Lecture 12: Continue the Band Theory of Solids
Band Structures
Two Dimensional Brillouin Zone:
We construct the first Brillouin zone from the shortest
lattice vector 𝐺1 as follows.
We construct the second Brillouin zone from the next
shortest vector 𝐺2 and so on.
Brillouin Zones- 2D
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BZ construction
 Reciprocal lattice
 Bisect vectors to the nearest neighbors
 Area defined by bisecting lines represents 1BZ
Three Dimensional Brillouin Zones:
 A 3-dimensional Brillouin zone can be constructed
in a similar way by bisecting all lattice vectors and
placing planes perpendicular to these points of
bisection.
 This is similar to the Wigner Seitz cell in the real
lattice.
Wigner Seitz Cell:
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 A primitive unit cell which shows the cubic
symmetry of the lattice( for the cubic system).(real
lattice)
 The First Brillouin zone is the Wigner Seitz cell in
the reciprocal lattice
Lets Study These Figures:
1. *First Brillouin zone of the bcc structure
2. ⇒Free electron bands for bcc structure
3. *First Brillouin zone of the fcc structure
4. ⇒ Free electron bands for fcc structure
Explanation of these symbols:
Look between the graph of bands and the first Brillouin
zone, you will find:
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Γ: center of the Brillouin zone
Χ: [100]intercept
Κ: [110]intercept
L: [111]intercept
Γ − Χ: path Δ
Γ − L: path Λ
Γ − Κ: path Σ
DRAWING
Next Figure:
Band structure of Al (fcc)
 Note the parabola shape bands:
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Γ−Χ Γ−𝐿 Γ−Κ
- Compare this graph with the free electron bands
of fcc
⇒ Looks close or similar which suggests that
electrons in Al behave like free electrons.
Important:
There are some band gaps between points
(𝑋4′ , 𝑋1 )
(𝑊3 , 𝑊2 ′)
But the individual energy bands overlap in different
directions
⇒No band gap exists as a whole
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Band Structure of Cu:
 The closely spaced bands are due to the 3d-bands
 4s bands: the heavily marked 4s, 3d bands overlap
 No band gap exists
Band structure of Si:
 Band gap exists of “1eV”
 The zero point of energy scale is placed under the
energy gap
 Note: the indirect band gap in silicon (Ill explain it
later on)
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Band structure of Ga As:
 Note the energy gap and note it is direct gap
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Direct and indirect band gaps:
1. Direct:
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
Direct band gap:
The maximum of valence band and the maximum
of conduction band have the same k. vector
2. Indirect:
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
Indirect band gap:
The maximum of valence band and the maximum
of conduction band have different k. vectors
Δ𝑘 ≠ 0
What is the implication??
We still need to understand more about the shape of
band structure.
To do that, we need to understand the effective mass:
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Effective mass of electron: m*
The mass of an electron in a solid is deviated from the
free electron mass due to interactions of electronelectron, and electrons-ions.
𝑚∗
𝑚
could be greater than 1 or smaller than 1
Let’s derive m*
The group velocity vg :
𝑑𝜔
vg =
𝑑𝑘
2π
ω = 2πυ
k=
λ
(2πE/h)
d(2πυ)
=
=d
dk
dk
1 𝑑𝐸
vg =
⇒
ℏ 𝑑k
Acceleration (a)=
dvg
dt
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1 𝑑 2 𝐸 𝑑k
=
∙
ℏ 𝑑k 2 𝑑t
Let’s find
𝑑k
𝑑t
dP
dt
=
1 d2 E
ℏ2 dΚ
2 ⋅
𝑃 = ℏ𝑘
=ℏ
d(mv)
dt
𝐸𝑞. 𝟏
dk
𝐸𝑞 𝟐
in Eq 1,
1 d2 E dP
a= 2 2
ℏ dΚ dt
dt
who Law?
1 d2 E
𝑎 = 2 2𝐹
ℏ dk
F
a=
m
2
d
E
∗
2
𝑚 = ℏ ( 2)
dk
−1
m* is inversely related to the curvature of E(K)
If the curvature of 𝐸 = 𝑓 (𝐾) at a given point is large,
the effective mass is small and vice versa.
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Look back into the band structures:
 Some regions have high curvature, near the center
of the boundary of a Brillouin zone. ⇒Effective
mass is reduced (sometimes up to less than 1% of
m)
- At points where there are more than one band,
than one effective mass
 A negative mass means electron travel in

 opposite directions to an electric field (electron
hole).
 Holes appear near the top of valence band.
 Go back to the band structures; find the valence
and conduction bands and the light and heavy
mass.
It reaches infinity at K
=
π
2a
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Memory Aid
“a hairpin is lighter than a frying pan”
light m*
(larger d2E/dK2)
heavy m*
(smaller d2E/dK2)
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