Download ECE874 Lecture 11, 04 Oct 12

ECE 874:
Physical Electronics
Prof. Virginia Ayres
Electrical & Computer Engineering
Michigan State University
[email protected]
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
VM Ayres, ECE874, F12
Finite Potential Well:
(eV)
Electron energy: E > U0
Electron energy: E < U0
(nm)
Regions:
-∞ to 0
0 to a
a to +∞
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Infinite Potential Well:
U (eV) = +∞
U (eV) = +∞
Electron energy: E < U0
(nm)
Regions:
-∞ to 0
0 to a
a to +∞
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Free (between scattering events) particle (e- in I):
Electron energy: E > U0
U (eV) = 0
(nm)
Region:
-∞
to
+∞
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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 < x < 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..
VM Ayres, ECE874, F12
Example problem: Find y(x) for a mixed U0 situation modelled as an
infinite/finite well. Consider the case where E < 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
VM Ayres, ECE874, F12
VM Ayres, ECE874, F12
VM Ayres, ECE874, F12
Example problem: Find y(x) for a mixed U0 situation modelled as an
infinite/triangular well
VM Ayres, ECE874, F12
VM Ayres, ECE874, F12
VM Ayres, ECE874, F12