Solid State Theory
Exercise 3
FS 2011
Prof. M. Sigrist
Exercise 3.1
6Orbital tightbinding model
(
j

1)
a
ja
(
j
+ 1)
a
Figure 1: Onedimensional chain of atom cores.
In this exercise we want to calculate the band structure of (fictitious) onedimensional
sodium in the tightbinding approximation.
The (singleparticle) Hamiltonian of the
system is given by
H
=
p
2
2
m
+
X
j

Ze
2

r

r
j

,
(1)
with
Z
= 11 for Na and
r
j
= (
x, y, ja
) and the lattice constant
a
.
a) As a starting point for the tightbinding approximation, a formulation in terms of
Wannier functions is more practicable. We define the Wannier function
w
n
(
x, y
;
z

ja
) of atom
j
in band
n
by
Ψ
n,k
(
x, y
;
z
) =
1
√
N
X
j
e
ikja
w
n
(
x, y
;
z

ja
)
.
(2)
Note that the potential is periodic in the
z
direction only.
The different bands
originate from the atomic orbitals. Since Na has 11 electrons we need to consider
the 6 orbitals
n
= (1
s,
2
s,
2
p
x
,
2
p
y
,
2
p
z
,
3
s
).
Show that within the tightbinding
approximation taking only nearestneighbor hopping into account, the hamiltonian
can be written as
H
=
X
n
H
n
+
X
n
6
=
n
0
H
n,n
0
,
(3)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Sigrist
 Condensed matter physics, Fext, tightbinding approximation, 6Orbital tightbinding model

Click to edit the document details