Banding in

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• Banding in
• Semiconductors

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In electrical engineering, we characterize a
device or system by it frequency response.
In semiconductor physics, materials are
characterized by Energy vs. k space.
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In our text book, the author uses the
Kroenig-Penny model to show that there are
forbidden regions, i.e., values of k at which a
particle cannot propagate. It shows that there
are forbidden bands where a particle in a lattice
cannot propagate.
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The forbidden k bands for a lattice with a
spacing of a are centered at
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Remembering that
I prefer to think of the forbidden bands at
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Therefore, the forbidden values of
i.e, any integer division of 2a is not allowed
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i.e, the forbidden bands occur when the
wavelength of the particle is i. twice lattice
spacing ii. the lattice spacing iii. 2/3 the
lattice spacing iv. ½ the lattice spacing v. 2/5
the lattice spaceing

• If a 20 A, this becomes
• 40 A
• 20 A
• 13.3 A
• 10 A
• 8 A

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Suppose I have a lattice of -0.1 eV points
spaced at 20 A (top graph). The graph on the
bottom shows the forbidden values of
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Notice that some of the positive frequency has
been attenuated, but it appears as negative
frequency. This corresponds to a wave that is
propagating in the negative direction, i.e., has
been reflected.
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Notice that there has been virtually no loss to
the positive frequency.
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Notice that there is some loss. But the loss
appears at the lambda 20 wavelength of the
propagation wave.
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In the following, notice that the input wave
packet is narrower, leading to a broader pulse in
the frequency domain.
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If thats the case, what happens when I use a
very narrow pulse for the input?
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The next question I might ask is, What if I have
the same lattice spacing but a different
arrangement of potentials?
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We have seen that the semiconductor lattice
results in certain forbidden regions where an
electron cannot propagate. Even when the
electron propagates, the lattice is going to have
some effect on how it propagates.
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Specifically, if there is an electric field
present, it will result in a force on the
particle due to its charge
In free space, this results in an acceleration
dictated by the rest mass of the electron
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We account for the lattice by an effective mass m
Appendix B Effective mass (density of states)
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If we had no forbidden regions, the plot of the
energy E vs. the wavenumber k would be a parabola
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GaAs Lattice
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I found Dattas diagram of a GaAs lattice to be
rather difficult to interpret.
Compare with Neaman.
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Near the nuclei of atoms, the electric fields are
very high leading to effects that can be
accounted for by the spin-orbit correction to the
Schroedinger equation.
,
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Start by using just the s orbitals for each atom.
Note that matrix elements are non-zero only when
both orbitals are centered on the same atom

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