Homework 10 Solution
Prob. 10.1
We will look at the density of states in a MOSFET inversion channel.
This can be
viewed as electrons being confined to a 3D box, but the gate direction (assigned as
z
here) has
much more confinement than the channel length and width direction.
Effectively, we can look at
the system as a quantum box of dimension (
a, b, c
), where
c << a, b
.
(a)
Write down the eigenfunctions and eigenenergy for this system.
Is there any difference
from Eqs. (10.8) and (10.9)?
(5 pts)
(
29
(
29
(
29
(
29
2
2
2
2
2
2
2
2
2
2
2
2
,
,
2
2
2
sin
sin
sin
8
,
,
mc
n
mb
n
ma
n
E
c
z
n
b
y
n
a
x
n
abc
z
y
x
z
y
x
z
y
x
nz
ny
nx
z
y
x
z
y
x
π
π
π
π
π
π
φ
φ
φ
φ
+
+
=
=
=
This is different from the plane waves in Eqs. (10.6) and (10.7), though similar in the way of
composition of 3D wavefunctions.
The eigenenergies are dominated by the last term since
c
<< a, b
.
The wave function is similar to Eq. (10.9) with a constant
z
.
There cannot be much
variation in
z
anyway, or there will be many cycles of the sine wave, which means very high
energy.
(b)
Construct the 3D plot of (
k
x
, k
y
, k
z
) and the allowable points.
What is the influence from
the relation of
c << a, b
?
(5 pts)
In the
k
x
and
k
y
plane, it is a dense uniform grid.
The spacing between the 2D dense grid in
the
k
z
direction is much larger.
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 '05
 TANG
 Gate, pts, NZ, quantum number nx

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