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ECE 3060 Fall 2008 Prelim Exam 2 Solution
Rules of the Exam (Please read carefully before start)
1.
This is an openbook, opennote exam.
You are allowed to use your computer as a browser
for downloaded course files, but you are NOT allowed to connect to Internet in any form.
Connection to Internet during the exam will be considered as violation of academic integrity.
2.
Grading will ONLY consider what you legibly put down on the exam paper.
References to
textbook or class notes will NOT count for credit.
Irrelevant answers, even though the
content is correct, will NOT receive any partial credit.
Wrong information will always cause
a deduction in total credit.
3.
The time for the exam will be exactly 90 minutes.
Do not be trapped in a question you
cannot answer, and use your time wisely for distributing your efforts in different problems.
Do not diverge into irrelevant answers, since this will negatively impact your performance.
1.
The goal is to design a realistic Flash memory cell in the potential below, using the
approximation we have learned.
The mass of the particle is 0.5
m
0
, where
m
0
is the rest mass
of electrons at 9.11
×
10
31
kg.
The barrier height is 3.1eV.
(a)
Find the ground state energy
E
0
in eV in the potential well assuming 3.1eV can be
approximated by infinite potentials.
(5 pts)
meV
mL
π
E
L
53
.
7
2
2
2
2
=
=
h
.
You can see that this is much smaller than the barrier height, due to
the relatively large well width.
(b)
Propose a variational method to solve for
E
0
.
Note that this function should account for
nontrivial wave function penetration into the barrier.
You do not need to solve it
numerically, but need to write down the steps.
(10 pts)
First you notice the left barrier and right barrier are 3 times thicker.
For evanescent
waves, it is much more difficult to penetrate the left barrier, and hence more likely to pile
up to the left.
The steps of the variational method are then:
x
3.1eV
d
3d
10nm
Left barrier
Right barrier
Well
x=0
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1.
Define the trial function:
()
2
0
4
/
1
)
(
exp
2
x
x
α
π
α
ψ
t
−
−
⎟
⎠
⎞
⎜
⎝
⎛
=
, where we expect
x
0
to
be negative (towards left from the center of the well), and
nm
α
10
/
1
≈
.
2.
Find the parametrical equation for the expected energy:
( ) >
=<
t
t
t
ψ
H
ψ
x
α
ε

ˆ

,
0
,
where
H
ˆ
contains the piecewiseconstant potential.
3.
Solve for
0
/
=
∂
∂
α
ε
t
and
0
/
0
=
∂
∂
x
ε
t
to obtain
α
opt
and
x
opt
.
4.
Use
opt
and
x
opt
to give
E
0
.
(c)
If there is one electron in the well, and the lifetime
τ
can give the decay probability as
exp(
t/
).
Find the barrier width
d
that can give
= 3.15
×
10
8
sec
(10
years).
Use
E
0
from (a). Justify the reason why only the right barrier needs to be considered. This is the
retention state of the Flash memory cell.
(10 pts)
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