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Fig. 8.2 Density of the energy states for a three-dimensional solid, quantum well, quantum wire,
and quantum dot
(
"
#
)
2
2
2
2
ℏ
∂
x
2
þ
∂
y
2
þ
∂
Ux
ð
y
z
Þ
Ψ
¼
E
,
;
;
Ψ
2
m
z
2
∂
∂
∂
where the potential energy in a box with sides
a
,
b
,
c
Ux
ð
y
z
Þ¼
Ux
ðÞþ
Uy
ðÞþ
Uz
ðÞ
:
;
;
Here
U
(
x
)
a
a
b
b
c
c
2
¼
U
(
y
)
¼
U
(
z
)
¼
0, if
2
x
2
,
2
y
2
,
2
z
or
U
(
x
)
U
0
at all other
x
,
y
, and
z
values.
The solution of this equation is
¼
U
(
y
)
¼
U
(
z
)
¼
,
2
2
n
1
n
2
n
3
c
2
E
n
1
,
n
2
,
n
3
¼
π
ℏ
a
2
þ
b
2
þ
where
n
1
,
n
2
,
n
3
¼
1, 2, 3
...
2
m
Thus, a discrete spectrum, generally similar to the spectrum of the atomic
system, corresponds to the quantum point.
In a quantum dot there can exist from a single to a large number of electrons,
whose distribution is determined by the Pauli principle.
Quantum dots can be created by the method of molecular beam epitaxy. Another
substance with a structure similar to that of the substrate is sprayed on a well-
prepared surface. Everything should happen in a high vacuum to avoid inclusion of
impurities in the object to be formed. The deposition rate must be carefully
controlled in order to avoid the formation of structural defects. Spontaneous growth
of quantum dots in a so-called Stranski-Krastanov mode has been well studied on
the example of InAs/GaAs. During the growth of the first monomolecular layer of
InAs on the GaAs surface, elastic stresses arise due to differences of permanent
