Homework # 7  Solution
Problem # 1
Consider the free electron energy bands of an fcc crystal lattice in the reduced zone
scheme in which all
k
's
are transformed to lie in the first Brillouin zone. Plot roughly in the [111]
direction the energies of all bands up to six times the lowest band energy at the zone boundary at
k
=
(2
π
/a)(½,½,½). Explain what happens with these bands in the presence of a weak crystal potential.
The reciprocal lattice is a bcc lattice with primitive translation vectors:
1
2
ˆˆˆ
()
a
π
=−
+
+
bx
y
z
,
2
2
a
+
y
,
z
3
2
a
=
+−
y
z
All the reciprocal lattice vectors are given by
11
2 2
33
2
(
nn n
h
k
l
a
)
=++ =
+
+
Gbbb
x
y
z
The energy within the first Brillouin zone:
2
22
2
xx
yy
zz
Ek
G
k
G
k
G
m
⎡⎤
=+
+
+
+
+
⎣⎦
=
2
z
Along [111] direction:
and therefore
xy
kk k k
===
2
2
G
k
G
k
G
m
2
z
⎡
⎤
=
+
++
⎣
⎦
=
;
k
varies from
a
−
to
a
Low energy bands are given in the table and plotted in the figure below.
bands
n
1
n
2
n
3
h k l
E
/
2
2
m
=
E
/
2
2
m
=
at
k
=0
E
/
2
2
m
=
at
k
=
a
1
000
000
2
3
k
0
2
3(
)
a
2,3,4
100, 010, 001
111,
111
2(
)
(
)
kk
aa
−
2
12(
)
a
2
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 Spring '08
 WORMER
 Energy, Work, Reciprocal lattice, Bragg plane, weak crystal potential

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