This homework assignment is optional. Its focus is on the nature of the magnetic susceptibility close to a phase transition from a paramagnetic state to a state with spontaneous magnetization.
1. Using the model of free energy of a metal which exhibits spontaneous magnetization* below a characteristic temperature Tc (as we discussed in class today) calculate the magnetic susceptibility in the normal metal state, that is, above Tc. Show that it becomes quite large near Tc and graph it as a function of T for T greater than Tc. What is the nature of its divergence?
(Note: Taking advantage of the restriction to linear response simplifies your calculation significantly.)
*A value of U-Bandwidth of about 0.1 eV should be fine for this problem. What is Tc for this case?
notes: Magnetic susceptibility is defined in terms of linear response. You can add a term to your free energy associated with a small applied magnetic field, something like \(\mu_B B\), where this B is applied magnetic field (not bandwidth). The position of the minimum of F in the presence of non-zero applied magnetic field, B, will tell you the induced magnetization and the susceptibility.
Friday, March 10, 2017
Sunday, March 5, 2017
Take home final: exploring eigenvectors and wave functions in graphene.
This is a closed book exam. You can do whatever preparation you like beforehand including discussing it with other people. Then please do this based on your own understanding and recollection without books, notes or computers. The one exception is obtaining eigenvectors and eigenvalues; please use whatever math resource you like for that; no need to show work going from matrixes to evs. Feel free to email me or comment here with any questions, corrections or anything else. Also, please feel free to email me if you get stuck.
preamble: Bloch theory addresses the nature of quantum wave-functions and energies for electrons in spatially periodic environments. The Bloch method provides a constructive approach for creating itinerant states from local atomic orbitals. While this approach fails dramatically in many important and interesting situations, it nevertheless provides a starting point for much of solid state physics.
In the following problems you are asked to explore and examine the nature of the quantum states of graphene, a hexagonal 2D arrangement of carbon atoms. In this structure \(2p_x\), \(2p_y\) and 2s orbitals hybridize into sp2 orbitals which are associated with the 3-fold, 120 degree covalent bonding of this hexagonal structure. \(2p_z\) orbital electrons, on the other hand, are free to move about the crystal. We will look at the nature of the bands associated with that \(2p_z\) orbital.
synopsis: Basically this problem is about exploring eigenvalues, eigenvectors and wave-functions of \(2p_z\) electron states in graphene. The specific problems are to provide some guidance for that endeavor.
1. a) Sketch a picture showing the positions of a dozen or so C atoms in graphene. Please use the orientation with a nearest-neighbor pair aligned along the x axis. Use "a" for your nearest neighbor distance in this and subsequent problems.
b) Show your Bravais lattice generating vectors, What are their x,y coordinates.
Note: It is important that you use the orientation indicated above and define "a" as the nearest-neighbor distance, otherwise the k vectors specified below won't make sense.
2. Consider a putative Bloch state made from the 2pz atomic orbital with undetermined coefficients \(c_A\) and \(c_B\).
a) Find the 2x2 matrix from which these coefficients and the energies of quantum eigenstates can be obtained. (This matrix can be expressed in terms of gamma (overlap integral) which one can presume to be real and negative. For the diagonal elements you may use either \(E_{2p_z}\) or zero. Note that the sign of gamma is plays a critical role in eigenvector order.)
b) Evaluate the matrix at the following points in k space:
1) at \(k_x=0\) , \(k_y=0\)
2) at \(k_x= 2 \pi/3a\) , \(k_y=0\)
3) at \(k_x= 4 \pi/3a\) , \(k_y=0\)
4) at \(k_x= 0\) , \(k_y= 4\pi/\sqrt{27}a\).
5) at \(k_x= 0\) , \(k_y= (4+\sqrt{2})\pi/\sqrt{27}a\).
6) at \(k_x= 0\) , \(k_y= (4+2\sqrt{2})\pi/\sqrt{27}a\).
c) Which of these 6 points are equivalent to others in this list? Explain and illustrate these equivalences. There are some intriguing and interesting differences between exploring the matrix along kx and ky. Feel free to discuss any that appeal to you.
d) (optional exploratory question) Consider that amazing simplicity of the matrix at \(k_x= 0\) , \(k_y= 4\pi/\sqrt{27}a\)? How does that come about? What does it mean?
3. Graph your eigenvalues:
a) as a function of \(k_x\) for \(k_y=0\). [Try going all the way to \(k_x= 4 \pi/3a\); discuss what your graph shows.]
b) as a function of \(k_y\) for \(k_x=0\). [Perhaps go all the way to \(k_y= (4+2\sqrt{2})\pi/\sqrt{27}a\). What is the significance of that stopping point? Feel free to illustrate and discuss for extra credit.]
4. At the point in k-space \(k_x=0\) , \(k_y=0\):
a) What is the eigenstate corresponding to the lower energy eigenstate? (Recall that \(\gamma\) is negative, i.e., around -2 or -3 eV, and that the negativity of gamma effects eigenvector order.) Use this eigenstate to illustrate the spatial nature of the Bloch state corresponding to this eigenstate and k value.
b) What is the eigenstate corresponding to the higher energy eigenstate? Use this eigenstate to illustrate the spatial nature of the Bloch state corresponding to this eigenstate and k value.
5. Do the same for the point in k-space \(k_x=2\pi/3a\) , \(k_y=0\).
extra credit:
6. Consider the point in k-space \(k_x=\pi/3a\) , \(k_y=0\).
a) What is the matrix there? Is this point more difficult than any of those in problem 2? Why?
b) What are the eigenstates and eigenvalues at this point? Do all the same things as in problem 4 for this point.
extra-extra credit:
7. Consider the point in k-space at \(k_x= 0\) , \(k_y= 4\pi/\sqrt{27}a\). Examine the eigenvectors and corresponding Bloch state wave-functions at that point. Illustrate the Bloch wavefunctions; use your illustrations to explain why they all have the same energy.
preamble: Bloch theory addresses the nature of quantum wave-functions and energies for electrons in spatially periodic environments. The Bloch method provides a constructive approach for creating itinerant states from local atomic orbitals. While this approach fails dramatically in many important and interesting situations, it nevertheless provides a starting point for much of solid state physics.
In the following problems you are asked to explore and examine the nature of the quantum states of graphene, a hexagonal 2D arrangement of carbon atoms. In this structure \(2p_x\), \(2p_y\) and 2s orbitals hybridize into sp2 orbitals which are associated with the 3-fold, 120 degree covalent bonding of this hexagonal structure. \(2p_z\) orbital electrons, on the other hand, are free to move about the crystal. We will look at the nature of the bands associated with that \(2p_z\) orbital.
synopsis: Basically this problem is about exploring eigenvalues, eigenvectors and wave-functions of \(2p_z\) electron states in graphene. The specific problems are to provide some guidance for that endeavor.
1. a) Sketch a picture showing the positions of a dozen or so C atoms in graphene. Please use the orientation with a nearest-neighbor pair aligned along the x axis. Use "a" for your nearest neighbor distance in this and subsequent problems.
b) Show your Bravais lattice generating vectors, What are their x,y coordinates.
Note: It is important that you use the orientation indicated above and define "a" as the nearest-neighbor distance, otherwise the k vectors specified below won't make sense.
2. Consider a putative Bloch state made from the 2pz atomic orbital with undetermined coefficients \(c_A\) and \(c_B\).
a) Find the 2x2 matrix from which these coefficients and the energies of quantum eigenstates can be obtained. (This matrix can be expressed in terms of gamma (overlap integral) which one can presume to be real and negative. For the diagonal elements you may use either \(E_{2p_z}\) or zero. Note that the sign of gamma is plays a critical role in eigenvector order.)
b) Evaluate the matrix at the following points in k space:
1) at \(k_x=0\) , \(k_y=0\)
2) at \(k_x= 2 \pi/3a\) , \(k_y=0\)
3) at \(k_x= 4 \pi/3a\) , \(k_y=0\)
4) at \(k_x= 0\) , \(k_y= 4\pi/\sqrt{27}a\).
5) at \(k_x= 0\) , \(k_y= (4+\sqrt{2})\pi/\sqrt{27}a\).
6) at \(k_x= 0\) , \(k_y= (4+2\sqrt{2})\pi/\sqrt{27}a\).
c) Which of these 6 points are equivalent to others in this list? Explain and illustrate these equivalences. There are some intriguing and interesting differences between exploring the matrix along kx and ky. Feel free to discuss any that appeal to you.
d) (optional exploratory question) Consider that amazing simplicity of the matrix at \(k_x= 0\) , \(k_y= 4\pi/\sqrt{27}a\)? How does that come about? What does it mean?
3. Graph your eigenvalues:
a) as a function of \(k_x\) for \(k_y=0\). [Try going all the way to \(k_x= 4 \pi/3a\); discuss what your graph shows.]
b) as a function of \(k_y\) for \(k_x=0\). [Perhaps go all the way to \(k_y= (4+2\sqrt{2})\pi/\sqrt{27}a\). What is the significance of that stopping point? Feel free to illustrate and discuss for extra credit.]
4. At the point in k-space \(k_x=0\) , \(k_y=0\):
a) What is the eigenstate corresponding to the lower energy eigenstate? (Recall that \(\gamma\) is negative, i.e., around -2 or -3 eV, and that the negativity of gamma effects eigenvector order.) Use this eigenstate to illustrate the spatial nature of the Bloch state corresponding to this eigenstate and k value.
b) What is the eigenstate corresponding to the higher energy eigenstate? Use this eigenstate to illustrate the spatial nature of the Bloch state corresponding to this eigenstate and k value.
5. Do the same for the point in k-space \(k_x=2\pi/3a\) , \(k_y=0\).
extra credit:
6. Consider the point in k-space \(k_x=\pi/3a\) , \(k_y=0\).
a) What is the matrix there? Is this point more difficult than any of those in problem 2? Why?
b) What are the eigenstates and eigenvalues at this point? Do all the same things as in problem 4 for this point.
extra-extra credit:
7. Consider the point in k-space at \(k_x= 0\) , \(k_y= 4\pi/\sqrt{27}a\). Examine the eigenvectors and corresponding Bloch state wave-functions at that point. Illustrate the Bloch wavefunctions; use your illustrations to explain why they all have the same energy.
Thursday, February 23, 2017
Graphene HW, part 2 & online problem.
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?
(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?
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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