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

Projecting this equation into φ(r − r 1 ) and φ(r − r 2 )
Plugging in Eqn. (1.6) to Eqn. (1.10), we obtain
< φ(r − r 1 )|H|Ψ(r) >= E(k) < φ(r − r 1 )|Ψ(r) > (1.9)
< φ(r − r 2 )|H|Ψ(r) >= E(k) < φ(r − r 2 )|Ψ(r) > (1.10)
φ(r − r 1 )|H|Ψ(r) >
=< φ(r − r 1 )|H 1 + ∆H 1 |Ψ(r) >
= E 1 < φ(r − r 1 )|Ψ(r) > + < φ(r − r 1 )|∆H 1 |Ψ(r) > (1.11)
Applying the orthogonality of the Bloch functions < φ(r − R)|φ(r − R ′ ) >= δ R,R ′,
< φ(r − r 1 )|Ψ(r) >
= Σ R e ik·R < φ(r − r 1 )|c 1 φ(r − r 1 − R) + c 2 φ(r − r 2 − R) >
= c 1 (1.12)
Considering only the onsite energy and the overlap between nearest neighbors (e.g. atom A and the
three B atoms surrounding it in Fig. 1.2, R = 0, a 1 , a 2 ), the second term in Eqn. 1.11 can be simplified as
< φ(r − r 1 )|∆H 1 |Ψ(r) >
where β, γ and f(k) are defined as follows:
= c 1 < φ(r − r 1 )|∆H 1 |φ(r − r 1 ) >
+c 2 (1 + e ik·a1 + e ik·a2 ) < φ(r − r 1 )|∆H 1 |φ(r − r 2 ) >
= c 1 β + c 2 γf(k) (1.13)
β ≡< φ(r − r 1 )|∆H 1 |φ(r − r 1 ) >
γ ≡< φ(r − r 1 )|∆H 1 |φ(r − r 2 ) >
f(k) ≡ 1 + e ik·a1 + e ik·a2 (1.14)
Here β is a correction to the onsite energy. γ is the next nearest neighbor hoping integral. Eqn. 1.11
can be simplified as
Similarly Eqn. 1.10 can be simplified as
Solving the secular equation
( ) ( )
β γf(k) c1
γf ∗ (k) β c 2
and E(k) is
E(k) = β ± γ|f(k)| = β ± γ
c 1 β + c 2 γf(k) = c 1 E(k) (1.15)
c 2 β + c 1 γf ∗ (k) = c 2 E(k) (1.16)
√
( )
c1
= E(k)
c 2
(1.17)
1 + 4cos( k √
xa
2 )cos( 3ky a
) + 4cos
2
2 ( k xa
2 ) (1.18)
Here a = √ 3a 0 . β is a small correction to the overall energy. Fig. 1.4(a) shows the π band dispersion of
graphene. The valence and conduction bands touch only at the six corners of the BZ. Since each carbon atom
contributes one electron, the conduction band is completely filled up to the Fermi level, where the valence
and conduction bands merge. Because of this, graphene is known as a semi-metal or zero-gap semiconductor.
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