ECE 874:
Physical Electronics
Prof. Virginia Ayres
Electrical & Computer Engineering
Michigan State University
ayresv@msu.edu
Lecture 15, 18 Oct 12
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Example problem:
(a) What are the allowed (normalized) energies and also the forbidden
energy gaps for the 1st-3rd energy bands of the crystal system shown below?
(b) What are the corresponding (energy, momentum) values? Take three
equally spaced k values from each energy band.
k=0
k=±
p
a+b
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k=0
0.5
k=±
p
a+b
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(a)
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(b)
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“Reduced zone”
representation of
allowed E-k states in a
1-D crystal
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k=0
k=±
p
a+b
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(b)
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“Reduced zone”
representation of
allowed E-k states in a
1-D crystal
This gave you the same allowed
energies paired with the same
momentum values, in the opposite
momentum vector direction.
Always remember that momentum
is a vector with magnitude and
direction. You can easily have the
same magnitude and a different
direction.
Energy is a scalar: single value.
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Can also show the same information as an “Extended zone representation”
to compare the crystal results with the free carrier results.
Assign a “next” k range when you move to a higher energy band.
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Example problem: There’s a band missing in this picture. Identify it and fill it
in in the reduced zone representation and show with arrows where it goes in
the extended zone representation.
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The missing band:
Band 2
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Notice that upper energy levels are getting closer to the free energy values.
Makes sense: the more energy an electron “has” the less it even notices the well
and barrier regions of the periodic potential as it transports past them.
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Note that at 0 and ±p/(a+b) the
tangent to each curve is flat:
dE/dk = 0
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A Brillouin zone is basically the allowed momentum range
associated with each allowed energy band
Allowed energy levels: if these are closely spaced energy levels they
are called “energy bands”
Allowed k values are the Brillouin zones
Both (E, k) are created by the crystal situation U(x). The allowed
energy levels are occupied – or not – by electrons
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VM Ayres, ECE874, F12
(b)
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What happens to the e- in response to the application of an external force:
example: a Coulomb force F = qE (Pr. 3.5):

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(d)
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(d)
Conduction energy bands
<111> type
8 of these
<100> type
6 of these
Symmetric
[100]
[100]
Warning: you will see a lot of literature in which people get
careless about <direction type> versus [specific direction]
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(d)
<111> and <100> type transport directions certainly have different
values for aBlock spacings of atomic cores. The G, X, and L labels
are a generic way to deal with this.
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Two points before moving on to effective mass:


Kronig-Penney boundary conditions
Crystal momentum, the Uncertainty Principle and
wavepackets
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Boundary conditions for Kronig-Penney model:
Can you write these blurry boundary conditions without looking them up?
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Locate the boundaries:
aKP + b = aBlock
b
aKP
[transport
direction p 56]
-b
a
0
a
-b
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Locate the boundaries: into and out of the well.
aKP + b = aBlock
b
aKP
[transport
direction p 56]
-b
a
0
a
-b
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Boundary conditions for Kronig-Penney model, p. 57:
Is the a in these equations aKP or aBl?
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Boundary conditions for Kronig-Penney model, p. 57:
Is the a in these equations aKP or aBl? It is aKP.
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Two points before moving on to effective mass:


Kronig-Penney boundary conditions
Crystal momentum, the Uncertainty Principle and
wavepackets
VM Ayres, ECE874, F12
VM Ayres, ECE874, F12
Chp. 04: learn how to find the probability that an e- actually makes it into “occupies” - a given energy level E.
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k2
k  wavenumber Chp. 02
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Suppose U(x) is a Kronig-Penney model for a crystal.
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On E-axis: Allowed energy levels
in a crystal, which an e- may
occupy
So a dispersion diagram is all about
crystal stuff but there is an easy to
understand connection between
crystal energy levels E and e- ‘s
occupying them.
The confusion with momentum is
that an e-’s real momentum is a
particle not a wave property.
Which brings us to the need for
wavepackets.
http://en.wikipedia.org/wiki/Crystal_momentum
hbark = crystal momentum
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