Biomedical Engineering Reference
In-Depth Information
where ς(
z
) is either the electron or hole wave function. Note that the param-
eters
m
*
and
e
and the dimensions
z
and
L
x
all depend on the particle being
investigated.
By making the substitution
1 3
/
=
⎡
*
⎤
⎥
m
e F
2
⊥
Z
(
W eF z
+
)
(3.23)
⎢
⊥
2
(
)
⊥
we can transform Equation 3.22 into an Airy differential equation:
d
d
Z
ς
( )
z Z z
−
ς
( )
=
0
(3.24)
2
The solutions have the form
ς
( )
z
=
bAi Z cBi Z
+
(3.25)
(
)
(
)
where
b
and
c
are constants and
Ai
(
Z
) and
Bi
(
Z
) are Airy functions
defined by
Ai Z
(
)
=
c f Z c g Z
(
)
−
(
)
1
2
Bi Z
(
)
=
3
1
[
c f Z c g Z
(
)
−
(
)]
2
Further definitions are
∞
∑
3
n
Z
f Z
(
)
= +
1
(
3
n
)(
3
n
−
1 3
)(
n
−
3 3
)(
n
−
4
)
3 2
⋅
n
=
1
∞
∑
3
n
+
1
Z
g Z
(
)
=
Z
+
(
3
n
+
1 3
)(
n
)(
3
n
−
2 3
)(
n
−
3
)
4 3
⋅
n
=
1
while
c
1
= 0.35503 and
c
2
= 0.25882.
It is convenient to work with dimensionless parameters for the energy and
the field. These are defined as
W
W
F
F
w
=
;
f
=
1
1
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