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.