Dirac Fermions in Graphene and Graphite—a view from angle ...

Figure 1.3. The band structure of graphene.
wave function for electrons can be approximated by a linear combination of the atomic wave functions. Thus
the tight binding model is also known as the linear combination of atomic orbitals (LCAO) approximation.
In the atomic description, the Hamiltonian for each carbon atom localized at r 1 and r 2 is
where V a is the atomic potential on each carbon atom.
H 1,2 = ∇2
2m + V a(r − r 1,2 ) (1.3)
The atomic wave functions centered on each carbon site φ(r − r 1,2 ) satisfy
H 1,2 φ(r − r 1,2 ) = E 1,2 φ(r − r 1,2 ) (1.4)
In graphene, the total Hamiltonian is
H = ∇2
2m + Σ R(V (r − r 1 − R) + V (r − r 2 − R)) (1.5)
The total Hamiltonian can be rewritten as the Hamiltonian for atom 1 or 2 with some corrections ∆H 1
and ∆H 2 which take care of the potential created by all other atoms. H = H 1 + ∆H 1 = H 2 + ∆H 2 .
The Bloch wave function is
Ψ(r) = Σ R e ik·R φ(r − R) (1.6)
where the wave function φ is a linear combination of the localized wave functions φ ( r − r 1,2 )
Here c 1 and c 2 are constants and c 2 1 + c 2 2 = 1.
φ(r) = c 1 φ(r − r 1 ) + c 2 φ(r − r 2 ) (1.7)
The eigenenergies can be obtained by solving the Shrödinger equation
HΨ(R) = E(k)Ψ(R) (1.8)
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