−1
−0.5
0
0.5
1
1.5
−0.3 −0.2 −0.1 0 0.1 0.2 0.3
Energy (eV)
k (1/Å)
Figure 8: Example dispersion relation
(Eq. (29)) for trans-polyacetylene, plotted
within the first BZ. In this example a =
10
˚
A,
p
= 0 eV and γ(|τ |) = 0.5 eV.
The solid lines show the band structure in
a calculation with γ(a) = 0 eV, and the
dashed lines show the 2 bands calculated
with γ(a) = 0.1 eV. In both cases the band
width is 4γ(|τ |) = 2 eV.
3.8 Contributions from more than one orbital
In general, bands will contain contributions from more than one type of
orbital. In graphene, for example, the lowest energy valence bands (which
form the bonds between atoms) and the highest energy conduction bands,
are constructed from a mixture of s, p
x
and p
y
orbitals (this is known as sp
2
hybridisation).
However, it is very easy to generalise the formalism in Eq.s (19) and (20) to
deal with multiple types of orbital. All we need to do is use the index i to
label both different basis sites and different orbital types.
For example, imagine a N
b
= 2 basis crystal in which s, p
x
, p
y
and p
z
orbitals
all contribute to the bands of interest. We would then expect 2×4 = 8 bands
(from N
b
= 2 atoms per unit cell, with 4 orbitals per atom) and we would
have to solve an 8×8 matrix eigenvalue equation to find the energies at each
k. In this case, if i = 1 labels an s-orbital on basis site A, and j = 5 labels
a p
x
orbital on basis site B the relevant hamiltonian matrix element would
be,
H
ij
=
1
N
X
R
A
X
R
B
e
ik·(R
A
−R
B
)
Z
φ
∗
s
(r − R
A
)Hφ
p
x
(r − R
B
)dr. (30)
The rest of the hamiltonian matrix elements could be calculated in a similar
way and then, as usual, the dispersion relation would be found by solving,
|H − E(k)I| = 0.
As an example of this type of calculation, Fig. 9 shows the dispersion relation
for graphene calculated within an orthogonal tight binding scheme using 4
types of orbital: s, p
x
, p
y
and p
z
. As graphene is a 2D hexagonal crystal
with 2 atoms in the basis we get 8 bands in total.
14