1. a) Using the 2x2 matrix we derived in class on Wednesday, find the shape of the Fermi boundary, that is, the constant energy contours for E= 0.1 eV, 0.2 eV...
(You can reference all energies to \(E_{2pz}\) and use \(\gamma = 2 eV\).)
b) what bandwidth do you get with \(\gamma = 2 eV\) ?
2. At what values of k do you find the centers of Dirac cones to be?
3. Using the 2x2 matrix, and a gamma of 2 eV, what is the speed associated with dispersion near a dirac point. Please go ahead and post your results, thoughts, questions and comments here. Try this as a group effort and work on it right here in the comments.
PS. I think you can do this in closed form (without numerical methods), once you understand the "landscape".
PPS. Like c is the speed of light and w=ck is a dispersion relationship. What is the analogous thing for graphene?
Thursday, February 23, 2017
Monday, February 20, 2017
sp2-graphene homework.
1. a) Construct the wave-function of an sp2 state.
b) Do a graph that shows the probability density for that state, e.g., a 2D contour plot or something.
c) Why is the coefficient of the 2s (200) state always \(\sqrt{1/3}\) for an sp2 state?
2. a) What is the Bravias lattic and basis for graphene? How many atoms are in a unit cell?
b) Sketch a picture of the graphene structure and find the coordinates of about 1/2 dozen atoms in that picture.
3. In a perfect hexagon, what are the lengths of all the "chords"? (In terms of the length of a side.) How many chords are there? (It is a small number.)
4. Construct a Bloch state for graphene using only the 2pz orbital. You don't need to solve for anything, but what is the form of such a Bloch state?
Please feel free to ask questions here.
b) Do a graph that shows the probability density for that state, e.g., a 2D contour plot or something.
c) Why is the coefficient of the 2s (200) state always \(\sqrt{1/3}\) for an sp2 state?
2. a) What is the Bravias lattic and basis for graphene? How many atoms are in a unit cell?
b) Sketch a picture of the graphene structure and find the coordinates of about 1/2 dozen atoms in that picture.
3. In a perfect hexagon, what are the lengths of all the "chords"? (In terms of the length of a side.) How many chords are there? (It is a small number.)
4. Construct a Bloch state for graphene using only the 2pz orbital. You don't need to solve for anything, but what is the form of such a Bloch state?
Please feel free to ask questions here.
Hubbard Homework.
Matrix for two sites and two electrons:$$
\begin{matrix}
U, &\gamma &-\gamma &0 &0 &0
\\ \gamma &0 &0 &\gamma &0 &0
\\ -\gamma &0 &0 &-\gamma &0 &0
\\0 &\gamma &-\gamma &U &0 &0
\\0 &0 &0 &0 &0 &0
\\0 &0 &0 &0 &0 &0
\\ \end{matrix} $$
1. a) Find the eigenvalues and eigenvectors of this matrix.
b) What are the 4 eigenvectors that do not have U in their eigenvalue. Which 3 eigenvectors are particularly related? What is the nature of their relationship? (Think about spin. Refer back to the definitions of the vectors of the initial basis states.)
U, &\gamma &-\gamma &0 &0 &0
\\ \gamma &0 &0 &\gamma &0 &0
\\ -\gamma &0 &0 &-\gamma &0 &0
\\0 &\gamma &-\gamma &U &0 &0
\\0 &0 &0 &0 &0 &0
\\0 &0 &0 &0 &0 &0
\\ \end{matrix} $$
1. a) Find the eigenvalues and eigenvectors of this matrix.
b) What are the 4 eigenvectors that do not have U in their eigenvalue. Which 3 eigenvectors are particularly related? What is the nature of their relationship? (Think about spin. Refer back to the definitions of the vectors of the initial basis states.)
Thursday, February 9, 2017
Hubbard Model
Matrix for two sites and one electron:
$$ \begin{matrix}
0 &-\gamma
\\ -\gamma &0
\\ \end{matrix} $$
Matrix for two sites and two electrons:
$$ \begin{matrix}
U, &\gamma &-\gamma &0 &0 &0
\\ \gamma &0 &0 &\gamma &0 &0
\\ -\gamma &0 &0 &-\gamma &0 &0
\\0 &\gamma &-\gamma &U &0 &0
\\0 &0 &0 &0 &0 &0
\\0 &0 &0 &0 &0 &0
\\ \end{matrix} $$
Things to think about:
What is the basis in each case?
What is the lowest eigenvalue in each case and what is its eigenstate?
What is the expectation value of its kinetic energy? (The lowest eigenstate)
What are the other eigenvectors and eigenvalues?
3 electrons and 2 sites: what would be an appropriate matrix (and basis) for that?
$$ \begin{matrix}
0 &-\gamma
\\ -\gamma &0
\\ \end{matrix} $$
Matrix for two sites and two electrons:
$$ \begin{matrix}
U, &\gamma &-\gamma &0 &0 &0
\\ \gamma &0 &0 &\gamma &0 &0
\\ -\gamma &0 &0 &-\gamma &0 &0
\\0 &\gamma &-\gamma &U &0 &0
\\0 &0 &0 &0 &0 &0
\\0 &0 &0 &0 &0 &0
\\ \end{matrix} $$
Things to think about:
What is the basis in each case?
What is the lowest eigenvalue in each case and what is its eigenstate?
What is the expectation value of its kinetic energy? (The lowest eigenstate)
What are the other eigenvectors and eigenvalues?
3 electrons and 2 sites: what would be an appropriate matrix (and basis) for that?
Friday, February 3, 2017
Homework 2. due Friday, Feb 10 (or Monday Feb 13 in my mailbox)
Edited Saturday, Feb 11.
Note added: I think maybe the expectation value of x, \(\langle x \rangle\) can be calculated as \(\langle n \rangle \times a\). Is that true? Similarly, the expectation value \(\langle x^2 \rangle\) is probably \(\langle n^2 \rangle \times a^2\). This could make the calculation of the "size" of the wave-function quicker. Please let me know if this seems correct or not. This is just off the top of my head. Also, please take the poll.
Note added: I think maybe the expectation value of x, \(\langle x \rangle\) can be calculated as \(\langle n \rangle \times a\). Is that true? Similarly, the expectation value \(\langle x^2 \rangle\) is probably \(\langle n^2 \rangle \times a^2\). This could make the calculation of the "size" of the wave-function quicker. Please let me know if this seems correct or not. This is just off the top of my head. Also, please take the poll.
Sunday, January 22, 2017
Homework 1 (due Feb 1)
1. At t=0, sketch the wave-function as a function of x over a few lattice spacings for Bloch states for which:
a) k=0; b) k=\(\pi/a\); c) k=\(\pi/2a\); d) k=\(- \pi/2a\)
e) sketch a graph of the atomic state wave-function you used in your Bloch state construction.
Note that, with time dependence included, a) and b) represent standing waves, while c) and d) are traveling waves, traveling to the right and left, respectively.
a) k=0; b) k=\(\pi/a\); c) k=\(\pi/2a\); d) k=\(- \pi/2a\)
e) sketch a graph of the atomic state wave-function you used in your Bloch state construction.
Note that, with time dependence included, a) and b) represent standing waves, while c) and d) are traveling waves, traveling to the right and left, respectively.
Tuesday, January 17, 2017
Physics 232 course outine
This is flexible and open to modification. Your thoughts and input are welcome.
- Semiconductor Physics: Band bending and the ability to manipulate the Fermi level (aka chemical potential) play an important role in semiconductor physics. We will look at characteristics of inhomogeneous semiconductor systems including the non-equilibrium dynamics of p-n junctions. We could also discuss FETs and 2DEGs if there is interest.
- Matrix approach to crystal quantum mechanics: -new ways to look at quantum state evolution, -role of disorder in crystals, disorder-driven metal-insulator transition (Anderson localization).
- Mott-Hubbard Model: -modeling e-e interaction, relevance to correlated phenomena, quasi-2D superconductors (cuprates), antiferromagnetism, quantum computing (2 dot qbit)
- Band theory: \(sp^2\) bonding and pz band in graphene and related boride structures?, Dirac point, bands in cuprates?, density functional theory
- Ferromagnetism: breakdown of band theory, role of e-e interaction in spin aligned states...
- ...
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