The tight-binding Hamiltonian for graphene can be written as:

,

.

Here,

and

denote the annihilation operators for the sites

and

, respectively, in the basis of the hexagonal lattice of graphene.

is known as the hopping term, representing the hopping of electrons from atoms at site

to atoms at site

:

.

Note that

is nonzero only for nearest neighbor atoms, and 0 otherwise, while

denotes the position vectors of the three nearest atoms in the lattice.

After diagonalizing

,

,

.

The expression for

is plotted against

and

to show the band structure of graphene. The lower band, colored blue, is the valence band, filled with electrons, while the upper conduction band is devoid of electrons. As a consequence of this type of band structure, graphene acts as a semi-metal, with its Fermi energy at

, where the conduction and valence bands meet. There are six such points in a hexagonal pattern in momentum space, called Dirac points. Near these Dirac points, the dispersion is linear and the bands are cone shaped, which is why they are known as Dirac cones.

Until now, hopping of electrons to atoms present in the same basis site (that is from site

to

or from

to

) was restricted, but if this restriction is done away with, the following changes to the previous equations are obtained:

.

Then after diagonalization of

,

,

.

This opens up gaps at the Dirac points, hence breaking the degeneracy. The band gap is given by

.

At Dirac points,

, so the band gap at the Dirac points is

.

This Demonstration shows how the band structure of graphene changes with

,

and

.

Snapshot 1: band structure with

and only

–

hopping allowed; in this case, the conduction and valence bands touch at the Dirac points

Snapshots 2, 3: the gap is opened up by increasing

and

[1] C. Kittel,

*Introduction to Solid State Physics*, 7th ed., New York: Wiley, 1996.