 1− 𝐵 2 / 𝐴 2
𝐵 𝐴 − 𝐵 𝐴 2
Or what percentage of the incident photons are reflected.
𝑃 reflect. = reflected prob. density incident prob. density
“Conservation of probability”: must either reflect or transmit
∴𝑅+𝑇=1