HW 8: 2D crystal carbon.

Hmm. I think it probably makes sense not to try to turn this in a get it graded. Just hold on to it after you have finished.

Two-dimensional crystalline carbon, know as graphene, has an interesting structure as well as very interesting and unusual electron properties. The structure is established by \(sp^2\) hybridized 2s and 2p wave-functions; the unusual electron properties are associated with a band formed from the left-over \( p_z\) wave-function.

Warm-up problems:
1. How many valence electrons does carbon have? How many (per atom) are locked up in bonding, and how many (per atom) are available to go into the non-local states of the \(p_z\) band?

2. a) Sketch the graphene structure.
b) How many nearest neighbors does each atom have?
c) If the C-C distance for nearest neighbors is a, then what is the distance to next-nearest neighbors? What is the distance to next-next-nearest neighbors?

3. a) What two vectors can be used to generate the Bravais lattice structure of graphene?
b) What is the difference between the Bravais lattice and the crystal lattice in this case?
c) what is a unit cell in this case?
============== end of warm-up====

4. For our usual 1D crystal (not graphene), the BZ can be chosen to cover any interval of width \(2 \pi/a \). The usual choice is from -pi/a to +pi/a.  The choice 0 to 2pi/a, for example, is equally valid.
a) graph E vs k for each of these interval choices.
b) Why is the range of \(k_x\) limited to one of these ranges? Why not less? Why not more?

5. For graphene the BZ extends from \( k_x = - 4 \pi /(\sqrt{27} a)\) to \( k_x = 4 \pi /(\sqrt{27} a)\) and from \( k_y = - 2 \pi /(3a)\) to \( k_y = 2 \pi /(3a)\). And it is a hexagon with a corner on the \(k_x\) axis. Draw this BZ.

6. For graphene, using the result of the nearest-neighbor band calculation we discussed in class (see video), plot E vs k along the \(k_x\) axis:
a)  from \( k_x = -4 \pi /(\sqrt{27} a)\) to \( k_x = +4 \pi /(\sqrt{27} a)\).
b)  from \( k_x = 0 \) to \( k_x = +8 \pi /(\sqrt{27} a)\).

7. Our nearest-neighbor band calculation yields two bands that arise from the \(2p_z\) "orbital".
      For one of the bands:
\( E_{2p_z,1}(k_x,k_y) = E_{2p_z} - \lvert t + 2t e^{-i (3/2) a k_y} cos ((\sqrt{3}/2)a k_x) \rvert\),
where the constant term is the atomic energy of the 2p state. The first term inside the absolute value symbols (t) comes from an atom in the same unit cell as the "starting atom" (a distance a above it). (It has no phase factor because it is chosen to be in the same unit cell.)  The other term, proportional to 2t, comes from the nearest-neighbors below-and-to-the-right and below-and-to-the-left which combine to create the 2 cos term.
      For the other \(2p_z\) -derived band:
\( E_{2p_z,2}(k_x,k_y) = E_{2p_z} + \lvert t + 2t e^{-i (3/2) a k_y} cos ((\sqrt{3}/2)a k_x) \rvert\),
a) Why are there two bands? Why not just one band?
b) If t = 1 eV, then what is the band width?
c) What is the Fermi energy?

8. interesting extra credit:  Given t=1 eV, as before, now suppose you find a way, via doping or a gate voltage of some sort, to increase the Fermi energy by .01 eV (10 meV) above its normal value. What is the size and shape of the Fermi surface and what is the density of electrons in the conduction band? For example, is the Fermi surface a circle or an ellipse?
Hint: If we assume a constant D(k) in 2D k-space, which is true, then the electron density will be proportional to the area inside the k-space Fermi surface. Also, I think the total area of the BZ times the k-space density of states should be related to the density of carbon atoms. (So hopefully you do NOT need D(E), only the much easier to work with D(k) (which is a constant and not dependent on k).
Note also that if one were to shift the choice of BZ boundary by \(\pi/3a\) in the \(k_y\) direction, then one would have two "Dirac cones" inside the BZ, instead of 6 on the corners (each shared 3 ways). Two inside might be nice for visualization, no?
b) What are some other cool things we can ask/explore here?
c) PS. If anyone can get D(E), that might be interesting too.

9. People talk about making p-n junctions with graphene, e.g., for solar energy. Based on these band theory calculations, what might be challenging about that (or at least different from Silicon)?

10. (see video, extra credit)
a) In the calculation of the 2pz-derived bands one gets a 2x2 matrix. (see video) Why is that? Why not just one band (from one atomic state)?
b) If one were to endeavor to calculate the Bloch (band) states associated with the sp2 bonding of graphene, how large a matrix would you expect? Explain. (Assume starting with ordinary orbitals and letting the hybridization emerge naturally in your calculation.)
c) In that calculation, where would you find the information expressing the nature of the hybridization?  (in the ________________. What is the missing word?)