A Two-Dimensional Periodic Example Graphite

A layer of graphite can be considered as infinite and periodic in two dimensions. Graphite has a planar hexagonal structure. It belongs to a layer group that is derivable from the P6/mmm space group, containing 24 symmetry operations. 

Also in this case, n electron bands can be studied with Eliickel s approximation by representing all matrices in a basis set of two Bloch functions from 

At each k point, H, S, C, and E in Eq. [25] are 2x2 matrices in the basis of and 4 b. Again, H and S must be computed to find the unknown matrices C and E. For symmetry reasons, only the asymmetric part of the Brillouin zone needs to be explored instead of the entire zone, and because the full representation of a 7i-type band structure for graphite would be three-dimensional, we can start by exploring a representative monodimensional path (as is usually done for three-dimensional structures). The most obvious choice is the triangle perimeter, and in particular, the special positions E, M, and K are expected to be topologically interesting for their symmetry properties. 

During the calculation of S and H, it must be taken into account that, because of the symmetry equivalence of the carbon atoms, SAA(k) = SBB(k) and ElAA(k) = HBB(k), whereas ElAB(k) = Hgy (k) for hermiticity. Moreover, Hiickel s rules imply that S(k) is the identity matrix of order 2, completely independent of k. In fact, diagonal elements can be computed in exactly the same way as for the linear chain, even though the geometry is different in this case, and the off-diagonal elements are zero because of Hiickel s orthogonality assumption. Consequently, Eq. [25] becomes 

We start by computing matrix H at E. This is a peculiar point because each of the two Bloch functions in Eq. [46] reduces to a simple sum of all AOs of that type (A or B) in the lattice (all factor phases are 1 when k = 0). 

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