Biomedical Engineering Reference
In-Depth Information
In summary, to calculate
E
b
for a particular filed we must first deduce β
e
and
β
h
for the field, using a variational calculation, and then calculate
E
KE
r
and
E
PE
r
, adjusting λ variationally to minimize
E
B
.
3.11.5 Effective Well Width Calculations and Computer Simulation
In order to perform all the previous calculations, we must substitute an effec-
tive QW width for the actual well width. Note that for an infinite well the
wave function has a value of zero outside the well walls. Because the wells
in the MQWS are finite, the effective well widths for the infinite well model
must be greater than the actual well widths to account for the significant
penetration of the wave function into the finite QW barriers. There are two
basic ways to find the effective well widths. One way is to try to match the
wave functions of the finite well as closely as possible with the infinite well.
Another way is to equate the energy of the first energy level in the finite well
with the infinite well.
3.11.5.1 Matching of Wave Functions
In order to understand how we match the wave functions, Figure 3.21 shows
a finite well wave function where
Ψ
=
Ψ Ψ Ψ
+
+
finite
I
II
III
a e
k z
( +α)
Ψ
I
= 2
B
cos(
κ
)
(3.35)
Ψ
II
= 2
B
cos(
κ
z
)
a e
k z a
(
−
)
Ψ
III
=
2
B
cos(
κ
)
κ
a tan(
κ
a
)
=
ka
Ψ
II
Ψ
I
Ψ
III
-
a
a
0
FIGURE 3.21
Finite well wave function.
2
a
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