Solid State Theory
Exercise 1
FS 11
Prof. M. Sigrist
KronigPenney model
We study a simple model for a onedimensional crystal lattice, which was introduced by
Kronig and Penney in 1931. The atomic potentials are taken to be rectangular, where the
minima correspond to the atomic cores. The model is simpliﬁed even more by replacing
the rectangular potentials by Dirac delta functions,
V
(
x
) =
V
0
∞
X
n
=
∞
δ
(
x

an
)
.
(1)
This is the socalled KronigPenney potential which is shown in Fig. 1A.
0 a 2 a
A
0 a 2 a
B
Figure 1:
A
KronigPenney potential
V
(
x
).
B
Interface between a constant potential
U
(
x
) and a KronigPenney potential.
Exercise 1.1
Energy bands
Using Bloch’s Ansatz for the wave function in a periodic potential
Ψ(
x
+
a
) = Ψ(
x
)
e
ika
,
(2)
show that the energy in the KronigPenney potential for a given
k
obeys the equation
cos
λ
=
v
2
β
sin
β
+ cos
β,
(3)
where
λ
=
ka
,
β
=
a
p
2
mE/
~
2
and
v
= 2
mV
0
a/
~
2
. In general, this equation can only be
solved graphically or numerically. Show that the resulting band structure has band gaps
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 Spring '11
 Sigrist
 Exponential Function, wave function, Condensed matter physics, Band gap, Prof. M. Sigrist

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