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Introduction to Quantum and Statistical Mechanics
Homework 5
Prob. 5.1
A quantum well laser is operated by electrons relaxing from the first excited state
E
2
to the ground state
E
1
in a finite-potential well, which is formed by a superlattice of
semiconductors with different bandgap.
Assume the electron has an effective mass
m
*
of
0.067
m
0
where m0 is the rest mass of electrons in vacuum.
Given the well width as
L
and well
depth as
V
o
,
(a)
To avoid transition in other levels, it is desirable to have exactly two levels in the
quantum well.
Express the relation of
V
o
with
L
and with
2
*
2
2
2
L
m
E
L
π
=
(8 pts)
(b)
Will
E
2
= 4E
1
?
Will this be a reasonable approximation?
(6 pts)
(c)
If the emitted wavelength is at 10.6µm (middle infrared), use
E
2
and
E
1
close to the
infinite well levels to obtain an estimate of the quantum well width in nm.
(8 pts)
(d)
Graphically justify that the approximation of using
E
2
and
E
1
from the infinite potential
well is acceptable from the transcendental equation plot.
Notice that this approximation

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