EE506 Semiconductor Physics
Homework Assignment #1
Prof. Steve Cronin
Due Sept. 13, 2011
Electronic Bandstructure:
1.)
Empty Lattice Approximation: Consider a twodimensional rectangular lattice:
a.
The real space lattice vectors are given by
R
i,j
=
(
ia, jb
). Write an
expression for the reciprocal lattice vectors,
G
n,m
.
b.
Draw the Brillouin zone for the two dimensional square lattice, and label
the high symmetry points,
Γ
= (0, 0),
X
= (
π
/a,
0) and
L
=(
/a
,
/b
).
Indicate the area enclosed by the irreducible Brillouin zone.
c.
Write a general expression for the
E
n,m
(k)
relations for the empty lattice
(
i.e
.,
V(r)
= 0) for a two dimensional rectangular lattice in terms of the
reciprocal lattice vectors. (
Hint: use parabolic dispersion relation for a
free electron.
)
d.
Make a table of the momenta and energies at the high symmetry points:
(
n,m
) (
G
k
+
)
Γ

E
Γ
 (
G
k
+
)
X

E
X
 (
G
k
+
)
L

E
L

where (
n,m
)=(0,0), (1,0), (0,1), (1,0)…etc.
e.
Find
E(k)
explicitly along
Γ
 X
and
X  L
for the lowest 3 energy levels,
indicating the degeneracies of each level. Plot
E(k)
for these levels
(preferably using MATLAB).
f.
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.
 '11
 CRONIN
 Crystallography, Photon, Reciprocal lattice, Brillouin zone, Brillouin

Click to edit the document details