Homework 4
(due Monday, October, 5)
1. Graphene is a two dimensional lattice of carbon atoms in a hexagonal lattice.
(a) A hexagonal lattice is not a Bravais lattice. Show that a hexagonal lattice can be
described by a triangular (Bravais) lattice with a two atom basis. If the distance
between nearest neighbor atoms in the hexagonal lattice is a, what shortest distance between sites in the triangular lattice? What are two basis vectors of the
triangular lattice?
(b) Using the results from part (a), what is the reciprocal lattice of the triangular
lattice? Give two basis vectors for the reciprocal lattice. Sketch the first Brillouin
zone.
(c) As you know, Carbon has four electrons available for bonding. Three of them are
used for bonding in the hexagonal lattice. This leaves one electron per atom. In
a tight binding model assume that the only nonzero term is the one responsible
for hopping between nearest neighbors (n.n.):
H=γ
X
|iihj|.
(1)
i,jn.n.
Solve this in a two atom basis and show that for a given wave vector, k, it reduces
to diagonalizing the matrix
H=γ
~
0
~
~
1 + e−ik·~a1 + e−ik·~a2
~
1 + eik·~a1 + eik·~a2
0
!
,
(2)
where ~a1 and ~a2 are basis vectors of the triangular lattice.
(d) Diagonalize this 2 by 2 matrix to find the energy eigenvalues.
(e) Make a plot of the energy vs. k in two dimensions in matlab. I will place a sample
code on the web site. Do a Google search to see if your results agrees the standard
plots of E vs. k for graphene.