Figure 36: Dispersion relationship for a bcc crystal.Figure 37: Surfaces of constant energy for a bcc crystal calculated with the Tight Binding
approximation.
7.2.4 Graphene
This part ist about graphene, which is one of the forms of carbon and has2 atoms per unit cell.
Other forms of carbon are diamond (a pretty hard material) and graphite (a very soft material). In
graphite there is a hexagonal lattice of carbon, and then the layers are just stepped. So graphene is
a single layer of graphite and it turns out that this causes quite different electronic properties. If we
look at the crystal structure in the left of fig. 38, we see that one of the carbon atoms has its bond to
the left and the other one to the right. So there are two inequivalent positions in the unit cell. The
reciprocal lattice is also a hexagonal lattice (can be seen in the right of fig. 38). Trying to calculate
the tight binding model, we see that there is a correlation in between the tight binding model and
phonon calculations. In phonon calculations we had two atoms per unit cell and got an acoustical
branch and an optical branch. Now we get two bands in the dispersion relationship, a lower energy
branch and a higher energy branch.
The wavefunction is calculated by taking the two wavefunctions|ψa >and|ψb>of the atomsa
andbof the unit cell and coupling them together like we were making a covalent bond. To calculate