Biomedical Engineering Reference
In-Depth Information
3.11.1 Description of the Quantum Well
The QW is formed by the discontinuities in the energy gap between the
GaAs and the Al
x
Ga
1 −
x
As layers. The energy gap difference is described by
2
3
Δ
E
g
eV
(
)
=
1 425
.
x
−
0 9
.
x
+
1 1
.
x
where
x
is the mole fraction of aluminum. This energy gap difference is
divided between the hole QWs and the electron QWs. Two theories exist
regarding the magnitude of this QW split. The first is an 85:15 split, which
means that 85% of the energy gap difference is at the conduction band dis-
continuity and 15% is at the valence band discontinuity. Another theory sug-
gests a 57:43 split. It is interesting to note that the use of either one of these
theories results in little or no difference in the magnitude of the energy shift.
Figure 3.17 shows a schematic diagram of a QW structure indicating the
band discontinuities.
To describe the exciton's energy levels in a QW, we must first know the
Hamiltonian. The Hamiltonian for this problem can be written as the sum of
terms due to the contribution from the individual electron and hole, and the
interaction between the two particles.
H
=
H
+
H
+
H
ex
(3.15)
e
h
where
H
=
H
+
V z
(
)
−
⊥
eF z
e
KE
z
e
e
e
e
H
=
H
+
V z
(
)
−
⊥
eF z
h
KE
z
h
h
h
h
H
=
H
+
V
( ,
r z z
,
)
ex
KE
r
e h
−
e
h
eh
The
z
direction is defined perpendicular to the MQWS layers and
z
e
and
z
h
are the positions of the electron and hole, while
r
is the relative position of
the electron and hole. The kinetic energy operators in the
z
direction are
defined by
2
4
2
−
d
d
H
=
(3.16)
KE
z
e
2
2
m
z
e x
⊥
e
2
2
−
d
d
H
=
KE
z
h
k
2
2
m
z
h z
⊥
h
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