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 11, 04 Oct 12
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Answers I can find
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Working tools
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Two unknowns y(x) and E in eV from one equation
1. You can find y(x) by inspection whenever the
Schroedinger equation takes a form with a known
solution like and exponential. The standard form
equation will also give you one relationship for
kx that contains E in eV. 2. Matching y(x) at a
boundary puts a different condition on kx and
setting kx kx enables you to also solve for E
in eV.
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Or equivalent Aexpikx Bexp-ikx form
Infinite potential well
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With B 0 tunnelling out of a finite well
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Finite Potential Well
(eV)
Electron energy E gt U0
Electron energy E lt U0
(nm)
-8 to 0
a to 8
0 to a
Regions
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Infinite Potential Well
U (eV) 8
U (eV) 8
Electron energy E lt U0
(nm)
-8 to 0
a to 8
0 to a
Regions
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Free (between scattering events) particle (e- in
I)
Electron energy E gt U0
U (eV) 0
(nm)
Region
-8 to 8
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For all three situations, found- y(x)- E
(free) or En (infinite and finite wells)
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Example problem Find energy levels in a finite
model for a SQW Consider a SQW of width a 10
nm that is fabricated in GaAs that operates at
300K. The SQW is modelled as a finite well. How
many energy levels for an e- exist for A) U0
0.7 eV half the size of the bandgap B) U0 1.4
eV just under the size of the bandgap C) What
is the practical meaning of the limit x E/U0,
0 lt x lt 1?
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Finite Potential Well
Advantage is you scale to important parameters
the height U0 and width a. Note Width a only
affects the LHS the number/spacing of tan
curves. Height U0 affects both sides but
practical advantage on RHS plot..
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Example problem Find y(x) for a mixed U0
situation modelled as an infinite/finite well.
Consider the case where E lt U0-RHS.
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Expected wavefunctions in each of three regions
are easy
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Energy levels set up the graphical solution
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units
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Example problem Find y(x) for a mixed U0
situation modelled as an infinite/triangular well
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