Biomedical Engineering Reference
In-Depth Information
To match these functions as closely as possible, the area between the func-
tions must be minimized.
3.11.5.2 Equating Energy
An alternative method of calculating
L
x
is to equate the energies of the first
energy levels and solve for the length of the infinite QW. The following sim-
ple formula is the result:
L
x
=
π
κ
where κ is determined from the finite wave function by Equation 3.35.
3.11.5.3 Computer Calculations
In order to complete the calculations described above, computer calculations
were used. The first step in calculating the effective width is to find the effec-
tive masses using the equations in Figure 3.20. Additionally, the height of the
potential well barrier must be calculated using the energy gap difference.
The split ratio must be taken into account when finding these two values.
By using equations and, the value of κ and
k
can be found. From these two
values, B can be calculated using the fact that
∞
∫
|
2
| Ψ
finite
d
x
=
1
−∞
At this point we can calculate the effective width using the equivalent energy
method (Section 3.11.5.2) because we know the value of κ. Assuming we have
completed the integration in Equation 3.36, we have sufficient information
to find the effective width that minimizes the area difference between the
two wave functions. The computer can find this effective width iteratively,
using Equation 3.36 with different effective well widths until the minimum
is found. The values we obtained for several well widths and aluminum con-
centrations are listed in Figure 3.24.
3.11.6 Summary of Calculation Procedure
With the full theory developed in the previous sections, a brief summary is
helpful to explain the steps needed to obtain the exciton energy. Equation
3.19 shows that the exciton energy is broken down into three energy values.
The quantum well is fully specified by what may be considered as three
“fundamental” values. The aluminum concentrations of the barrier and the
well material determine the band gap difference. The other parameter is the
actual width of the well.
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