Solid State Theory
Exercise 4
SS 11
Prof. M. Sigrist
Excitons
Exercise 4.1
OneDimensional Model of a Semiconductor
Let us consider electrons moving on a onedimensional chain. We use the socalled tight
binding approximation. Thus, we assume that each atom has a localized electron state
and that the electrons are able to hop between neighboring atoms. This hopping process
describes the kinetic energy term.
It is most convenient to use a secondquantized language.
For simplicity, we assume
spinless electrons. Let
c
i
and
c
†
i
be the creation and annihilation operators for an electron
at site
i
, respectively. The overlap integral between neighboring electron states is denoted
by

t
. Then, the kinetic energy operator is written as
H
0
=

t
X
i
c
†
i
c
i
+1
+
c
†
i
+1
c
i
.
(1)
We assume that the chain contains
N
atoms and in the following we set the lattice constant
a
= 1. Furthermore, we assume that two consecutive atoms are nonequivalent which is
modeled by an alternating potential of the form
V
=
v
X
i
(

1)
i
c
†
i
c
i
.
(2)
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 Spring '11
 Sigrist
 Electrons, Electric charge, Brillouin zone, ek, spinless electrons

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