Biomedical Engineering Reference
In-Depth Information
E
pE
r
is the Coulomb potential energy of the electron-hole relative motion.
E
r
= 〈
Ψ
|
V
|
Ψ
〉
pE
e h
−
In this case, we must use variational wave functions for ψ
e
and ψ
h
in Equation
3.20 [43]. These are written as
⎡
⎢
⎤
⎥
⎡
⎢
π
z
L
⎤
⎥
⎡
⎢
z
L
1
2
⎤
⎥
ψ
=
N
( )cos
β
exp
−
β
+
(3.30)
e,h
x
x
Note that β (β
e
or β
h
),
L
x
(
L
e
or
L
h
) and
z
(
z
e
or
z
h
) all depend on whether
we are using ψ
e
or ψ
h
.
N
(β) is a normalization function and is defined such
that
2
2
4
β
β
+
n
2
N
( )
β
=
2
L
[
1
−
exp(
−
2
β
)
]
β
x
β is a variational parameter and is calculated by minimizing the function
E
(β) with respect to β
2
⎡
⎤
2
⎡
⎢
⎤
⎥
β
π
1
2
β
1
2
⎛
⎜
β
⎞
⎟
( )
0
E
( )
β
=
E
1
+
+
χ
+
−
coth
(3.31)
⎢
⎥
1
2
2
2
4
β
4
π
+
β
2
⎣
⎦
where the ground state energy at zero field is
2
2
π
( )
0
E
=
1
*
2
2
m L
⊥
x
and the dimensionless electro-static energy is
| |
e F L
E
⊥
x
χ =
( )
0
1
This is calculated separately for both the electron and hole.
Using Equations 3.18, 3.20, and 3.30 Equation 3.20 becomes
+
L
L
h
/
2
2
π
∞
e
/
2
2
2
2
⎡
⎢
⎤
⎥
=
−
e
N
π
z
L
⎡
⎢
z
L
1
2
⎤
⎥
∫
∫
∫
∫
2
2
2
e
e
e
E
(
β
)
N
(
β
)
cos
exp
−
2
β
e
+
pE
e
h
r
πελ
e
θ
=
0
r
=
0
z
=−
L
z
=−
L
e
e
/
2
h
h/2
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