Biomedical Engineering Reference
In-Depth Information
where
E
e
= 〈
Ψ
|
H
e
|
Ψ
〉
E
= 〈
Ψ
|
H
|
Ψ
〉
h
h
E
= 〈
Ψ
|
H
|
Ψ
〉
E
ex
E
e
and
E
h
are the energies of the electron and hole respectively, which can be
found using Equation 3.15.
E
B
is the exciton-binding energy which is found
using Equation 3.15.
The wave function Ψ is written in a separable form as
Ψ ψ
=
(
z
)
ψ
(
z
)
∅
( )
r
(3.20)
e
e
h
h
e h
−
where
1 2
/
2
1
=
⎛
⎞
⎟
[
]
−
r
∅
e h
( )
λ
r
e
⎜
−
π
λ
The parameter λ is used as a variational parameter describing the amplitude
radius of the exciton.
3.11.3 Determination of
E
e
and
E
h
The problem of determining the energies of the electron and the hole in the
QW is subject to a great deal of simplification. Initially it may be noted that,
due to the separable nature of the wave function Ψ, Equations 3.19 reduce to
the following:
E
= 〈
ψ
|
H
|
ψ
〉
(3.21)
e
e
e
e
E
= 〈
ψ
|
H
|
ψ
〉
h
h
h
h
The next simplification is that a wave function describing an electron and
hole in an infinite QW can be substituted for the wave function ψ
e
or ψ
h
if an
effective well width
L
x
is used in place of the actual QW width. The longer
well width effectively accounts for the fact that there is significant penetra-
tion of the wave function into the QW barriers.
The Schrödinger equation for the infinite well barrier in a uniform perpen-
dicular electric field is
2
2
−
d
d
ς
( )
z W eF z
−
(
+
) ( )
ς
z
=
0
(3.22)
⊥
*
2
2
m z
⊥
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