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Lecture 11: Continue Band Theory of solids
Recall from last lecture We have derived the energy levels for free
electrons, bound electrons, and electrons in
crystals.
 From mathematical model and Schrodinger
solutions: we have seen that there are constraints
that lead to energy bands and forbidden gaps.
 We have also seen from the mathematical solutions
that:
 By decreasing separation distance, P
decreases, energy levels become energy bands,
and then overlapping.
 Question:
 How can you explain that just from a physics
point of view? (not based on derived from
equations)
 When the atoms get loser, electron waves
interfere (scatter log the potential).
 Constructive interference, destructive
interference creates energy bands and band
gaps.
 Electron energy dispersion:
E − k curve
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Now, we are interested in constructing the E-R curve.
You will see that we can draw important conclusions
about electronic properties of materials from this
diagram.
Let’s start
We know that:
E=
h2 k2
2m
for free electrons
k=√
2m 1⁄
E 2
2
h
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√
2m 1⁄
E 2
2
h
Here:
sin(αa)
cos(ka) = P
+ cos(αa)
αa
we have free electrons when P → 0
∴ cos(αa) = cos(ka)
But, that is not all
cos is a periodic function in 2𝜋.
∴ cos(αa) = cos(ka) = cos(ka + n2π)
αa = ka + n2π
𝛼2 =
Then,
2𝑚𝐸
ℎ2
𝛼=
√2𝑚
ℏ
𝐸
1⁄
2
2m 1⁄
ka + n2π =
E 2a
h
Divide by a:
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2π
2m 1⁄
k+n
=√ 2 E 2
a
h
What does this mean?
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Curve 1 repeats itself in increments of
(n=±1, ±2, ±3)
2π
a
∙n
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Energies are not continuous at
cos(𝑘𝑎) = ±1
ka = nπ
n = ±1, ±2, ±3
nπ
⟹k=
a
at k =
nπ
, E deviates from the parabolic curve
a
For most k x values, electrons in crystals behave like free
nπ
electrons except k close to
a
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π π
Look between k = − & :
a a
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Reduced Zone Scheme
2 nd Brillioun Zone
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Look between k = −
2π 2π
&
:
a
a
Let’s look to the extended zone scheme again:
Energy gaps (band gaps) occur at Brillouin zone
boundaries
1D
The first Brillouin zone occupies:
π
π
− ≤ k ≤ (Corresponds to n-band)
a
a
The second Brillouin zone occupies:
π
2π
2π
π
− ≤ k ≤ − and
≤k≤
a
a
a
a
(Corresponds to m-band)
We get bands of allowed electron energies separated by
smaller regions of forbidden energies at the Brillouin
zone boundaries.
We will call these bounds: n, m,… (The valence and
conduction bands)
 Note: we don’t need to plot E versus k for all
Brillouin zones? Why?
Because of the periodicity, the interesting
information is already contained in the first
Brillouin zones.
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Plot free electrons in a reduced zone scheme:
2π
2m 1⁄
√
k+n
=
E 2
2
a
ℏ
ℏ2
2π
E=
(k x + n )
2m
a
2
ℏ
2
n=0 ⇒ E=
k
x
2m
(Parabola at the origin)
2
ℏ
2π 2
𝑛 = −1 ⇒ E = 2m (kx − a )
(Parabola with origin at
2π
a
)
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Free electron bands plotted in the reduced zone
scheme.
Let’s compare this graph with Fig. (b) above
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