(or, I guess Tuesday morning would be okay)
1. Considering graphene, suppose you are given the (x,y) positions of two carbon atoms. One is at (0,0) and the other is at (b,0). (Aside: b = .142 nm I think.)
a) do a drawing that shows these two atoms and all the nearest neighbors of these two atoms. (how many are there. email that to me as soon as you work on this!)
b) calculate all atom to atom distances in your drawing. (how many are there. email that to me as soon as you work on this!)
b') extra credit. What are the Bravais lattice generating vectors for the graphene lattice structure?
c) Find the location of the "first Brillouin boundary" along the ky axis. (There will be a spot along the +ky axis and another point, mirroring that, along the -ky axis.) Where exactly is that relative to like \(\pi/b\)? (email that to me ) (Feel free to use online resources for this. It is not that easy. Just so long as you get the answer as it applies to our lattice orientation and lattice parameter, b.)
d) extra credit. Find the location of the "first Brillouin boundary" along the kx axis. (There will be a spot along the +kx axis and another point, mirroring that, along the -kx axis.) Where exactly is that relative to like \(\pi/b\)?
b') extra credit. What are the Bravais lattice generating vectors for the graphene lattice structure?
c) Find the location of the "first Brillouin boundary" along the ky axis. (There will be a spot along the +ky axis and another point, mirroring that, along the -ky axis.) Where exactly is that relative to like \(\pi/b\)? (email that to me ) (Feel free to use online resources for this. It is not that easy. Just so long as you get the answer as it applies to our lattice orientation and lattice parameter, b.)
d) extra credit. Find the location of the "first Brillouin boundary" along the kx axis. (There will be a spot along the +kx axis and another point, mirroring that, along the -kx axis.) Where exactly is that relative to like \(\pi/b\)?
e) extra credit. What is the shape of the first Brillouin zone (BZ)? (Note. Make sure you find the BZ for our orientation of the spatial lattice, that is, with an atom-pair bond parallel to the spatial x axis, not the y axis.)
2. Major extra credit project problem!
Calculate the hopping integral, gamma, associated with an electron in the 2pz state on one lattice site hopping to the 2pz orbital on an adjacent site. For example, I think the integrand could be:
\(\psi_{2pz}(\:\vec{r}) \: \frac{4e^2}{4 \pi \epsilon_{o} \epsilon r} \: \psi_{2pz}(\:\vec{r}-b \hat{x})\). Does that integrand look correct? Do the units work out? Over what region would you integrate? (You can email me to discuss if your would like to work on this.) (We can use this result in our band structure calculation for the 2pz band of graphene.)
Well, you use a two atom basis.
ReplyDeleteyes. 2 of them are.
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