ECE 874: Physical Electronics

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ECE 874Physical Electronics

• Prof. Virginia Ayres
• Electrical Computer Engineering
• Michigan State University
• ayresv_at_msu.edu

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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
4
k 0
0.5
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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
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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 Theres 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/(ab) 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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(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)
Symmetric
Conduction energy bands
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100
lt100gt type 6 of these
lt111gt type 8 of these
Warning you will see a lot of literature in
which people get careless about ltdirection typegt
versus specific direction
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lt111gt and lt100gt 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.
(d)
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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
-b
a
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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
-b
a
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Boundary conditions for Kronig-Penney model, p.
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Is the a in these equations aKP or aBl?
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Boundary conditions for Kronig-Penney model, p.
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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

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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.
hbark crystal momentum
http//en.wikipedia.org/wiki/Crystal_momentum