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.
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 '11
 CRONIN
 Crystallography, Photon, Reciprocal lattice, Brillouin zone, Brillouin

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