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Free Electron Bands
Energy
n=-2
ε k = ε k (k + G ) =
n=1
2 
2nπ  2
 kx +

2m 
a 
n=-1
n=0
k
Brillouin zones - 2D
BZ construction
• reciprocal lattice • bisect vectors to the nearest neighbors • area defined by bisecting lines
represents 1BZ
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Brillouin zones -2D
• higher order zones can be
mapped directly onto the 1st
BZ by simple translation
• all BZs have exactly the same
area/volume
• 1BZ corresponds to the
primitive lattice cell in
reciprocal space
Free Electrons in a square lattice k :0 →
π
a
In the [1,0] direction
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Max Energy 1st BZ
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Brillouin zones - 3D Example: FCC Lattice
Primitive Lattice Vectors
a
(0,1,1)
2
a
t 2 = (1,0,1)
2
a
t 3 = (1,1,0)
2
t1 =
Primitive Reciprocal Lattice Vectors
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2π
(−1,1,1)
a
2π
b2 =
(1,−1,1)
a
2π
b3 =
(1,1,−1)
a
b1 =
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Brillouin zones - 3D
fcc
1st BZ
2nd BZ
3rd BZ
bcc
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Free electron bands for fcc structure
Γ – center of the BZ
X – [100] intercept; Γ - X path Δ
K – [110] intercept; Γ - K path Σ
L – [111] intercept; Γ - L path Λ
Band structure of Al (fcc)
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Band structure of Cu (fcc)
Band structure of Si (diamond)
Bandgap
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Band structure of GaAs (zb)
Bandgap
Band Structure of AlN (wurtzite)
L-M
K
Γ
Bandgap
First Brillouin Zone Nomenclature
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Curves of equal energy
• 2D square lattice
• 1st BZ
• deviation from circles
a we approach critical
points (X, K)
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