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Homework # 8  Solution
Problem #1
A twodimensional metal has one atom of valence one in a simple rectangular
primitive cell of
a
1
= 2Å and
a
2
= 4Å.
(a)
Draw the first and the second Brillouin zones.
(b)
Calculate the radius of the free electron Fermi sphere and draw this sphere to scale on the
drawing of the Brillouin zones.
(c)
Draw the Fermi surface in reduced zone scheme and show schematically the effect of a
weak crystal potential.
(a)
2
π
/
a
1
2BZ
1BZ
2
π
/
a
2
(b)
In two dimensions the “volume” of the Fermi “sphere” is
2
F
k
π
. The “volume” of kspace per
kpoint is
12
22
aa
×
. Therefore, taking into account spin degeneracy we have
2
2
FF
ka
a
k
N
ππ
==
⎛⎞
⎜⎟
⎝⎠
,
which results in
1/2
1
1
2
Å0
.
8
9
Å
2
F
N
k
≈
.
This value is bigger than the shortest distance from the origin to the boundary of the first
Brillouin zone (0.79Å), so that the Fermi sphere appears in the second zone.
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The Fermi surface in the reduced zone scheme is
shown by the red line, and the effect of a weak crystal
potential is shown by the blue line.
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 Spring '08
 WORMER
 Work

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