1. a) Using the matrix we derived in class today, find the energy of the 2pz orbital-related states in graphene as a function of \(k_y\), for \(k_x=0\). That is, calculate and graph \(E_\vec{k}\) along the \(k_y\) axis from (0,0) to the edge of the Brillouin zone. (You can assume that the hoping integral is about -2 eV.
extra credit. Why can we assume that the hopping integral, \(\gamma\), is negative in this case?
b) Where (that is, at what value of \(k_y\), is the edge of the Brillouin zone along ky? (post here in a comment or email me when you get to this.) (no derivation or shown work needed. Just the answer. (Which will be proportional to like \(\pi/b\), right?)
c) extra credit. (This is an interesting question, not just a throwaway.) What sort of information do you get if you keep going along the \(k_y\) axis past the edge of the 1st BZ? What do you learn from that?? hint: Draw the pattern of BZ's in k space. (thinking outside the box...)
2. extra credit. Suppose you would wish to calculate the dispersion relation for graphene from \(\Gamma\) to K to M and back to \(\Gamma\). Which parts of that can you do using the above matrix? Which part can you not do without considering a non-zero \(k_x\)? hint: use symmetry.
3. extra credit. What are the eigenvectors for the two branches of \(E_\vec{k}\) along the \(k_y\) axis?
4. deep extra credit. What are the eigenvectors for the two branches of \(E_\vec{k}\) along the \(k_x\) axis?
-see below for Matrix and Video
Some of my notes related to graphene dispersion:
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